# Theory Live_True

theory Live_True
imports While_Combinator Vars Big_Step
```(* Author: Tobias Nipkow *)

theory Live_True
imports "~~/src/HOL/Library/While_Combinator" Vars Big_Step
begin

subsection "True Liveness Analysis"

fun L :: "com ⇒ vname set ⇒ vname set" where
"L SKIP X = X" |
"L (x ::= a) X = (if x ∈ X then vars a ∪ (X - {x}) else X)" |
"L (c⇩1;; c⇩2) X = L c⇩1 (L c⇩2 X)" |
"L (IF b THEN c⇩1 ELSE c⇩2) X = vars b ∪ L c⇩1 X ∪ L c⇩2 X" |
"L (WHILE b DO c) X = lfp(λY. vars b ∪ X ∪ L c Y)"

lemma L_mono: "mono (L c)"
proof-
{ fix X Y have "X ⊆ Y ⟹ L c X ⊆ 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 ∪ X ∪ L c Z ⊆ vars b ∪ Y ∪ 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 (λY. X ∪ 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 ∪ X ∪ L c (L (WHILE b DO c) X)"
by(metis lfp_unfold[OF mono_union_L] L.simps(5))

lemma L_While_pfp: "L c (L (WHILE b DO c) X) ⊆ L (WHILE b DO c) X"
using L_While_unfold by blast

lemma L_While_vars: "vars b ⊆ L (WHILE b DO c) X"
using L_While_unfold by blast

lemma L_While_X: "X ⊆ L (WHILE b DO c) X"
using L_While_unfold by blast

text{* Disable @{text "L WHILE"} equation and reason only with @{text "L WHILE"} constraints: *}
declare L.simps(5)[simp del]

subsection "Correctness"

theorem L_correct:
"(c,s) ⇒ s'  ⟹ s = t on L c X ⟹
∃ t'. (c,t) ⇒ 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) ⇒ t2" and s2t2: "s2 = t2 on L c2 X"
by simp blast
from Seq.IH(2)[OF s2t2] obtain t3 where
t23: "(c2, t2) ⇒ 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" and "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.IH[OF `s = t on L c1 X`] obtain t' where
"(c1, t) ⇒ 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.IH[OF `s = t on L c2 X`] obtain t' where
"(c2, t) ⇒ 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_vars bval_eq_if_eq_on_vars set_mp)
thus ?case using WhileFalse.prems L_While_X[of X b c] 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_vars bval_eq_if_eq_on_vars set_mp)
have "s1 = t1 on L c (L ?w X)" using  L_While_pfp WhileTrue.prems
by (blast)
from WhileTrue.IH(1)[OF this] obtain t2 where
"(c, t1) ⇒ t2" "s2 = t2 on L ?w X" by auto
from WhileTrue.IH(2)[OF this(2)] obtain t3 where "(?w,t2) ⇒ t3" "s3 = t3 on X"
by auto
with `bval b t1` `(c, t1) ⇒ t2` show ?case by auto
qed

subsection "Executability"

lemma L_subset_vars: "L c X ⊆ rvars c ∪ X"
proof(induction c arbitrary: X)
case (While b c)
have "lfp(λY. vars b ∪ X ∪ L c Y) ⊆ vars b ∪ rvars c ∪ X"
using While.IH[of "vars b ∪ rvars c ∪ X"]
by (auto intro!: lfp_lowerbound)
thus ?case by (simp add: L.simps(5))
qed auto

text{* Make @{const L} executable by replacing @{const lfp} with the @{const
while} combinator from theory @{theory While_Combinator}. The @{const while}
combinator obeys the recursion equation
@{thm[display] While_Combinator.while_unfold[no_vars]}
and is thus executable. *}

lemma L_While: fixes b c X
assumes "finite X" defines "f == λY. vars b ∪ X ∪ L c Y"
shows "L (WHILE b DO c) X = while (λY. f Y ≠ Y) f {}" (is "_ = ?r")
proof -
let ?V = "vars b ∪ rvars c ∪ X"
have "lfp f = ?r"
proof(rule lfp_while[where C = "?V"])
show "mono f" by(simp add: f_def mono_union_L)
next
fix Y show "Y ⊆ ?V ⟹ f Y ⊆ ?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 L.simps(5))
qed

lemma L_While_let: "finite X ⟹ L (WHILE b DO c) X =
(let f = (λY. vars b ∪ X ∪ L c Y)
in while (λY. f Y ≠ Y) f {})"

lemma L_While_set: "L (WHILE b DO c) (set xs) =
(let f = (λY. vars b ∪ set xs ∪ L c Y)
in while (λY. f Y ≠ Y) f {})"
by(rule L_While_let, simp)

text{* Replace the equation for @{text "L (WHILE …)"} by the executable @{thm[source] L_While_set}: *}
lemmas [code] = L.simps(1-4) L_While_set
text{* Sorry, this syntax is odd. *}

text{* A test: *}
lemma "(let b = Less (N 0) (V ''y''); c = ''y'' ::= V ''x'';; ''x'' ::= V ''z''
in L (WHILE b DO c) {''y''}) = {''x'', ''y'', ''z''}"
by eval

subsection "Limiting the number of iterations"

text{* The final parameter is the default value: *}

fun iter :: "('a ⇒ 'a) ⇒ nat ⇒ 'a ⇒ 'a ⇒ '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)"

text{* A version of @{const L} with a bounded number of iterations (here: 2)
in the WHILE case: *}

fun Lb :: "com ⇒ vname set ⇒ vname set" where
"Lb SKIP X = X" |
"Lb (x ::= a) X = (if x ∈ X then X - {x} ∪ vars a else X)" |
"Lb (c⇩1;; c⇩2) X = (Lb c⇩1 ∘ Lb c⇩2) X" |
"Lb (IF b THEN c⇩1 ELSE c⇩2) X = vars b ∪ Lb c⇩1 X ∪ Lb c⇩2 X" |
"Lb (WHILE b DO c) X = iter (λA. vars b ∪ X ∪ Lb c A) 2 {} (vars b ∪ rvars c ∪ X)"

text{* @{const Lb} (and @{const iter}) is not monotone! *}
lemma "let w = WHILE Bc False DO (''x'' ::= V ''y'';; ''z'' ::= V ''x'')
in ¬ (Lb w {''z''} ⊆ Lb w {''y'',''z''})"
by eval

lemma lfp_subset_iter:
"⟦ mono f; !!X. f X ⊆ f' X; lfp f ⊆ D ⟧ ⟹ lfp f ⊆ iter f' n A D"
proof(induction n arbitrary: A)
case 0 thus ?case by simp
next
case Suc thus ?case by simp (metis lfp_lowerbound)
qed

lemma "L c X ⊆ Lb c X"
proof(induction c arbitrary: X)
case (While b c)
let ?f  = "λA. vars b ∪ X ∪ L  c A"
let ?fb = "λA. vars b ∪ X ∪ Lb c A"
show ?case
proof (simp add: L.simps(5), rule lfp_subset_iter[OF mono_union_L])
show "!!X. ?f X ⊆ ?fb X" using While.IH by blast
show "lfp ?f ⊆ vars b ∪ rvars c ∪ X"
by (metis (full_types) L.simps(5) L_subset_vars rvars.simps(5))
qed
next
case Seq thus ?case by simp (metis (full_types) L_mono monoD subset_trans)
qed auto

end
```