Shortened names and added new thm.

--- a/src/HOL/Induct/ABexp.ML	Wed Nov 04 13:00:15 1998 +0100
+++ b/src/HOL/Induct/ABexp.ML	Thu Nov 05 14:05:57 1998 +0100
@@ -6,11 +6,20 @@
 Arithmetic and boolean expressions
 *)
 
-(** substitution theorem for arithmetic and boolean expressions **)
+(** substitution theorems for arithmetic and boolean expressions **)
 
+(* One variable *)
 Goal
-  "eval_aexp env (subst_aexp (Var(v := a')) a) = eval_aexp (env(v := eval_aexp env a')) a &  \
- \ eval_bexp env (subst_bexp (Var(v := a')) b) = eval_bexp (env(v := eval_aexp env a')) b";
+  "evala env (substa (Var(v := a')) a) = evala (env(v := evala env a')) a &  \
+ \ evalb env (substb (Var(v := a')) b) = evalb (env(v := evala env a')) b";
 by (mutual_induct_tac ["a","b"] 1);
 by (ALLGOALS Asm_full_simp_tac);
-qed "subst_aexp_bexp";
+qed "subst1_aexp_bexp";
+
+(* All variables *)
+Goal
+  "evala env (substa s a) = evala (%x. evala env (s x)) a &  \
+ \ evalb env (substb s b) = evalb (%x. evala env (s x)) b";
+by (mutual_induct_tac ["a","b"] 1);
+by (Auto_tac);
+qed "subst_all_aexp_bexp";

--- a/src/HOL/Induct/ABexp.thy	Wed Nov 04 13:00:15 1998 +0100
+++ b/src/HOL/Induct/ABexp.thy	Thu Nov 05 14:05:57 1998 +0100
@@ -9,7 +9,7 @@
 ABexp = Main +
 
 datatype
-  'a aexp = If_then_else ('a bexp) ('a aexp) ('a aexp)
+  'a aexp = IF ('a bexp) ('a aexp) ('a aexp)
           | Sum ('a aexp) ('a aexp)
           | Diff ('a aexp) ('a aexp)
           | Var 'a
@@ -17,44 +17,44 @@
 and
   'a bexp = Less ('a aexp) ('a aexp)
           | And ('a bexp) ('a bexp)
-          | Or ('a bexp) ('a bexp)
+          | Neg ('a bexp)
 
 
 (** evaluation of arithmetic and boolean expressions **)
 
 consts
-  eval_aexp :: "['a => nat, 'a aexp] => nat"
-  eval_bexp :: "['a => nat, 'a bexp] => bool"
+  evala :: ('a => nat) => 'a aexp => nat
+  evalb :: ('a => nat) => 'a bexp => bool
 
 primrec
-  "eval_aexp env (If_then_else b a1 a2) =
-     (if eval_bexp env b then eval_aexp env a1 else eval_aexp env a2)"
-  "eval_aexp env (Sum a1 a2) = eval_aexp env a1 + eval_aexp env a2"
-  "eval_aexp env (Diff a1 a2) = eval_aexp env a1 - eval_aexp env a2"
-  "eval_aexp env (Var v) = env v"
-  "eval_aexp env (Num n) = n"
+  "evala env (IF b a1 a2) =
+     (if evalb env b then evala env a1 else evala env a2)"
+  "evala env (Sum a1 a2) = evala env a1 + evala env a2"
+  "evala env (Diff a1 a2) = evala env a1 - evala env a2"
+  "evala env (Var v) = env v"
+  "evala env (Num n) = n"
 
-  "eval_bexp env (Less a1 a2) = (eval_aexp env a1 < eval_aexp env a2)"
-  "eval_bexp env (And b1 b2) = (eval_bexp env b1 & eval_bexp env b2)"
-  "eval_bexp env (Or b1 b2) = (eval_bexp env b1 & eval_bexp env b2)"
+  "evalb env (Less a1 a2) = (evala env a1 < evala env a2)"
+  "evalb env (And b1 b2) = (evalb env b1 & evalb env b2)"
+  "evalb env (Neg b) = (~ evalb env b)"
 
 
 (** substitution on arithmetic and boolean expressions **)
 
 consts
-  subst_aexp :: "['a => 'b aexp, 'a aexp] => 'b aexp"
-  subst_bexp :: "['a => 'b aexp, 'a bexp] => 'b bexp"
+  substa :: ('a => 'b aexp) => 'a aexp => 'b aexp
+  substb :: ('a => 'b aexp) => 'a bexp => 'b bexp
 
 primrec
-  "subst_aexp f (If_then_else b a1 a2) =
-     If_then_else (subst_bexp f b) (subst_aexp f a1) (subst_aexp f a2)"
-  "subst_aexp f (Sum a1 a2) = Sum (subst_aexp f a1) (subst_aexp f a2)"
-  "subst_aexp f (Diff a1 a2) = Diff (subst_aexp f a1) (subst_aexp f a2)"
-  "subst_aexp f (Var v) = f v"
-  "subst_aexp f (Num n) = Num n"
+  "substa f (IF b a1 a2) =
+     IF (substb f b) (substa f a1) (substa f a2)"
+  "substa f (Sum a1 a2) = Sum (substa f a1) (substa f a2)"
+  "substa f (Diff a1 a2) = Diff (substa f a1) (substa f a2)"
+  "substa f (Var v) = f v"
+  "substa f (Num n) = Num n"
 
-  "subst_bexp f (Less a1 a2) = Less (subst_aexp f a1) (subst_aexp f a2)"
-  "subst_bexp f (And b1 b2) = And (subst_bexp f b1) (subst_bexp f b2)"
-  "subst_bexp f (Or b1 b2) = Or (subst_bexp f b1) (subst_bexp f b2)"
+  "substb f (Less a1 a2) = Less (substa f a1) (substa f a2)"
+  "substb f (And b1 b2) = And (substb f b1) (substb f b2)"
+  "substb f (Neg b) = Neg (substb f b)"
 
 end