src/HOL/Induct/Com.thy

     Copyright   1997  University of Cambridge
 
 Example of Mutual Induction via Iteratived Inductive Definitions: Commands
 *)
 
-Com = Main +
-
-types loc
-      state = "loc => nat"
-      n2n2n = "nat => nat => nat"
+theory Com = Main:
+
+typedecl loc
+
+types  state = "loc => nat"
+       n2n2n = "nat => nat => nat"
 
 arities loc :: type
 
 datatype
   exp = N nat

       | ":="  loc exp          (infixl  60)
       | Semi  com com          ("_;;_"  [60, 60] 60)
       | Cond  exp com com      ("IF _ THEN _ ELSE _"  60)
       | While exp com          ("WHILE _ DO _"  60)
 
-(** Execution of commands **)
+text{* Execution of commands *}
 consts  exec    :: "((exp*state) * (nat*state)) set => ((com*state)*state)set"
        "@exec"  :: "((exp*state) * (nat*state)) set => 
                     [com*state,state] => bool"     ("_/ -[_]-> _" [50,0,50] 50)
 
-translations  "csig -[eval]-> s" == "(csig,s) : exec eval"
+translations  "csig -[eval]-> s" == "(csig,s) \<in> exec eval"
 
 syntax  eval'   :: "[exp*state,nat*state] => 
 		    ((exp*state) * (nat*state)) set => bool"
-						   ("_/ -|[_]-> _" [50,0,50] 50)
+					   ("_/ -|[_]-> _" [50,0,50] 50)
+
 translations
-    "esig -|[eval]-> ns" => "(esig,ns) : eval"
-
-(*Command execution.  Natural numbers represent Booleans: 0=True, 1=False*)
+    "esig -|[eval]-> ns" => "(esig,ns) \<in> eval"
+
+text{*Command execution.  Natural numbers represent Booleans: 0=True, 1=False*}
 inductive "exec eval"
-  intrs
-    Skip    "(SKIP,s) -[eval]-> s"
-
-    Assign  "(e,s) -|[eval]-> (v,s') ==> (x := e, s) -[eval]-> s'(x:=v)"
-
-    Semi    "[| (c0,s) -[eval]-> s2; (c1,s2) -[eval]-> s1 |] 
-            ==> (c0 ;; c1, s) -[eval]-> s1"
-
-    IfTrue "[| (e,s) -|[eval]-> (0,s');  (c0,s') -[eval]-> s1 |] 
-            ==> (IF e THEN c0 ELSE c1, s) -[eval]-> s1"
-
-    IfFalse "[| (e,s) -|[eval]->  (Suc 0, s');  (c1,s') -[eval]-> s1 |] 
+  intros
+    Skip:    "(SKIP,s) -[eval]-> s"
+
+    Assign:  "(e,s) -|[eval]-> (v,s') ==> (x := e, s) -[eval]-> s'(x:=v)"
+
+    Semi:    "[| (c0,s) -[eval]-> s2; (c1,s2) -[eval]-> s1 |] 
+             ==> (c0 ;; c1, s) -[eval]-> s1"
+
+    IfTrue: "[| (e,s) -|[eval]-> (0,s');  (c0,s') -[eval]-> s1 |] 
              ==> (IF e THEN c0 ELSE c1, s) -[eval]-> s1"
 
-    WhileFalse "(e,s) -|[eval]-> (Suc 0, s1) ==> (WHILE e DO c, s) -[eval]-> s1"
-
-    WhileTrue  "[| (e,s) -|[eval]-> (0,s1);
-                (c,s1) -[eval]-> s2;  (WHILE e DO c, s2) -[eval]-> s3 |] 
-                ==> (WHILE e DO c, s) -[eval]-> s3"
+    IfFalse: "[| (e,s) -|[eval]->  (Suc 0, s');  (c1,s') -[eval]-> s1 |] 
+              ==> (IF e THEN c0 ELSE c1, s) -[eval]-> s1"
+
+    WhileFalse: "(e,s) -|[eval]-> (Suc 0, s1) 
+                 ==> (WHILE e DO c, s) -[eval]-> s1"
+
+    WhileTrue:  "[| (e,s) -|[eval]-> (0,s1);
+                    (c,s1) -[eval]-> s2;  (WHILE e DO c, s2) -[eval]-> s3 |] 
+                 ==> (WHILE e DO c, s) -[eval]-> s3"
+
+declare exec.intros [intro]
+
+
+inductive_cases
+	[elim!]: "(SKIP,s) -[eval]-> t"
+    and [elim!]: "(x:=a,s) -[eval]-> t"
+    and	[elim!]: "(c1;;c2, s) -[eval]-> t"
+    and	[elim!]: "(IF e THEN c1 ELSE c2, s) -[eval]-> t"
+    and	exec_WHILE_case: "(WHILE b DO c,s) -[eval]-> t"
+
+
+text{*Justifies using "exec" in the inductive definition of "eval"*}
+lemma exec_mono: "A<=B ==> exec(A) <= exec(B)"
+apply (unfold exec.defs )
+apply (rule lfp_mono)
+apply (assumption | rule basic_monos)+
+done
+
+ML {*
+Unify.trace_bound := 30;
+Unify.search_bound := 60;
+*}
+
+text{*Command execution is functional (deterministic) provided evaluation is*}
+theorem single_valued_exec: "single_valued ev ==> single_valued(exec ev)"
+apply (simp add: single_valued_def)
+apply (intro allI) 
+apply (rule impI)
+apply (erule exec.induct)
+apply (blast elim: exec_WHILE_case)+
+done
+
+
+section {* Expressions *}
+
+text{* Evaluation of arithmetic expressions *}
+consts  eval    :: "((exp*state) * (nat*state)) set"
+       "-|->"   :: "[exp*state,nat*state] => bool"         (infixl 50)
+
+translations
+    "esig -|-> (n,s)" <= "(esig,n,s) \<in> eval"
+    "esig -|-> ns"    == "(esig,ns ) \<in> eval"
+  
+inductive eval
+  intros 
+    N [intro!]: "(N(n),s) -|-> (n,s)"
+
+    X [intro!]: "(X(x),s) -|-> (s(x),s)"
+
+    Op [intro]: "[| (e0,s) -|-> (n0,s0);  (e1,s0)  -|-> (n1,s1) |] 
+                 ==> (Op f e0 e1, s) -|-> (f n0 n1, s1)"
+
+    valOf [intro]: "[| (c,s) -[eval]-> s0;  (e,s0)  -|-> (n,s1) |] 
+                    ==> (VALOF c RESULTIS e, s) -|-> (n, s1)"
+
+  monos exec_mono
+
+
+inductive_cases
+	[elim!]: "(N(n),sigma) -|-> (n',s')"
+    and [elim!]: "(X(x),sigma) -|-> (n,s')"
+    and	[elim!]: "(Op f a1 a2,sigma)  -|-> (n,s')"
+    and	[elim!]: "(VALOF c RESULTIS e, s) -|-> (n, s1)"
+
+
+lemma var_assign_eval [intro!]: "(X x, s(x:=n)) -|-> (n, s(x:=n))"
+by (rule fun_upd_same [THEN subst], fast)
+
+
+text{* Make the induction rule look nicer -- though eta_contract makes the new
+    version look worse than it is...*}
+
+lemma split_lemma:
+     "{((e,s),(n,s')). P e s n s'} = Collect (split (%v. split (split P v)))"
+by auto
+
+text{*New induction rule.  Note the form of the VALOF induction hypothesis*}
+lemma eval_induct:
+  "[| (e,s) -|-> (n,s');                                          
+      !!n s. P (N n) s n s;                                       
+      !!s x. P (X x) s (s x) s;                                   
+      !!e0 e1 f n0 n1 s s0 s1.                                    
+         [| (e0,s) -|-> (n0,s0); P e0 s n0 s0;                    
+            (e1,s0) -|-> (n1,s1); P e1 s0 n1 s1                   
+         |] ==> P (Op f e0 e1) s (f n0 n1) s1;                    
+      !!c e n s s0 s1.                                            
+         [| (c,s) -[eval Int {((e,s),(n,s')). P e s n s'}]-> s0;  
+            (c,s) -[eval]-> s0;                                   
+            (e,s0) -|-> (n,s1); P e s0 n s1 |]                    
+         ==> P (VALOF c RESULTIS e) s n s1                        
+   |] ==> P e s n s'"
+apply (erule eval.induct, blast) 
+apply blast 
+apply blast 
+apply (frule Int_lower1 [THEN exec_mono, THEN subsetD])
+apply (auto simp add: split_lemma)
+done
+
+
+text{*Lemma for Function_eval.  The major premise is that (c,s) executes to s1
+  using eval restricted to its functional part.  Note that the execution
+  (c,s) -[eval]-> s2 can use unrestricted eval!  The reason is that 
+  the execution (c,s) -[eval Int {...}]-> s1 assures us that execution is
+  functional on the argument (c,s).
+*}
+lemma com_Unique:
+ "(c,s) -[eval Int {((e,s),(n,t)). \<forall>nt'. (e,s) -|-> nt' --> (n,t)=nt'}]-> s1  
+  ==> \<forall>s2. (c,s) -[eval]-> s2 --> s2=s1"
+apply (erule exec.induct, simp_all)
+      apply blast
+     apply force
+    apply blast
+   apply blast
+  apply blast
+ apply (blast elim: exec_WHILE_case)
+apply (erule_tac V = "(?c,s2) -[?ev]-> s3" in thin_rl)
+apply clarify
+apply (erule exec_WHILE_case, blast+) 
+done
+
+
+text{*Expression evaluation is functional, or deterministic*}
+theorem single_valued_eval: "single_valued eval"
+apply (unfold single_valued_def)
+apply (intro allI, rule impI) 
+apply (simp (no_asm_simp) only: split_tupled_all)
+apply (erule eval_induct)
+apply (drule_tac [4] com_Unique)
+apply (simp_all (no_asm_use))
+apply blast+
+done
+
+
+lemma eval_N_E_lemma: "(e,s) -|-> (v,s') ==> (e = N n) --> (v=n & s'=s)"
+by (erule eval_induct, simp_all)
+
+lemmas eval_N_E [dest!] = eval_N_E_lemma [THEN mp, OF _ refl]
+
+
+text{*This theorem says that "WHILE TRUE DO c" cannot terminate*}
+lemma while_true_E [rule_format]:
+     "(c', s) -[eval]-> t ==> (c' = WHILE (N 0) DO c) --> False"
+by (erule exec.induct, auto)
+
+
+subsection{* Equivalence of IF e THEN c;;(WHILE e DO c) ELSE SKIP  and  
+       WHILE e DO c *}
+
+lemma while_if1 [rule_format]:
+     "(c',s) -[eval]-> t 
+      ==> (c' = WHILE e DO c) -->  
+          (IF e THEN c;;c' ELSE SKIP, s) -[eval]-> t"
+by (erule exec.induct, auto)
+
+lemma while_if2 [rule_format]:
+     "(c',s) -[eval]-> t
+      ==> (c' = IF e THEN c;;(WHILE e DO c) ELSE SKIP) -->  
+          (WHILE e DO c, s) -[eval]-> t"
+by (erule exec.induct, auto)
+
+
+theorem while_if:
+     "((IF e THEN c;;(WHILE e DO c) ELSE SKIP, s) -[eval]-> t)  =   
+      ((WHILE e DO c, s) -[eval]-> t)"
+by (blast intro: while_if1 while_if2)
+
+
+
+subsection{* Equivalence of  (IF e THEN c1 ELSE c2);;c
+                         and  IF e THEN (c1;;c) ELSE (c2;;c)   *}
+
+lemma if_semi1 [rule_format]:
+     "(c',s) -[eval]-> t
+      ==> (c' = (IF e THEN c1 ELSE c2);;c) -->  
+          (IF e THEN (c1;;c) ELSE (c2;;c), s) -[eval]-> t"
+by (erule exec.induct, auto)
+
+lemma if_semi2 [rule_format]:
+     "(c',s) -[eval]-> t
+      ==> (c' = IF e THEN (c1;;c) ELSE (c2;;c)) -->  
+          ((IF e THEN c1 ELSE c2);;c, s) -[eval]-> t"
+by (erule exec.induct, auto)
+
+theorem if_semi: "(((IF e THEN c1 ELSE c2);;c, s) -[eval]-> t)  =   
+                  ((IF e THEN (c1;;c) ELSE (c2;;c), s) -[eval]-> t)"
+by (blast intro: if_semi1 if_semi2)
+
+
+
+subsection{* Equivalence of  VALOF c1 RESULTIS (VALOF c2 RESULTIS e)
+                  and  VALOF c1;;c2 RESULTIS e
+ *}
+
+lemma valof_valof1 [rule_format]:
+     "(e',s) -|-> (v,s')  
+      ==> (e' = VALOF c1 RESULTIS (VALOF c2 RESULTIS e)) -->  
+          (VALOF c1;;c2 RESULTIS e, s) -|-> (v,s')"
+by (erule eval_induct, auto)
+
+
+lemma valof_valof2 [rule_format]:
+     "(e',s) -|-> (v,s')
+      ==> (e' = VALOF c1;;c2 RESULTIS e) -->  
+          (VALOF c1 RESULTIS (VALOF c2 RESULTIS e), s) -|-> (v,s')"
+by (erule eval_induct, auto)
+
+theorem valof_valof:
+     "((VALOF c1 RESULTIS (VALOF c2 RESULTIS e), s) -|-> (v,s'))  =   
+      ((VALOF c1;;c2 RESULTIS e, s) -|-> (v,s'))"
+by (blast intro: valof_valof1 valof_valof2)
+
+
+subsection{* Equivalence of  VALOF SKIP RESULTIS e  and  e *}
+
+lemma valof_skip1 [rule_format]:
+     "(e',s) -|-> (v,s')
+      ==> (e' = VALOF SKIP RESULTIS e) -->  
+          (e, s) -|-> (v,s')"
+by (erule eval_induct, auto)
+
+lemma valof_skip2:
+     "(e,s) -|-> (v,s') ==> (VALOF SKIP RESULTIS e, s) -|-> (v,s')"
+by blast
+
+theorem valof_skip:
+     "((VALOF SKIP RESULTIS e, s) -|-> (v,s'))  =  ((e, s) -|-> (v,s'))"
+by (blast intro: valof_skip1 valof_skip2)
+
+
+subsection{* Equivalence of  VALOF x:=e RESULTIS x  and  e *}
+
+lemma valof_assign1 [rule_format]:
+     "(e',s) -|-> (v,s'')
+      ==> (e' = VALOF x:=e RESULTIS X x) -->  
+          (\<exists>s'. (e, s) -|-> (v,s') & (s'' = s'(x:=v)))"
+apply (erule eval_induct)
+apply (simp_all del: fun_upd_apply, clarify, auto) 
+done
+
+lemma valof_assign2:
+     "(e,s) -|-> (v,s') ==> (VALOF x:=e RESULTIS X x, s) -|-> (v,s'(x:=v))"
+by blast
+
+
 end