# Theory Kildall

theory Kildall
imports SemilatAlg
(*  Title:      HOL/MicroJava/DFA/Kildall.thy
Author:     Tobias Nipkow, Gerwin Klein
*)

section ‹Kildall's Algorithm \label{sec:Kildall}›

theory Kildall
imports SemilatAlg "HOL-Library.While_Combinator"
begin

primrec propa :: "'s binop ⇒ (nat × 's) list ⇒ 's list ⇒ nat set ⇒ 's list * nat set" where
"propa f []      ss w = (ss,w)"
| "propa f (q'#qs) ss w = (let (q,t) = q';
u = t +_f ss!q;
w' = (if u = ss!q then w else insert q w)
in propa f qs (ss[q := u]) w')"

definition iter :: "'s binop ⇒ 's step_type ⇒ 's list ⇒ nat set ⇒ 's list × nat set" where
"iter f step ss w == while (%(ss,w). w ≠ {})
(%(ss,w). let p = SOME p. p ∈ w
in propa f (step p (ss!p)) ss (w-{p}))
(ss,w)"

definition unstables :: "'s ord ⇒ 's step_type ⇒ 's list ⇒ nat set" where
"unstables r step ss == {p. p < size ss ∧ ¬stable r step ss p}"

definition kildall :: "'s ord ⇒ 's binop ⇒ 's step_type ⇒ 's list ⇒ 's list" where
"kildall r f step ss == fst(iter f step ss (unstables r step ss))"

primrec merges :: "'s binop ⇒ (nat × 's) list ⇒ 's list ⇒ 's list" where
"merges f []      ss = ss"
| "merges f (p'#ps) ss = (let (p,s) = p' in merges f ps (ss[p := s +_f ss!p]))"

lemmas [simp] = Let_def Semilat.le_iff_plus_unchanged [OF Semilat.intro, symmetric]

lemma (in Semilat) nth_merges:
"⋀ss. ⟦p < length ss; ss ∈ list n A; ∀(p,t)∈set ps. p<n ∧ t∈A ⟧ ⟹
(merges f ps ss)!p = map snd [(p',t') ← ps. p'=p] ++_f ss!p"
(is "⋀ss. ⟦_; _; ?steptype ps⟧ ⟹ ?P ss ps")
proof (induct ps)
show "⋀ss. ?P ss []" by simp

fix ss p' ps'
assume ss: "ss ∈ list n A"
assume l:  "p < length ss"
assume "?steptype (p'#ps')"
then obtain a b where
p': "p'=(a,b)" and ab: "a<n" "b∈A" and ps': "?steptype ps'"
by (cases p') auto
assume "⋀ss. p< length ss ⟹ ss ∈ list n A ⟹ ?steptype ps' ⟹ ?P ss ps'"
from this [OF _ _ ps'] have IH: "⋀ss. ss ∈ list n A ⟹ p < length ss ⟹ ?P ss ps'" .

from ss ab
have "ss[a := b +_f ss!a] ∈ list n A" by (simp add: closedD)
moreover
from calculation l
have "p < length (ss[a := b +_f ss!a])" by simp
ultimately
have "?P (ss[a := b +_f ss!a]) ps'" by (rule IH)
with p' l
show "?P ss (p'#ps')" by simp
qed

(** merges **)

lemma length_merges [simp]: "size(merges f ps ss) = size ss"
by (induct ps arbitrary: ss) auto

lemma (in Semilat) merges_preserves_type_lemma:
shows "∀xs. xs ∈ list n A ⟶ (∀(p,x) ∈ set ps. p<n ∧ x∈A)
⟶ merges f ps xs ∈ list n A"
apply (insert closedI)
apply (unfold closed_def)
apply (induct_tac ps)
apply simp
apply clarsimp
done

lemma (in Semilat) merges_preserves_type [simp]:
"⟦ xs ∈ list n A; ∀(p,x) ∈ set ps. p<n ∧ x∈A ⟧
⟹ merges f ps xs ∈ list n A"

lemma (in Semilat) merges_incr_lemma:
"∀xs. xs ∈ list n A ⟶ (∀(p,x)∈set ps. p<size xs ∧ x ∈ A) ⟶ xs <=[r] merges f ps xs"
apply (induct_tac ps)
apply simp
apply simp
apply clarify
apply (rule order_trans)
apply simp
apply (erule list_update_incr)
apply simp
apply simp
apply (blast intro!: listE_set intro: closedD listE_length [THEN nth_in])
done

lemma (in Semilat) merges_incr:
"⟦ xs ∈ list n A; ∀(p,x)∈set ps. p<size xs ∧ x ∈ A ⟧
⟹ xs <=[r] merges f ps xs"

lemma (in Semilat) merges_same_conv [rule_format]:
"(∀xs. xs ∈ list n A ⟶ (∀(p,x)∈set ps. p<size xs ∧ x∈A) ⟶
(merges f ps xs = xs) = (∀(p,x)∈set ps. x <=_r xs!p))"
apply (induct_tac ps)
apply simp
apply clarsimp
apply (rename_tac p x ps xs)
apply (rule iffI)
apply (rule context_conjI)
apply (subgoal_tac "xs[p := x +_f xs!p] <=[r] xs")
apply (drule_tac p = p in le_listD)
apply simp
apply simp
apply (erule subst, rule merges_incr)
apply (blast intro!: listE_set intro: closedD listE_length [THEN nth_in])
apply clarify
apply (rule conjI)
apply simp
apply (blast dest: boundedD)
apply blast
apply clarify
apply (erule allE)
apply (erule impE)
apply assumption
apply (drule bspec)
apply assumption
apply (simp add: le_iff_plus_unchanged [THEN iffD1] list_update_same_conv [THEN iffD2])
apply blast
apply clarify
apply (simp add: le_iff_plus_unchanged [THEN iffD1] list_update_same_conv [THEN iffD2])
done

lemma (in Semilat) list_update_le_listI [rule_format]:
"set xs <= A ⟶ set ys <= A ⟶ xs <=[r] ys ⟶ p < size xs ⟶
x <=_r ys!p ⟶ x∈A ⟶ xs[p := x +_f xs!p] <=[r] ys"
apply(insert semilat)
apply (unfold Listn.le_def lesub_def semilat_def)
done

lemma (in Semilat) merges_pres_le_ub:
assumes "set ts <= A" and "set ss <= A"
and "∀(p,t)∈set ps. t <=_r ts!p ∧ t ∈ A ∧ p < size ts" and "ss <=[r] ts"
shows "merges f ps ss <=[r] ts"
proof -
{ fix t ts ps
have
"⋀qs. ⟦set ts <= A; ∀(p,t)∈set ps. t <=_r ts!p ∧ t ∈ A ∧ p< size ts ⟧ ⟹
set qs <= set ps  ⟶
(∀ss. set ss <= A ⟶ ss <=[r] ts ⟶ merges f qs ss <=[r] ts)"
apply (induct_tac qs)
apply simp
apply (simp (no_asm_simp))
apply clarify
apply (rotate_tac -2)
apply simp
apply (erule allE, erule impE, erule_tac [2] mp)
apply (drule bspec, assumption)
apply (drule bspec, assumption)
done
} note this [dest]

from assms show ?thesis by blast
qed

(** propa **)

lemma decomp_propa:
"⋀ss w. (∀(q,t)∈set qs. q < size ss) ⟹
propa f qs ss w =
(merges f qs ss, {q. ∃t. (q,t)∈set qs ∧ t +_f ss!q ≠ ss!q} Un w)"
apply (induct qs)
apply simp
apply (simp (no_asm))
apply clarify
apply simp
apply (rule conjI)
apply blast
apply blast
done

(** iter **)

lemma (in Semilat) stable_pres_lemma:
shows "⟦pres_type step n A; bounded step n;
ss ∈ list n A; p ∈ w; ∀q∈w. q < n;
∀q. q < n ⟶ q ∉ w ⟶ stable r step ss q; q < n;
∀s'. (q,s') ∈ set (step p (ss ! p)) ⟶ s' +_f ss ! q = ss ! q;
q ∉ w ∨ q = p ⟧
⟹ stable r step (merges f (step p (ss!p)) ss) q"
apply (unfold stable_def)
apply (subgoal_tac "∀s'. (q,s') ∈ set (step p (ss!p)) ⟶ s' : A")
prefer 2
apply clarify
apply (erule pres_typeD)
prefer 3 apply assumption
apply (rule listE_nth_in)
apply assumption
apply simp
apply simp
apply simp
apply clarify
apply (subst nth_merges)
apply simp
apply (blast dest: boundedD)
apply assumption
apply clarify
apply (rule conjI)
apply (blast dest: boundedD)
apply (erule pres_typeD)
prefer 3 apply assumption
apply simp
apply simp
apply(subgoal_tac "q < length ss")
prefer 2 apply simp
apply (frule nth_merges [of q _ _ "step p (ss!p)"]) (* fixme: why does method subst not work?? *)
apply assumption
apply clarify
apply (rule conjI)
apply (blast dest: boundedD)
apply (erule pres_typeD)
prefer 3 apply assumption
apply simp
apply simp
apply (drule_tac P = "λx. (a, b) ∈ set (step q x)" in subst)
apply assumption

apply (cases "q ∈ w")
apply simp
apply (rule ub1')
apply (rule semilat)
apply clarify
apply (rule pres_typeD)
apply assumption
prefer 3 apply assumption
apply (blast intro: listE_nth_in dest: boundedD)
apply (blast intro: pres_typeD dest: boundedD)
apply (blast intro: listE_nth_in dest: boundedD)
apply assumption

apply simp
apply (erule allE, erule impE, assumption, erule impE, assumption)
apply (rule order_trans)
apply simp
defer
apply (rule pp_ub2)(*
apply assumption*)
apply simp
apply clarify
apply simp
apply (rule pres_typeD)
apply assumption
prefer 3 apply assumption
apply (blast intro: listE_nth_in dest: boundedD)
apply (blast intro: pres_typeD dest: boundedD)
apply (blast intro: listE_nth_in dest: boundedD)
apply blast
done

lemma (in Semilat) merges_bounded_lemma:
"⟦ mono r step n A; bounded step n;
∀(p',s') ∈ set (step p (ss!p)). s' ∈ A; ss ∈ list n A; ts ∈ list n A; p < n;
ss <=[r] ts; ∀p. p < n ⟶ stable r step ts p ⟧
⟹ merges f (step p (ss!p)) ss <=[r] ts"
apply (unfold stable_def)
apply (rule merges_pres_le_ub)
apply simp
apply simp
prefer 2 apply assumption

apply clarsimp
apply (drule boundedD, assumption+)
apply (erule allE, erule impE, assumption)
apply (drule bspec, assumption)
apply simp

apply (drule monoD [of _ _ _ _ p "ss!p"  "ts!p"])
apply assumption
apply simp

apply (drule lesub_step_typeD, assumption)
apply clarify
apply (drule bspec, assumption)
apply simp
apply (blast intro: order_trans)
done

lemma termination_lemma:
assumes semilat: "semilat (A, r, f)"
shows "⟦ ss ∈ list n A; ∀(q,t)∈set qs. q<n ∧ t∈A; p∈w ⟧ ⟹
ss <[r] merges f qs ss ∨
merges f qs ss = ss ∧ {q. ∃t. (q,t)∈set qs ∧ t +_f ss!q ≠ ss!q} Un (w-{p}) < w" (is "PROP ?P")
proof -
interpret Semilat A r f using assms by (rule Semilat.intro)
show "PROP ?P" apply(insert semilat)
apply (unfold lesssub_def)
apply (rule impI)
apply (rule merges_same_conv [THEN iffD1, elim_format])
apply assumption+
defer
apply (rule sym, assumption)
defer apply simp
apply (subgoal_tac "∀q t. ¬((q, t) ∈ set qs ∧ t +_f ss ! q ≠ ss ! q)")
apply (blast elim: equalityE)
apply clarsimp
apply (drule bspec, assumption)
apply (drule bspec, assumption)
apply clarsimp
done
qed

lemma iter_properties[rule_format]:
assumes semilat: "semilat (A, r, f)"
shows "⟦ acc r ; pres_type step n A; mono r step n A;
bounded step n; ∀p∈w0. p < n; ss0 ∈ list n A;
∀p<n. p ∉ w0 ⟶ stable r step ss0 p ⟧ ⟹
iter f step ss0 w0 = (ss',w')
⟶
ss' ∈ list n A ∧ stables r step ss' ∧ ss0 <=[r] ss' ∧
(∀ts∈list n A. ss0 <=[r] ts ∧ stables r step ts ⟶ ss' <=[r] ts)"
(is "PROP ?P")
proof -
interpret Semilat A r f using assms by (rule Semilat.intro)
show "PROP ?P" apply(insert semilat)
apply (unfold iter_def stables_def)
apply (rule_tac P = "%(ss,w).
ss ∈ list n A ∧ (∀p<n. p ∉ w ⟶ stable r step ss p) ∧ ss0 <=[r] ss ∧
(∀ts∈list n A. ss0 <=[r] ts ∧ stables r step ts ⟶ ss <=[r] ts) ∧
(∀p∈w. p < n)" and
r = "{(ss',ss) . ss <[r] ss'} <*lex*> finite_psubset"
in while_rule)

― "Invariant holds initially:"

― "Invariant is preserved:"
apply(rename_tac ss w)
apply(subgoal_tac "(SOME p. p ∈ w) ∈ w")
prefer 2 apply (fast intro: someI)
apply(subgoal_tac "∀(q,t) ∈ set (step (SOME p. p ∈ w) (ss ! (SOME p. p ∈ w))). q < length ss ∧ t ∈ A")
prefer 2
apply clarify
apply (rule conjI)
apply(clarsimp, blast dest!: boundedD)
apply (erule pres_typeD)
prefer 3
apply assumption
apply (erule listE_nth_in)
apply simp
apply simp
apply (subst decomp_propa)
apply fast
apply simp
apply (rule conjI)
apply (rule merges_preserves_type)
apply blast
apply clarify
apply (rule conjI)
apply(clarsimp, fast dest!: boundedD)
apply (erule pres_typeD)
prefer 3
apply assumption
apply (erule listE_nth_in)
apply blast
apply blast
apply (rule conjI)
apply clarify
apply (blast intro!: stable_pres_lemma)
apply (rule conjI)
apply (blast intro!: merges_incr intro: le_list_trans)
apply (rule conjI)
apply clarsimp
apply (blast intro!: merges_bounded_lemma)
apply (blast dest!: boundedD)

― "Postcondition holds upon termination:"

― "Well-foundedness of the termination relation:"
apply (rule wf_lex_prod)
apply (insert orderI [THEN acc_le_listI])
apply (simp add: acc_def lesssub_def wfP_wf_eq [symmetric])
apply (rule wf_finite_psubset)

― "Loop decreases along termination relation:"
apply(rename_tac ss w)
apply(subgoal_tac "(SOME p. p ∈ w) ∈ w")
prefer 2 apply (fast intro: someI)
apply(subgoal_tac "∀(q,t) ∈ set (step (SOME p. p ∈ w) (ss ! (SOME p. p ∈ w))). q < length ss ∧ t ∈ A")
prefer 2
apply clarify
apply (rule conjI)
apply(clarsimp, blast dest!: boundedD)
apply (erule pres_typeD)
prefer 3
apply assumption
apply (erule listE_nth_in)
apply blast
apply blast
apply (subst decomp_propa)
apply blast
apply clarify
apply (simp del: listE_length
bounded_nat_set_is_finite)
apply (rule termination_lemma)
apply assumption+
defer
apply assumption
apply clarsimp
done

qed

lemma kildall_properties:
assumes semilat: "semilat (A, r, f)"
shows "⟦ acc r; pres_type step n A; mono r step n A;
bounded step n; ss0 ∈ list n A ⟧ ⟹
kildall r f step ss0 ∈ list n A ∧
stables r step (kildall r f step ss0) ∧
ss0 <=[r] kildall r f step ss0 ∧
(∀ts∈list n A. ss0 <=[r] ts ∧ stables r step ts ⟶
kildall r f step ss0 <=[r] ts)"
(is "PROP ?P")
proof -
interpret Semilat A r f using assms by (rule Semilat.intro)
show "PROP ?P"
apply (unfold kildall_def)
apply(case_tac "iter f step ss0 (unstables r step ss0)")
apply(simp)
apply (rule iter_properties)
apply (rule semilat)
done
qed

lemma is_bcv_kildall:
assumes semilat: "semilat (A, r, f)"
shows "⟦ acc r; top r T; pres_type step n A; bounded step n; mono r step n A ⟧
⟹ is_bcv r T step n A (kildall r f step)"
(is "PROP ?P")
proof -
interpret Semilat A r f using assms by (rule Semilat.intro)
show "PROP ?P"
apply(unfold is_bcv_def wt_step_def)
apply(insert semilat kildall_properties[of A])
apply clarify
apply(subgoal_tac "kildall r f step ss ∈ list n A")
prefer 2 apply (simp(no_asm_simp))
apply (rule iffI)
apply (rule_tac x = "kildall r f step ss" in bexI)
apply (rule conjI)
apply (blast)
apply (simp  (no_asm_simp))
apply(assumption)
apply clarify
apply(subgoal_tac "kildall r f step ss!p <=_r ts!p")
apply simp
apply (blast intro!: le_listD less_lengthI)
done
qed

end