--- a/src/HOL/Subst/Subst.thy Mon Mar 28 16:19:56 2005 +0200
+++ b/src/HOL/Subst/Subst.thy Tue Mar 29 12:30:48 2005 +0200
@@ -2,11 +2,13 @@
ID: $Id$
Author: Martin Coen, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
-
-Substitutions on uterms
*)
-Subst = AList + UTerm +
+header{*Substitutions on uterms*}
+
+theory Subst
+imports AList UTerm
+begin
consts
@@ -19,22 +21,170 @@
primrec
- 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)"
+ 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)"
defs
- subst_eq_def "r =$= s == ALL t. t <| r = t <| s"
+ subst_eq_def: "r =$= s == ALL t. t <| r = t <| s"
- comp_def "al <> bl == alist_rec al bl (%x y xs g. (x,y <| bl)#g)"
+ comp_def: "al <> bl == alist_rec al bl (%x y xs g. (x,y <| bl)#g)"
- sdom_def
+ 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_def:
"srange(al) == Union({y. EX x:sdom(al). y=vars_of(Var(x) <| al)})"
+
+
+subsection{*Basic Laws*}
+
+lemma subst_Nil [simp]: "t <| [] = t"
+by (induct_tac "t", auto)
+
+lemma subst_mono [rule_format]: "t <: u --> t <| s <: u <| s"
+by (induct_tac "u", auto)
+
+lemma Var_not_occs [rule_format]:
+ "~ (Var(v) <: t) --> t <| (v,t <| s) # s = t <| s"
+apply (case_tac "t = Var v")
+apply (erule_tac [2] rev_mp)
+apply (rule_tac [2] P = "%x.~x=Var (v) --> ~ (Var (v) <: x) --> x <| (v,t<|s) #s=x<|s" in uterm.induct)
+apply auto
+done
+
+lemma agreement: "(t <|r = t <|s) = (\<forall>v. v : vars_of(t) --> Var(v) <|r = Var(v) <|s)"
+by (induct_tac "t", auto)
+
+lemma repl_invariance: "~ v: vars_of(t) ==> t <| (v,u)#s = t <| s"
+by (simp add: agreement)
+
+lemma Var_in_subst [rule_format]:
+ "v : vars_of(t) --> w : vars_of(t <| (v,Var(w))#s)"
+by (induct_tac "t", auto)
+
+
+subsection{*Equality between Substitutions*}
+
+lemma subst_eq_iff: "r =$= s = (\<forall>t. t <| r = t <| s)"
+by (simp add: subst_eq_def)
+
+lemma subst_refl [iff]: "r =$= r"
+by (simp add: subst_eq_iff)
+
+lemma subst_sym: "r =$= s ==> s =$= r"
+by (simp add: subst_eq_iff)
+
+lemma subst_trans: "[| q =$= r; r =$= s |] ==> q =$= s"
+by (simp add: subst_eq_iff)
+
+lemma subst_subst2:
+ "[| r =$= s; P (t <| r) (u <| r) |] ==> P (t <| s) (u <| s)"
+by (simp add: subst_eq_def)
+
+lemmas ssubst_subst2 = subst_sym [THEN subst_subst2]
+
+
+subsection{*Composition of Substitutions*}
+
+lemma [simp]:
+ "[] <> bl = bl"
+ "((a,b)#al) <> bl = (a,b <| bl) # (al <> 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 <> [] = s"
+by (induct "s", auto)
+
+lemma subst_comp_Nil: "s =$= s <> []"
+by simp
+
+lemma subst_comp [simp]: "(t <| r <> s) = (t <| r <| s)"
+apply (induct_tac "t")
+apply (simp_all (no_asm_simp))
+apply (induct "r", auto)
+done
+
+lemma comp_assoc: "(q <> r) <> s =$= q <> (r <> s)"
+apply (simp (no_asm) add: subst_eq_iff)
+done
+
+lemma subst_cong:
+ "[| theta =$= theta1; sigma =$= sigma1|]
+ ==> (theta <> sigma) =$= (theta1 <> sigma1)"
+by (simp add: subst_eq_def)
+
+
+lemma Cons_trivial: "(w, Var(w) <| s) # s =$= s"
+apply (simp add: subst_eq_iff)
+apply (rule allI)
+apply (induct_tac "t", simp_all)
+done
+
+
+lemma comp_subst_subst: "q <> r =$= s ==> t <| q <| r = t <| s"
+by (simp add: subst_eq_iff)
+
+
+subsection{*Domain and range of Substitutions*}
+
+lemma sdom_iff: "(v : sdom(s)) = (Var(v) <| s ~= Var(v))"
+apply (induct "s")
+apply (case_tac [2] a, auto)
+done
+
+
+lemma srange_iff:
+ "v : srange(s) = (\<exists>w. w : sdom(s) & v : vars_of(Var(w) <| 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 <| 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 : sdom(s) --> v : vars_of(t <| s) --> v : srange(s)"
+apply (induct_tac "t")
+apply (case_tac "a : sdom (s) ")
+apply (auto simp add: sdom_iff srange_iff)
+done
+
+lemma Var_elim: "[| v : sdom(s); v ~: srange(s) |] ==> v ~: vars_of(t <| s)"
+by (blast intro: Var_in_srange)
+
+lemma Var_intro [rule_format]:
+ "v : vars_of(t <| s) --> v : srange(s) | v : 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 [rule_format]:
+ "v : srange(s) --> (\<exists>w. w : sdom(s) & v : vars_of(Var(w) <| s))"
+apply (simp (no_asm) add: srange_iff)
+done
+
+lemma dom_range_disjoint:
+ "sdom(s) Int srange(s) = {} = (\<forall>t. sdom(s) Int vars_of(t <| s) = {})"
+apply (simp (no_asm) add: empty_iff_all_not)
+apply (force intro: Var_in_srange dest: srangeD)
+done
+
+lemma subst_not_empty: "~ u <| s = u ==> (\<exists>x. x : sdom(s))"
+by (auto simp add: empty_iff_all_not invariance)
+
+
+lemma id_subst_lemma [simp]: "(M <| [(x, Var x)]) = M"
+by (induct_tac "M", auto)
+
+
end