(* Author: Tobias Nipkow *)
theory Live_True
imports "~~/src/HOL/Library/While_Combinator" Vars Big_Step
begin
subsection "True Liveness Analysis"
fun L :: "com \<Rightarrow> vname set \<Rightarrow> vname set" where
"L SKIP X = X" |
"L (x ::= a) X = (if x:X then X-{x} \<union> vars a else X)" |
"L (c\<^isub>1; c\<^isub>2) X = (L c\<^isub>1 \<circ> L c\<^isub>2) X" |
"L (IF b THEN c\<^isub>1 ELSE c\<^isub>2) X = vars b \<union> L c\<^isub>1 X \<union> L c\<^isub>2 X" |
"L (WHILE b DO c) X = lfp(%Y. vars b \<union> X \<union> L c Y)"
lemma L_mono: "mono (L c)"
proof-
{ fix X Y have "X \<subseteq> Y \<Longrightarrow> L c X \<subseteq> L c Y"
proof(induction c arbitrary: X Y)
case (While b c)
show ?case
proof(simp, rule lfp_mono)
fix Z show "vars b \<union> X \<union> L c Z \<subseteq> vars b \<union> Y \<union> L c Z"
using While by auto
qed
next
case If thus ?case by(auto simp: subset_iff)
qed auto
} thus ?thesis by(rule monoI)
qed
lemma mono_union_L:
"mono (%Y. X \<union> L c Y)"
by (metis (no_types) L_mono mono_def order_eq_iff set_eq_subset sup_mono)
lemma L_While_unfold:
"L (WHILE b DO c) X = vars b \<union> X \<union> L c (L (WHILE b DO c) X)"
by(metis lfp_unfold[OF mono_union_L] L.simps(5))
subsection "Soundness"
theorem L_sound:
"(c,s) \<Rightarrow> s' \<Longrightarrow> s = t on L c X \<Longrightarrow>
\<exists> t'. (c,t) \<Rightarrow> t' & s' = t' on X"
proof (induction arbitrary: X t rule: big_step_induct)
case Skip then show ?case by auto
next
case Assign then show ?case
by (auto simp: ball_Un)
next
case (Seq c1 s1 s2 c2 s3 X t1)
from Seq.IH(1) Seq.prems obtain t2 where
t12: "(c1, t1) \<Rightarrow> t2" and s2t2: "s2 = t2 on L c2 X"
by simp blast
from Seq.IH(2)[OF s2t2] obtain t3 where
t23: "(c2, t2) \<Rightarrow> t3" and s3t3: "s3 = t3 on X"
by auto
show ?case using t12 t23 s3t3 by auto
next
case (IfTrue b s c1 s' c2)
hence "s = t on vars b" "s = t on L c1 X" by auto
from bval_eq_if_eq_on_vars[OF this(1)] IfTrue(1) have "bval b t" by simp
from IfTrue(3)[OF `s = t on L c1 X`] obtain t' where
"(c1, t) \<Rightarrow> t'" "s' = t' on X" by auto
thus ?case using `bval b t` by auto
next
case (IfFalse b s c2 s' c1)
hence "s = t on vars b" "s = t on L c2 X" by auto
from bval_eq_if_eq_on_vars[OF this(1)] IfFalse(1) have "~bval b t" by simp
from IfFalse(3)[OF `s = t on L c2 X`] obtain t' where
"(c2, t) \<Rightarrow> t'" "s' = t' on X" by auto
thus ?case using `~bval b t` by auto
next
case (WhileFalse b s c)
hence "~ bval b t"
by (metis L_While_unfold UnI1 bval_eq_if_eq_on_vars)
thus ?case using WhileFalse.prems L_While_unfold[of b c X] by auto
next
case (WhileTrue b s1 c s2 s3 X t1)
let ?w = "WHILE b DO c"
from `bval b s1` WhileTrue.prems have "bval b t1"
by (metis L_While_unfold UnI1 bval_eq_if_eq_on_vars)
have "s1 = t1 on L c (L ?w X)" using L_While_unfold WhileTrue.prems
by (blast)
from WhileTrue.IH(1)[OF this] obtain t2 where
"(c, t1) \<Rightarrow> t2" "s2 = t2 on L ?w X" by auto
from WhileTrue.IH(2)[OF this(2)] obtain t3 where "(?w,t2) \<Rightarrow> t3" "s3 = t3 on X"
by auto
with `bval b t1` `(c, t1) \<Rightarrow> t2` show ?case by auto
qed
instantiation com :: vars
begin
fun vars_com :: "com \<Rightarrow> vname set" where
"vars SKIP = {}" |
"vars (x::=e) = vars e" |
"vars (c\<^isub>1; c\<^isub>2) = vars c\<^isub>1 \<union> vars c\<^isub>2" |
"vars (IF b THEN c\<^isub>1 ELSE c\<^isub>2) = vars b \<union> vars c\<^isub>1 \<union> vars c\<^isub>2" |
"vars (WHILE b DO c) = vars b \<union> vars c"
instance ..
end
lemma L_subset_vars: "L c X \<subseteq> vars c \<union> X"
proof(induction c arbitrary: X)
case (While b c)
have "lfp(%Y. vars b \<union> X \<union> L c Y) \<subseteq> vars b \<union> vars c \<union> X"
using While.IH[of "vars b \<union> vars c \<union> X"]
by (auto intro!: lfp_lowerbound)
thus ?case by simp
qed auto
lemma afinite[simp]: "finite(vars(a::aexp))"
by (induction a) auto
lemma bfinite[simp]: "finite(vars(b::bexp))"
by (induction b) auto
lemma cfinite[simp]: "finite(vars(c::com))"
by (induction c) auto
(* move to Inductive; call Kleene? *)
lemma lfp_finite_iter: assumes "mono f" and "(f^^Suc k) bot = (f^^k) bot"
shows "lfp f = (f^^k) bot"
proof(rule antisym)
show "lfp f \<le> (f^^k) bot"
proof(rule lfp_lowerbound)
show "f ((f^^k) bot) \<le> (f^^k) bot" using assms(2) by simp
qed
next
show "(f^^k) bot \<le> lfp f"
proof(induction k)
case 0 show ?case by simp
next
case Suc
from monoD[OF assms(1) Suc] lfp_unfold[OF assms(1)]
show ?case by simp
qed
qed
(* move to While_Combinator *)
lemma while_option_stop2:
"while_option b c s = Some t \<Longrightarrow> EX k. t = (c^^k) s \<and> \<not> b t"
apply(simp add: while_option_def split: if_splits)
by (metis (lifting) LeastI_ex)
(* move to While_Combinator *)
lemma while_option_finite_subset_Some: fixes C :: "'a set"
assumes "mono f" and "!!X. X \<subseteq> C \<Longrightarrow> f X \<subseteq> C" and "finite C"
shows "\<exists>P. while_option (\<lambda>A. f A \<noteq> A) f {} = Some P"
proof(rule measure_while_option_Some[where
f= "%A::'a set. card C - card A" and P= "%A. A \<subseteq> C \<and> A \<subseteq> f A" and s= "{}"])
fix A assume A: "A \<subseteq> C \<and> A \<subseteq> f A" "f A \<noteq> A"
show "(f A \<subseteq> C \<and> f A \<subseteq> f (f A)) \<and> card C - card (f A) < card C - card A"
(is "?L \<and> ?R")
proof
show ?L by(metis A(1) assms(2) monoD[OF `mono f`])
show ?R by (metis A assms(2,3) card_seteq diff_less_mono2 equalityI linorder_le_less_linear rev_finite_subset)
qed
qed simp
(* move to While_Combinator *)
lemma lfp_eq_while_option:
assumes "mono f" and "!!X. X \<subseteq> C \<Longrightarrow> f X \<subseteq> C" and "finite C"
shows "lfp f = the(while_option (\<lambda>A. f A \<noteq> A) f {})"
proof-
obtain P where "while_option (\<lambda>A. f A \<noteq> A) f {} = Some P"
using while_option_finite_subset_Some[OF assms] by blast
with while_option_stop2[OF this] lfp_finite_iter[OF assms(1)]
show ?thesis by auto
qed
text{* For code generation: *}
lemma L_While: fixes b c X
assumes "finite X" defines "f == \<lambda>A. vars b \<union> X \<union> L c A"
shows "L (WHILE b DO c) X = the(while_option (\<lambda>A. f A \<noteq> A) f {})" (is "_ = ?r")
proof -
let ?V = "vars b \<union> vars c \<union> X"
have "lfp f = ?r"
proof(rule lfp_eq_while_option[where C = "?V"])
show "mono f" by(simp add: f_def mono_union_L)
next
fix Y show "Y \<subseteq> ?V \<Longrightarrow> f Y \<subseteq> ?V"
unfolding f_def using L_subset_vars[of c] by blast
next
show "finite ?V" using `finite X` by simp
qed
thus ?thesis by (simp add: f_def)
qed
text{* An approximate computation of the WHILE-case: *}
fun iter :: "('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a"
where
"iter f 0 p d = d" |
"iter f (Suc n) p d = (if f p = p then p else iter f n (f p) d)"
lemma iter_pfp:
"f d \<le> d \<Longrightarrow> mono f \<Longrightarrow> x \<le> f x \<Longrightarrow> f(iter f i x d) \<le> iter f i x d"
apply(induction i arbitrary: x)
apply simp
apply (simp add: mono_def)
done
lemma iter_While_pfp:
fixes b c X W k f
defines "f == \<lambda>A. vars b \<union> X \<union> L c A" and "W == vars b \<union> vars c \<union> X"
and "P == iter f k {} W"
shows "f P \<subseteq> P"
proof-
have "f W \<subseteq> W" unfolding f_def W_def using L_subset_vars[of c] by blast
have "mono f" by(simp add: f_def mono_union_L)
from iter_pfp[of f, OF `f W \<subseteq> W` `mono f` empty_subsetI]
show ?thesis by(simp add: P_def)
qed
end