src/HOL/Subst/Subst.thy

--- a/src/HOL/Subst/Subst.thy	Fri Apr 01 18:59:17 2005 +0200
+++ b/src/HOL/Subst/Subst.thy	Fri Apr 01 21:04:00 2005 +0200
@@ -20,70 +20,80 @@
   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 <| 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 \<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 =$= s == ALL t. t <| r = t <| s"
+  subst_eq_def:  "r =$= s == ALL t. t \<lhd> r = t \<lhd> s"
 
-  comp_def:    "al <> bl == alist_rec al bl (%x y xs g. (x,y <| bl)#g)"
+  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. EX x:sdom(al). y=vars_of(Var(x) <| al)})"
+   "srange(al) == Union({y. \<exists>x \<in> sdom(al). y = vars_of(Var(x) \<lhd> al)})"
 
 
 
 subsection{*Basic Laws*}
 
-lemma subst_Nil [simp]: "t <| [] = t"
+lemma subst_Nil [simp]: "t \<lhd> [] = t"
 by (induct_tac "t", auto)
 
-lemma subst_mono [rule_format]: "t <: u --> t <| s <: u <| s"
+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) <: t) --> t <| (v,t <| s) # s = t <| s"
+     "~ (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=Var (v) --> ~ (Var (v) <: x) --> x <| (v,t<|s) #s=x<|s" in uterm.induct)
+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 <|r = t <|s) = (\<forall>v. v : vars_of(t) --> Var(v) <|r = Var(v) <|s)"
+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 <| (v,u)#s = t <| s"
+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 : vars_of(t) --> w : vars_of(t <| (v,Var(w))#s)"
+     "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 =$= s = (\<forall>t. t <| r = t <| s)"
+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 =$= r"
+lemma subst_refl [iff]: "r \<doteq> r"
 by (simp add: subst_eq_iff)
 
-lemma subst_sym: "r =$= s ==> s =$= r"
+lemma subst_sym: "r \<doteq> s ==> s \<doteq> r"
 by (simp add: subst_eq_iff)
 
-lemma subst_trans: "[| q =$= r; r =$= s |] ==> q =$= s"
+lemma subst_trans: "[| q \<doteq> r; r \<doteq> s |] ==> q \<doteq> s"
 by (simp add: subst_eq_iff)
 
 lemma subst_subst2:
-    "[| r =$= s; P (t <| r) (u <| r) |] ==> P (t <| s) (u <| s)"
+    "[| 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]
@@ -92,98 +102,95 @@
 subsection{*Composition of Substitutions*}
 
 lemma [simp]: 
-     "[] <> bl = bl"
-     "((a,b)#al) <> bl = (a,b <| bl) # (al <> bl)"
+     "[] \<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 <> [] = s"
+lemma comp_Nil [simp]: "s \<lozenge> [] = s"
 by (induct "s", auto)
 
-lemma subst_comp_Nil: "s =$= s <> []"
+lemma subst_comp_Nil: "s \<doteq> s \<lozenge> []"
 by simp
 
-lemma subst_comp [simp]: "(t <| r <> s) = (t <| r <| s)"
+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 <> r) <> s =$= q <> (r <> s)"
-apply (simp (no_asm) add: subst_eq_iff)
-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 =$= theta1; sigma =$= sigma1|] 
-      ==> (theta <> sigma) =$= (theta1 <> sigma1)"
+     "[| 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) <| s) # s =$= s"
+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 <> r =$= s ==>  t <| q <| r = t <| s"
+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 : sdom(s)) = (Var(v) <| s ~= Var(v))"
+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 : srange(s) = (\<exists>w. w : sdom(s) & v : vars_of(Var(w) <| s))"
+   "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 <| s = t) = (sdom(s) Int vars_of(t) = {})"
+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 : sdom(s) -->  v : vars_of(t <| s) --> v : srange(s)"
+     "v \<in> sdom(s) -->  v \<in> vars_of(t \<lhd> s) --> v \<in> srange(s)"
 apply (induct_tac "t")
-apply (case_tac "a : sdom (s) ")
+apply (case_tac "a \<in> 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)"
+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 : vars_of(t <| s) --> v : srange(s) | v : vars_of(t)"
+     "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 [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 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 <| s) = {})"
-apply (simp (no_asm) add: empty_iff_all_not)
+     "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 <| s = u ==> (\<exists>x. x : sdom(s))"
+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 <| [(x, Var x)]) = M"
+lemma id_subst_lemma [simp]: "(M \<lhd> [(x, Var x)]) = M"
 by (induct_tac "M", auto)