src/HOL/Subst/Subst.thy

merged

(*  Title:      HOL/Subst/Subst.thy
    Author:     Martin Coen, Cambridge University Computer Laboratory
    Copyright   1993  University of Cambridge
*)

header {* Substitutions on uterms *}

theory Subst
imports AList UTerm
begin

primrec
  subst :: "'a uterm => ('a * 'a uterm) list => 'a uterm"  (infixl "<|" 55)
where
  subst_Var: "(Var v <| s) = assoc v (Var v) s"
| subst_Const: "(Const c <| s) = Const c"
| subst_Comb: "(Comb M N <| s) = Comb (M <| s) (N <| s)"

notation (xsymbols)
  subst  (infixl "\<lhd>" 55)

definition
  subst_eq :: "[('a*('a uterm)) list,('a*('a uterm)) list] => bool"  (infixr "=$=" 52)
  where "r =$= s \<longleftrightarrow> (\<forall>t. t \<lhd> r = t \<lhd> s)"

notation (xsymbols)
  subst_eq  (infixr "\<doteq>" 52)

definition
  comp :: "('a * 'a uterm) list \<Rightarrow> ('a * 'a uterm) list \<Rightarrow> ('a* 'a uterm) list"
    (infixl "<>" 56)
  where "al <> bl = alist_rec al bl (%x y xs g. (x,y \<lhd> bl) # g)"

notation (xsymbols)
  comp  (infixl "\<lozenge>" 56)

definition
  sdom :: "('a*('a uterm)) list => 'a set" where
  "sdom al = alist_rec al {} (%x y xs g. if Var(x)=y then g - {x} else g Un {x})"

definition
  srange :: "('a*('a uterm)) list => 'a set" where
  "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 t) auto

lemma subst_mono: "t \<prec> u \<Longrightarrow> t \<lhd> s \<prec> u \<lhd> s"
  by (induct u) auto

lemma Var_not_occs: "~ (Var(v) \<prec> t) \<Longrightarrow> t \<lhd> (v,t \<lhd> s) # s = t \<lhd> s"
  apply (case_tac "t = Var v")
   prefer 2
   apply (erule rev_mp)+
   apply (rule_tac P = "%x. x \<noteq> Var v \<longrightarrow> ~(Var v \<prec> x) \<longrightarrow> 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 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:
    "v \<in> vars_of(t) --> w \<in> vars_of(t \<lhd> (v,Var(w))#s)"
  by (induct 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)

lemma ssubst_subst2:
    "[| s \<doteq> r; P (t \<lhd> r) (u \<lhd> r) |] ==> P (t \<lhd> s) (u \<lhd> s)"
  by (simp add: subst_eq_def)


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 t)
  apply simp_all
  apply (induct r)
   apply 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)
    apply 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)
   apply 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)"
  unfolding empty_def by blast

lemma invariance: "(t \<lhd> s = t) = (sdom(s) Int vars_of(t) = {})"
  apply (induct t)
    apply (auto simp add: empty_iff_all_not sdom_iff)
  done

lemma Var_in_srange:
    "v \<in> sdom(s) \<Longrightarrow>  v \<in> vars_of(t \<lhd> s) \<Longrightarrow> v \<in> srange(s)"
  apply (induct 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:
    "v \<in> vars_of(t \<lhd> s) \<Longrightarrow> v \<in> srange(s) | v \<in> vars_of(t)"
  apply (induct t)
    apply (auto simp add: sdom_iff srange_iff)
  apply (rule_tac x=a in exI)
  apply 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 M) auto

end