# Theory LBVComplete

```(*  Title:      HOL/MicroJava/DFA/LBVComplete.thy
Author:     Gerwin Klein
*)

section ‹Completeness of the LBV›

theory LBVComplete
imports LBVSpec Typing_Framework
begin

definition is_target :: "['s step_type, 's list, nat] ⇒ bool" where
"is_target step phi pc' ⟷
(∃pc s'. pc' ≠ pc+1 ∧ pc < length phi ∧ (pc',s') ∈ set (step pc (phi!pc)))"

definition make_cert :: "['s step_type, 's list, 's] ⇒ 's certificate" where
"make_cert step phi B =
map (λpc. if is_target step phi pc then phi!pc else B) [0..<length phi] @ [B]"

lemma [code]:
"is_target step phi pc' =
list_ex (λpc. pc' ≠ pc+1 ∧ List.member (map fst (step pc (phi!pc))) pc') [0..<length phi]"
by (force simp: list_ex_iff member_def is_target_def)

locale lbvc = lbv +
fixes phi :: "'a list" ("φ")
fixes c   :: "'a list"
defines cert_def: "c ≡ make_cert step φ ⊥"

assumes mono: "mono r step (length φ) A"
assumes pres: "pres_type step (length φ) A"
assumes phi:  "∀pc < length φ. φ!pc ∈ A ∧ φ!pc ≠ ⊤"
assumes bounded: "bounded step (length φ)"

assumes B_neq_T: "⊥ ≠ ⊤"

lemma (in lbvc) cert: "cert_ok c (length φ) ⊤ ⊥ A"
proof (unfold cert_ok_def, intro strip conjI)
note [simp] = make_cert_def cert_def nth_append

show "c!length φ = ⊥" by simp

fix pc assume pc: "pc < length φ"
from pc phi B_A show "c!pc ∈ A" by simp
from pc phi B_neq_T show "c!pc ≠ ⊤" by simp
qed

lemmas [simp del] = split_paired_Ex

lemma (in lbvc) cert_target [intro?]:
"⟦ (pc',s') ∈ set (step pc (φ!pc));
pc' ≠ pc+1; pc < length φ; pc' < length φ ⟧
⟹ c!pc' = φ!pc'"
by (auto simp add: cert_def make_cert_def nth_append is_target_def)

lemma (in lbvc) cert_approx [intro?]:
"⟦ pc < length φ; c!pc ≠ ⊥ ⟧
⟹ c!pc = φ!pc"
by (auto simp add: cert_def make_cert_def nth_append)

lemma (in lbv) le_top [simp, intro]:
"x <=_r ⊤"
using top by simp

lemma (in lbv) merge_mono:
assumes less:  "ss2 ≤|r| ss1"
assumes x:     "x ∈ A"
assumes ss1:   "snd`set ss1 ⊆ A"
assumes ss2:   "snd`set ss2 ⊆ A"
shows "merge c pc ss2 x <=_r merge c pc ss1 x" (is "?s2 <=_r ?s1")
proof-
have "?s1 = ⊤ ⟹ ?thesis" by simp
moreover {
assume merge: "?s1 ≠ T"
from x ss1 have "?s1 =
(if ∀(pc', s')∈set ss1. pc' ≠ pc + 1 ⟶ s' <=_r c!pc'
then (map snd [(p', t') ← ss1 . p'=pc+1]) ++_f x
else ⊤)"
by (rule merge_def)
with merge obtain
app: "∀(pc',s')∈set ss1. pc' ≠ pc+1 ⟶ s' <=_r c!pc'"
(is "?app ss1") and
sum: "(map snd [(p',t') ← ss1 . p' = pc+1] ++_f x) = ?s1"
(is "?map ss1 ++_f x = _" is "?sum ss1 = _")
by (simp split: if_split_asm)
from app less
have "?app ss2" by (blast dest: trans_r lesub_step_typeD)
moreover {
from ss1 have map1: "set (?map ss1) ⊆ A" by auto
with x have "?sum ss1 ∈ A" by (auto intro!: plusplus_closed semilat)
with sum have "?s1 ∈ A" by simp
moreover
have mapD: "⋀x ss. x ∈ set (?map ss) ⟹ ∃p. (p,x) ∈ set ss ∧ p=pc+1" by auto
from x map1
have "∀x ∈ set (?map ss1). x <=_r ?sum ss1"
by clarify (rule pp_ub1)
with sum have "∀x ∈ set (?map ss1). x <=_r ?s1" by simp
with less have "∀x ∈ set (?map ss2). x <=_r ?s1"
by (fastforce dest!: mapD lesub_step_typeD intro: trans_r)
moreover
from map1 x have "x <=_r (?sum ss1)" by (rule pp_ub2)
with sum have "x <=_r ?s1" by simp
moreover
from ss2 have "set (?map ss2) ⊆ A" by auto
ultimately
have "?sum ss2 <=_r ?s1" using x by - (rule pp_lub)
}
moreover
from x ss2 have
"?s2 =
(if ∀(pc', s')∈set ss2. pc' ≠ pc + 1 ⟶ s' <=_r c!pc'
then map snd [(p', t') ← ss2 . p' = pc + 1] ++_f x
else ⊤)"
by (rule merge_def)
ultimately have ?thesis by simp
}
ultimately show ?thesis by (cases "?s1 = ⊤") auto
qed

lemma (in lbvc) wti_mono:
assumes less: "s2 <=_r s1"
assumes pc:   "pc < length φ"
assumes s1:   "s1 ∈ A"
assumes s2:   "s2 ∈ A"
shows "wti c pc s2 <=_r wti c pc s1" (is "?s2' <=_r ?s1'")
proof -
from mono pc s2 less have "step pc s2 ≤|r| step pc s1" by (rule monoD)
moreover
from cert B_A pc have "c!Suc pc ∈ A" by (rule cert_okD3)
moreover
from pres s1 pc
have "snd`set (step pc s1) ⊆ A" by (rule pres_typeD2)
moreover
from pres s2 pc
have "snd`set (step pc s2) ⊆ A" by (rule pres_typeD2)
ultimately
show ?thesis by (simp add: wti merge_mono)
qed

lemma (in lbvc) wtc_mono:
assumes less: "s2 <=_r s1"
assumes pc:   "pc < length φ"
assumes s1:   "s1 ∈ A"
assumes s2:   "s2 ∈ A"
shows "wtc c pc s2 <=_r wtc c pc s1" (is "?s2' <=_r ?s1'")
proof (cases "c!pc = ⊥")
case True
moreover from less pc s1 s2 have "wti c pc s2 <=_r wti c pc s1" by (rule wti_mono)
ultimately show ?thesis by (simp add: wtc)
next
case False
have "?s1' = ⊤ ⟹ ?thesis" by simp
moreover {
assume "?s1' ≠ ⊤"
with False have c: "s1 <=_r c!pc" by (simp add: wtc split: if_split_asm)
with less have "s2 <=_r c!pc" ..
with False c have ?thesis by (simp add: wtc)
}
ultimately show ?thesis by (cases "?s1' = ⊤") auto
qed

lemma (in lbv) top_le_conv [simp]:
"⊤ <=_r x = (x = ⊤)"
using semilat by (simp add: top)

lemma (in lbv) neq_top [simp, elim]:
"⟦ x <=_r y; y ≠ ⊤ ⟧ ⟹ x ≠ ⊤"
by (cases "x = T") auto

lemma (in lbvc) stable_wti:
assumes stable:  "stable r step φ pc"
assumes pc:      "pc < length φ"
shows "wti c pc (φ!pc) ≠ ⊤"
proof -
let ?step = "step pc (φ!pc)"
from stable
have less: "∀(q,s')∈set ?step. s' <=_r φ!q" by (simp add: stable_def)

from cert B_A pc
have cert_suc: "c!Suc pc ∈ A" by (rule cert_okD3)
moreover
from phi pc have "φ!pc ∈ A" by simp
from pres this pc
have stepA: "snd`set ?step ⊆ A" by (rule pres_typeD2)
ultimately
have "merge c pc ?step (c!Suc pc) =
(if ∀(pc',s')∈set ?step. pc'≠pc+1 ⟶ s' <=_r c!pc'
then map snd [(p',t') ← ?step.p'=pc+1] ++_f c!Suc pc
else ⊤)" unfolding mrg_def by (rule lbv.merge_def [OF lbvc.axioms(1), OF lbvc_axioms])
moreover {
fix pc' s' assume s': "(pc', s') ∈ set ?step" and suc_pc: "pc' ≠ pc+1"
with less have "s' <=_r φ!pc'" by auto
also
from bounded pc s' have "pc' < length φ" by (rule boundedD)
with s' suc_pc pc have "c!pc' = φ!pc'" ..
hence "φ!pc' = c!pc'" ..
finally have "s' <=_r c!pc'" .
} hence "∀(pc',s')∈set ?step. pc'≠pc+1 ⟶ s' <=_r c!pc'" by auto
moreover
from pc have "Suc pc = length φ ∨ Suc pc < length φ" by auto
hence "map snd [(p',t') ← ?step.p'=pc+1] ++_f c!Suc pc ≠ ⊤"
(is "?map ++_f _ ≠ _")
proof (rule disjE)
assume pc': "Suc pc = length φ"
with cert have "c!Suc pc = ⊥" by (simp add: cert_okD2)
moreover
from pc' bounded pc
have "∀(p',t')∈set ?step. p'≠pc+1" by clarify (drule boundedD, auto)
hence "[(p',t') ← ?step.p'=pc+1] = []" by (blast intro: filter_False)
hence "?map = []" by simp
ultimately show ?thesis by (simp add: B_neq_T)
next
assume pc': "Suc pc < length φ"
from pc' phi have "φ!Suc pc ∈ A" by simp
moreover note cert_suc
moreover from stepA
have "set ?map ⊆ A" by auto
moreover
have "⋀s. s ∈ set ?map ⟹ ∃t. (Suc pc, t) ∈ set ?step" by auto
with less have "∀s' ∈ set ?map. s' <=_r φ!Suc pc" by auto
moreover
from pc' have "c!Suc pc <=_r φ!Suc pc"
by (cases "c!Suc pc = ⊥") (auto dest: cert_approx)
ultimately
have "?map ++_f c!Suc pc <=_r φ!Suc pc" by (rule pp_lub)
moreover
from pc' phi have "φ!Suc pc ≠ ⊤" by simp
ultimately
show ?thesis by auto
qed
ultimately
have "merge c pc ?step (c!Suc pc) ≠ ⊤" by simp
thus ?thesis by (simp add: wti)
qed

lemma (in lbvc) wti_less:
assumes stable:  "stable r step φ pc"
assumes suc_pc:   "Suc pc < length φ"
shows "wti c pc (φ!pc) <=_r φ!Suc pc" (is "?wti <=_r _")
proof -
let ?step = "step pc (φ!pc)"

from stable
have less: "∀(q,s')∈set ?step. s' <=_r φ!q" by (simp add: stable_def)

from suc_pc have pc: "pc < length φ" by simp
with cert B_A have cert_suc: "c!Suc pc ∈ A" by (rule cert_okD3)
moreover
from phi pc have "φ!pc ∈ A" by simp
with pres pc have stepA: "snd`set ?step ⊆ A" by - (rule pres_typeD2)
moreover
from stable pc have "?wti ≠ ⊤" by (rule stable_wti)
hence "merge c pc ?step (c!Suc pc) ≠ ⊤" by (simp add: wti)
ultimately
have "merge c pc ?step (c!Suc pc) =
map snd [(p',t')← ?step.p'=pc+1] ++_f c!Suc pc" by (rule merge_not_top_s)
hence "?wti = …" (is "_ = (?map ++_f _)" is "_ = ?sum") by (simp add: wti)
also {
from suc_pc phi have "φ!Suc pc ∈ A" by simp
moreover note cert_suc
moreover from stepA have "set ?map ⊆ A" by auto
moreover
have "⋀s. s ∈ set ?map ⟹ ∃t. (Suc pc, t) ∈ set ?step" by auto
with less have "∀s' ∈ set ?map. s' <=_r φ!Suc pc" by auto
moreover
from suc_pc have "c!Suc pc <=_r φ!Suc pc"
by (cases "c!Suc pc = ⊥") (auto dest: cert_approx)
ultimately
have "?sum <=_r φ!Suc pc" by (rule pp_lub)
}
finally show ?thesis .
qed

lemma (in lbvc) stable_wtc:
assumes stable:  "stable r step phi pc"
assumes pc:      "pc < length φ"
shows "wtc c pc (φ!pc) ≠ ⊤"
proof -
from stable pc have wti: "wti c pc (φ!pc) ≠ ⊤" by (rule stable_wti)
show ?thesis
proof (cases "c!pc = ⊥")
case True with wti show ?thesis by (simp add: wtc)
next
case False
with pc have "c!pc = φ!pc" ..
with False wti show ?thesis by (simp add: wtc)
qed
qed

lemma (in lbvc) wtc_less:
assumes stable: "stable r step φ pc"
assumes suc_pc: "Suc pc < length φ"
shows "wtc c pc (φ!pc) <=_r φ!Suc pc" (is "?wtc <=_r _")
proof (cases "c!pc = ⊥")
case True
moreover from stable suc_pc have "wti c pc (φ!pc) <=_r φ!Suc pc"
by (rule wti_less)
ultimately show ?thesis by (simp add: wtc)
next
case False
from suc_pc have pc: "pc < length φ" by simp
with stable have "?wtc ≠ ⊤" by (rule stable_wtc)
with False have "?wtc = wti c pc (c!pc)"
by (unfold wtc) (simp split: if_split_asm)
also from pc False have "c!pc = φ!pc" ..
finally have "?wtc = wti c pc (φ!pc)" .
also from stable suc_pc have "wti c pc (φ!pc) <=_r φ!Suc pc" by (rule wti_less)
finally show ?thesis .
qed

lemma (in lbvc) wt_step_wtl_lemma:
assumes wt_step: "wt_step r ⊤ step φ"
shows "⋀pc s. pc+length ls = length φ ⟹ s <=_r φ!pc ⟹ s ∈ A ⟹ s≠⊤ ⟹
wtl ls c pc s ≠ ⊤"
(is "⋀pc s. _ ⟹ _ ⟹ _ ⟹ _ ⟹ ?wtl ls pc s ≠ _")
proof (induct ls)
fix pc s assume "s≠⊤" thus "?wtl [] pc s ≠ ⊤" by simp
next
fix pc s i ls
assume "⋀pc s. pc+length ls=length φ ⟹ s <=_r φ!pc ⟹ s ∈ A ⟹ s≠⊤ ⟹
?wtl ls pc s ≠ ⊤"
moreover
assume pc_l: "pc + length (i#ls) = length φ"
hence suc_pc_l: "Suc pc + length ls = length φ" by simp
ultimately
have IH: "⋀s. s <=_r φ!Suc pc ⟹ s ∈ A ⟹ s ≠ ⊤ ⟹ ?wtl ls (Suc pc) s ≠ ⊤" .

from pc_l obtain pc: "pc < length φ" by simp
with wt_step have stable: "stable r step φ pc" by (simp add: wt_step_def)
from this pc have wt_phi: "wtc c pc (φ!pc) ≠ ⊤" by (rule stable_wtc)
assume s_phi: "s <=_r φ!pc"
from phi pc have phi_pc: "φ!pc ∈ A" by simp
assume s: "s ∈ A"
with s_phi pc phi_pc have wt_s_phi: "wtc c pc s <=_r wtc c pc (φ!pc)" by (rule wtc_mono)
with wt_phi have wt_s: "wtc c pc s ≠ ⊤" by simp
moreover
assume s': "s ≠ ⊤"
ultimately
have "ls = [] ⟹ ?wtl (i#ls) pc s ≠ ⊤" by simp
moreover {
assume "ls ≠ []"
with pc_l have suc_pc: "Suc pc < length φ" by (auto simp add: neq_Nil_conv)
with stable have "wtc c pc (phi!pc) <=_r φ!Suc pc" by (rule wtc_less)
with wt_s_phi have "wtc c pc s <=_r φ!Suc pc" by (rule trans_r)
moreover
from cert suc_pc have "c!pc ∈ A" "c!(pc+1) ∈ A"
from pres this s pc have "wtc c pc s ∈ A" by (rule wtc_pres)
ultimately
have "?wtl ls (Suc pc) (wtc c pc s) ≠ ⊤" using IH wt_s by blast
with s' wt_s have "?wtl (i#ls) pc s ≠ ⊤" by simp
}
ultimately show "?wtl (i#ls) pc s ≠ ⊤" by (cases ls) blast+
qed

theorem (in lbvc) wtl_complete:
assumes wt: "wt_step r ⊤ step φ"
and s: "s <=_r φ!0" "s ∈ A" "s ≠ ⊤"
and len: "length ins = length phi"
shows "wtl ins c 0 s ≠ ⊤"
proof -
from len have "0+length ins = length phi" by simp
from wt this s show ?thesis by (rule wt_step_wtl_lemma)
qed

end
```