header{* Hoare Logic for Total Correctness *}

theory HoareT imports Hoare_Sound_Complete begin

text{*
Now that we have termination, we can define
total validity, @{text"\<Turnstile>\<^sub>t"}, as partial validity and guaranteed termination:*}

definition hoare_tvalid :: "assn \<Rightarrow> com \<Rightarrow> assn \<Rightarrow> bool"
  ("\<Turnstile>\<^sub>t {(1_)}/ (_)/ {(1_)}" 50) where
"\<Turnstile>\<^sub>t {P}c{Q}  \<equiv>  \<forall>s. P s \<longrightarrow> (\<exists>t. (c,s) \<Rightarrow> t \<and> Q t)"

text{* Proveability of Hoare triples in the proof system for total
correctness is written @{text"\<turnstile>\<^sub>t {P}c{Q}"} and defined
inductively. The rules for @{text"\<turnstile>\<^sub>t"} differ from those for
@{text"\<turnstile>"} only in the one place where nontermination can arise: the
@{term While}-rule. *}

inductive
  hoaret :: "assn \<Rightarrow> com \<Rightarrow> assn \<Rightarrow> bool" ("\<turnstile>\<^sub>t ({(1_)}/ (_)/ {(1_)})" 50)
where
Skip:  "\<turnstile>\<^sub>t {P} SKIP {P}" |
Assign:  "\<turnstile>\<^sub>t {\<lambda>s. P(s[a/x])} x::=a {P}" |
Semi: "\<lbrakk> \<turnstile>\<^sub>t {P\<^isub>1} c\<^isub>1 {P\<^isub>2}; \<turnstile>\<^sub>t {P\<^isub>2} c\<^isub>2 {P\<^isub>3} \<rbrakk> \<Longrightarrow> \<turnstile>\<^sub>t {P\<^isub>1} c\<^isub>1;c\<^isub>2 {P\<^isub>3}" |
If: "\<lbrakk> \<turnstile>\<^sub>t {\<lambda>s. P s \<and> bval b s} c\<^isub>1 {Q}; \<turnstile>\<^sub>t {\<lambda>s. P s \<and> \<not> bval b s} c\<^isub>2 {Q} \<rbrakk>
  \<Longrightarrow> \<turnstile>\<^sub>t {P} IF b THEN c\<^isub>1 ELSE c\<^isub>2 {Q}" |
While:
  "\<lbrakk> \<And>n::nat. \<turnstile>\<^sub>t {\<lambda>s. P s \<and> bval b s \<and> f s = n} c {\<lambda>s. P s \<and> f s < n}\<rbrakk>
   \<Longrightarrow> \<turnstile>\<^sub>t {P} WHILE b DO c {\<lambda>s. P s \<and> \<not>bval b s}" |
conseq: "\<lbrakk> \<forall>s. P' s \<longrightarrow> P s; \<turnstile>\<^sub>t {P}c{Q}; \<forall>s. Q s \<longrightarrow> Q' s  \<rbrakk> \<Longrightarrow>
           \<turnstile>\<^sub>t {P'}c{Q'}"

text{* The @{term While}-rule is like the one for partial correctness but it
requires additionally that with every execution of the loop body some measure
function @{term[source]"f :: state \<Rightarrow> nat"} decreases. *}

lemma strengthen_pre:
  "\<lbrakk> \<forall>s. P' s \<longrightarrow> P s;  \<turnstile>\<^sub>t {P} c {Q} \<rbrakk> \<Longrightarrow> \<turnstile>\<^sub>t {P'} c {Q}"
by (metis conseq)

lemma weaken_post:
  "\<lbrakk> \<turnstile>\<^sub>t {P} c {Q};  \<forall>s. Q s \<longrightarrow> Q' s \<rbrakk> \<Longrightarrow>  \<turnstile>\<^sub>t {P} c {Q'}"
by (metis conseq)

lemma Assign': "\<forall>s. P s \<longrightarrow> Q(s[a/x]) \<Longrightarrow> \<turnstile>\<^sub>t {P} x ::= a {Q}"
by (simp add: strengthen_pre[OF _ Assign])

lemma While':
assumes "\<And>n::nat. \<turnstile>\<^sub>t {\<lambda>s. P s \<and> bval b s \<and> f s = n} c {\<lambda>s. P s \<and> f s < n}"
    and "\<forall>s. P s \<and> \<not> bval b s \<longrightarrow> Q s"
shows "\<turnstile>\<^sub>t {P} WHILE b DO c {Q}"
by(blast intro: assms(1) weaken_post[OF While assms(2)])

text{* Our standard example: *}

abbreviation "w n ==
  WHILE Less (V 1) (N n)
  DO ( 1 ::= Plus (V 1) (N 1); 0 ::= Plus (V 0) (V 1) )"

lemma "\<turnstile>\<^sub>t {\<lambda>s. True} 0 ::= N 0; 1 ::= N 0; w n {\<lambda>s. s 0 = \<Sum> {1 .. n}}"
apply(rule Semi)
prefer 2
apply(rule While'
  [where P = "\<lambda>s. s 0 = \<Sum> {1..s 1} \<and> s 1 \<le> n"
   and f = "\<lambda>s. n - s 1"])
apply(rule Semi)
prefer 2
apply(rule Assign)
apply(rule Assign')
apply simp
apply arith
apply fastsimp
apply(rule Semi)
prefer 2
apply(rule Assign)
apply(rule Assign')
apply simp
done


text{* The soundness theorem: *}

theorem hoaret_sound: "\<turnstile>\<^sub>t {P}c{Q}  \<Longrightarrow>  \<Turnstile>\<^sub>t {P}c{Q}"
proof(unfold hoare_tvalid_def, induct rule: hoaret.induct)
  case (While P b f c)
  show ?case
  proof
    fix s
    show "P s \<longrightarrow> (\<exists>t. (WHILE b DO c, s) \<Rightarrow> t \<and> P t \<and> \<not> bval b t)"
    proof(induct "f s" arbitrary: s rule: less_induct)
      case (less n)
      thus ?case by (metis While(2) WhileFalse WhileTrue)
    qed
  qed
next
  case If thus ?case by auto blast
qed fastsimp+


text{*
The completeness proof proceeds along the same lines as the one for partial
correctness. First we have to strengthen our notion of weakest precondition
to take termination into account: *}

definition wpt :: "com \<Rightarrow> assn \<Rightarrow> assn" ("wp\<^sub>t") where
"wp\<^sub>t c Q  \<equiv>  \<lambda>s. \<exists>t. (c,s) \<Rightarrow> t \<and> Q t"

lemma [simp]: "wp\<^sub>t SKIP Q = Q";
by(auto intro!: ext simp: wpt_def)

lemma [simp]: "wp\<^sub>t (x ::= e) Q = (\<lambda>s. Q(s(x := aval e s)))"
by(auto intro!: ext simp: wpt_def)

lemma [simp]: "wp\<^sub>t (c\<^isub>1;c\<^isub>2) Q = wp\<^sub>t c\<^isub>1 (wp\<^sub>t c\<^isub>2 Q)";
unfolding wpt_def
apply(rule ext)
apply auto
done

lemma [simp]:
 "wp\<^sub>t (IF b THEN c\<^isub>1 ELSE c\<^isub>2) Q = (\<lambda>s. wp\<^sub>t (if bval b s then c\<^isub>1 else c\<^isub>2) Q s)";
apply(unfold wpt_def)
apply(rule ext)
apply auto
done


text{* Now we define the number of iterations @{term "WHILE b DO c"} needs to
terminate when started in state @{text s}. Because this is a truly partial
function, we define it as an (inductive) relation first: *}

inductive Its :: "bexp \<Rightarrow> com \<Rightarrow> state \<Rightarrow> nat \<Rightarrow> bool" where
Its_0: "\<not> bval b s \<Longrightarrow> Its b c s 0" |
Its_Suc: "\<lbrakk> bval b s;  (c,s) \<Rightarrow> s';  Its b c s' n \<rbrakk> \<Longrightarrow> Its b c s (Suc n)"

text{* The relation is in fact a function: *}

lemma Its_fun: "Its b c s n \<Longrightarrow> Its b c s n' \<Longrightarrow> n=n'"
proof(induct arbitrary: n' rule:Its.induct)
(* new release:
  case Its_0 thus ?case by(metis Its.cases)
next
  case Its_Suc thus ?case by(metis Its.cases big_step_determ)
qed
*)
  case Its_0
  from this(1) Its.cases[OF this(2)] show ?case by metis
next
  case (Its_Suc b s c s' n n')
  note C = this
  from this(5) show ?case
  proof cases
    case Its_0 with Its_Suc(1) show ?thesis by blast
  next
    case Its_Suc with C show ?thesis by(metis big_step_determ)
  qed
qed

text{* For all terminating loops, @{const Its} yields a result: *}

lemma WHILE_Its: "(WHILE b DO c,s) \<Rightarrow> t \<Longrightarrow> \<exists>n. Its b c s n"
proof(induct "WHILE b DO c" s t rule: big_step_induct)
  case WhileFalse thus ?case by (metis Its_0)
next
  case WhileTrue thus ?case by (metis Its_Suc)
qed

text{* Now the relation is turned into a function with the help of
the description operator @{text THE}: *}

definition its :: "bexp \<Rightarrow> com \<Rightarrow> state \<Rightarrow> nat" where
"its b c s = (THE n. Its b c s n)"

text{* The key property: every loop iteration increases @{const its} by 1. *}

lemma its_Suc: "\<lbrakk> bval b s; (c, s) \<Rightarrow> s'; (WHILE b DO c, s') \<Rightarrow> t\<rbrakk>
       \<Longrightarrow> its b c s = Suc(its b c s')"
by (metis its_def WHILE_Its Its.intros(2) Its_fun the_equality)

lemma wpt_is_pre: "\<turnstile>\<^sub>t {wp\<^sub>t c Q} c {Q}"
proof (induct c arbitrary: Q)
  case SKIP show ?case by simp (blast intro:hoaret.Skip)
next
  case Assign show ?case by simp (blast intro:hoaret.Assign)
next
  case Semi thus ?case by simp (blast intro:hoaret.Semi)
next
  case If thus ?case by simp (blast intro:hoaret.If hoaret.conseq)
next
  case (While b c)
  let ?w = "WHILE b DO c"
  { fix n
    have "\<forall>s. wp\<^sub>t ?w Q s \<and> bval b s \<and> its b c s = n \<longrightarrow>
              wp\<^sub>t c (\<lambda>s'. wp\<^sub>t ?w Q s' \<and> its b c s' < n) s"
      unfolding wpt_def by (metis WhileE its_Suc lessI)
    note strengthen_pre[OF this While]
  } note hoaret.While[OF this]
  moreover have "\<forall>s. wp\<^sub>t ?w Q s \<and> \<not> bval b s \<longrightarrow> Q s" by (auto simp add:wpt_def)
  ultimately show ?case by(rule weaken_post)
qed


text{*\noindent In the @{term While}-case, @{const its} provides the obvious
termination argument.

The actual completeness theorem follows directly, in the same manner
as for partial correctness: *}

theorem hoaret_complete: "\<Turnstile>\<^sub>t {P}c{Q} \<Longrightarrow> \<turnstile>\<^sub>t {P}c{Q}"
apply(rule strengthen_pre[OF _ wpt_is_pre])
apply(auto simp: hoare_tvalid_def hoare_valid_def wpt_def)
done

end