src/HOL/Induct/Com.thy
 author haftmann Wed, 16 Jun 2021 08:19:09 +0000 changeset 73853 52b829b18066 parent 63167 0909deb8059b permissions -rw-r--r--
more lemmas
```
(*  Title:      HOL/Induct/Com.thy
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

Example of Mutual Induction via Iteratived Inductive Definitions: Commands
*)

section\<open>Mutual Induction via Iteratived Inductive Definitions\<close>

theory Com imports Main begin

typedecl loc
type_synonym state = "loc => nat"

datatype
exp = N nat
| X loc
| Op "nat => nat => nat" exp exp
| valOf com exp          ("VALOF _ RESULTIS _"  60)
and
com = SKIP
| Assign loc exp         (infixl ":=" 60)
| Semi com com           ("_;;_"  [60, 60] 60)
| Cond exp com com       ("IF _ THEN _ ELSE _"  60)
| While exp com          ("WHILE _ DO _"  60)

subsection \<open>Commands\<close>

text\<open>Execution of commands\<close>

abbreviation (input)
generic_rel  ("_/ -|[_]-> _" [50,0,50] 50)  where
"esig -|[eval]-> ns == (esig,ns) \<in> eval"

text\<open>Command execution.  Natural numbers represent Booleans: 0=True, 1=False\<close>

inductive_set
exec :: "((exp*state) * (nat*state)) set => ((com*state)*state)set"
and exec_rel :: "com * state => ((exp*state) * (nat*state)) set => state => bool"
("_/ -[_]-> _" [50,0,50] 50)
for eval :: "((exp*state) * (nat*state)) set"
where
"csig -[eval]-> s == (csig,s) \<in> exec eval"

| 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 |]
==> (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\<open>Justifies using "exec" in the inductive definition of "eval"\<close>
lemma exec_mono: "A<=B ==> exec(A) <= exec(B)"
apply (rule subsetI)
apply (erule exec.induct)
apply blast+
done

lemma [pred_set_conv]:
"((\<lambda>x x' y y'. ((x, x'), (y, y')) \<in> R) <= (\<lambda>x x' y y'. ((x, x'), (y, y')) \<in> S)) = (R <= S)"
unfolding subset_eq

lemma [pred_set_conv]:
"((\<lambda>x x' y. ((x, x'), y) \<in> R) <= (\<lambda>x x' y. ((x, x'), y) \<in> S)) = (R <= S)"
unfolding subset_eq

text\<open>Command execution is functional (deterministic) provided evaluation is\<close>
theorem single_valued_exec: "single_valued ev ==> single_valued(exec ev)"
apply (intro allI)
apply (rule impI)
apply (erule exec.induct)
apply (blast elim: exec_WHILE_case)+
done

subsection \<open>Expressions\<close>

text\<open>Evaluation of arithmetic expressions\<close>

inductive_set
eval    :: "((exp*state) * (nat*state)) set"
and eval_rel :: "[exp*state,nat*state] => bool"  (infixl "-|->" 50)
where
"esig -|-> ns == (esig, ns) \<in> eval"

| 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\<open>Make the induction rule look nicer -- though \<open>eta_contract\<close> makes the new
version look worse than it is...\<close>

lemma split_lemma: "{((e,s),(n,s')). P e s n s'} = Collect (case_prod (%v. case_prod (case_prod P v)))"
by auto

text\<open>New induction rule.  Note the form of the VALOF induction hypothesis\<close>
lemma eval_induct
[case_names N X Op valOf, consumes 1, induct set: eval]:
"[| (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 (induct set: eval)
apply blast
apply blast
apply blast
apply (frule Int_lower1 [THEN exec_mono, THEN subsetD])
done

text\<open>Lemma for \<open>Function_eval\<close>.  The major premise is that \<open>(c,s)\<close> executes to \<open>s1\<close>
using eval restricted to its functional part.  Note that the execution
\<open>(c,s) -[eval]-> s2\<close> can use unrestricted \<open>eval\<close>!  The reason is that
the execution \<open>(c,s) -[eval Int {...}]-> s1\<close> assures us that execution is
functional on the argument \<open>(c,s)\<close>.
\<close>
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 (induct set: exec)
apply 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" for c ev in thin_rl)
apply clarify
apply (erule exec_WHILE_case, blast+)
done

text\<open>Expression evaluation is functional, or deterministic\<close>
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 [dest!]: "(N n, s) -|-> (v, s') ==> (v = n & s' = s)"
by (induct e == "N n" s v s' set: eval) simp_all

text\<open>This theorem says that "WHILE TRUE DO c" cannot terminate\<close>
lemma while_true_E:
"(c', s) -[eval]-> t ==> c' = WHILE (N 0) DO c ==> False"
by (induct set: exec) auto

subsection\<open>Equivalence of IF e THEN c;;(WHILE e DO c) ELSE SKIP  and
WHILE e DO c\<close>

lemma while_if1:
"(c',s) -[eval]-> t
==> c' = WHILE e DO c ==>
(IF e THEN c;;c' ELSE SKIP, s) -[eval]-> t"
by (induct set: exec) auto

lemma while_if2:
"(c',s) -[eval]-> t
==> c' = IF e THEN c;;(WHILE e DO c) ELSE SKIP ==>
(WHILE e DO c, s) -[eval]-> t"
by (induct set: exec) 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\<open>Equivalence of  (IF e THEN c1 ELSE c2);;c
and  IF e THEN (c1;;c) ELSE (c2;;c)\<close>

lemma if_semi1:
"(c',s) -[eval]-> t
==> c' = (IF e THEN c1 ELSE c2);;c ==>
(IF e THEN (c1;;c) ELSE (c2;;c), s) -[eval]-> t"
by (induct set: exec) auto

lemma if_semi2:
"(c',s) -[eval]-> t
==> c' = IF e THEN (c1;;c) ELSE (c2;;c) ==>
((IF e THEN c1 ELSE c2);;c, s) -[eval]-> t"
by (induct set: exec) 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\<open>Equivalence of  VALOF c1 RESULTIS (VALOF c2 RESULTIS e)
and  VALOF c1;;c2 RESULTIS e
\<close>

lemma valof_valof1:
"(e',s) -|-> (v,s')
==> e' = VALOF c1 RESULTIS (VALOF c2 RESULTIS e) ==>
(VALOF c1;;c2 RESULTIS e, s) -|-> (v,s')"
by (induct set: eval) auto

lemma valof_valof2:
"(e',s) -|-> (v,s')
==> e' = VALOF c1;;c2 RESULTIS e ==>
(VALOF c1 RESULTIS (VALOF c2 RESULTIS e), s) -|-> (v,s')"
by (induct set: eval) 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\<open>Equivalence of  VALOF SKIP RESULTIS e  and  e\<close>

lemma valof_skip1:
"(e',s) -|-> (v,s')
==> e' = VALOF SKIP RESULTIS e ==>
(e, s) -|-> (v,s')"
by (induct set: eval) 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\<open>Equivalence of  VALOF x:=e RESULTIS x  and  e\<close>

lemma valof_assign1:
"(e',s) -|-> (v,s'')
==> e' = VALOF x:=e RESULTIS X x ==>
(\<exists>s'. (e, s) -|-> (v,s') & (s'' = s'(x:=v)))"
by (induct set: eval) (simp_all del: fun_upd_apply, clarify, auto)

lemma valof_assign2:
"(e,s) -|-> (v,s') ==> (VALOF x:=e RESULTIS X x, s) -|-> (v,s'(x:=v))"
by blast

end
```