Theory Com

theory Com
imports Main
(*  Title:      HOL/Induct/Com.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1997 University of Cambridge

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


header{*Mutual Induction via Iteratived Inductive Definitions*}

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 {* Commands *}

text{* Execution of commands *}

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

text{*Command execution. Natural numbers represent Booleans: 0=True, 1=False*}

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) ∈ 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{*Justifies using "exec" in the inductive definition of "eval"*}
lemma exec_mono: "A<=B ==> exec(A) <= exec(B)"
apply (rule subsetI)
apply (simp add: split_paired_all)
apply (erule exec.induct)
apply blast+
done

lemma [pred_set_conv]:
"((λx x' y y'. ((x, x'), (y, y')) ∈ R) <= (λx x' y y'. ((x, x'), (y, y')) ∈ S)) = (R <= S)"
unfolding subset_eq
by (auto simp add: le_fun_def)

lemma [pred_set_conv]:
"((λx x' y. ((x, x'), y) ∈ R) <= (λx x' y. ((x, x'), y) ∈ S)) = (R <= S)"
unfolding subset_eq
by (auto simp add: le_fun_def)

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


subsection {* Expressions *}

text{* Evaluation of arithmetic expressions *}

inductive_set
eval :: "((exp*state) * (nat*state)) set"
and eval_rel :: "[exp*state,nat*state] => bool" (infixl "-|->" 50)
where
"esig -|-> ns == (esig, ns) ∈ 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{* Make the induction rule look nicer -- though @{text 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
[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])
apply (auto simp add: split_lemma)
done


text{*Lemma for @{text Function_eval}. The major premise is that @{text "(c,s)"} executes to @{text "s1"}
using eval restricted to its functional part. Note that the execution
@{text "(c,s) -[eval]-> s2"} can use unrestricted @{text eval}! The reason is that
the execution @{text "(c,s) -[eval Int {...}]-> s1"} assures us that execution is
functional on the argument @{text "(c,s)"}.
*}

lemma com_Unique:
"(c,s) -[eval Int {((e,s),(n,t)). ∀nt'. (e,s) -|-> nt' --> (n,t)=nt'}]-> s1
==> ∀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" 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 [dest!]: "(N n, s) -|-> (v, s') ==> (v = n & s' = s)"
by (induct e == "N n" s v s' set: eval) simp_all

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


subsection{* Equivalence of IF e THEN c;;(WHILE e DO c) ELSE SKIP and
WHILE e DO c *}


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{* Equivalence of (IF e THEN c1 ELSE c2);;c
and IF e THEN (c1;;c) ELSE (c2;;c) *}


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{* Equivalence of VALOF c1 RESULTIS (VALOF c2 RESULTIS e)
and VALOF c1;;c2 RESULTIS e
*}


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{* Equivalence of VALOF SKIP RESULTIS e and e *}

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{* Equivalence of VALOF x:=e RESULTIS x and e *}

lemma valof_assign1:
"(e',s) -|-> (v,s'')
==> e' = VALOF x:=e RESULTIS X x ==>
(∃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