Theory OG_Tran

section ‹Operational Semantics›

theory OG_Tran imports OG_Com begin

type_synonym 'a ann_com_op = "('a ann_com) option"
type_synonym 'a ann_triple_op = "('a ann_com_op × 'a assn)"

primrec com :: "'a ann_triple_op ⇒ 'a ann_com_op" where
  "com (c, q) = c"

primrec post :: "'a ann_triple_op ⇒ 'a assn" where
  "post (c, q) = q"

definition All_None :: "'a ann_triple_op list ⇒ bool" where
  "All_None Ts ≡ ∀(c, q) ∈ set Ts. c = None"

subsection ‹The Transition Relation›

inductive_set
  ann_transition :: "(('a ann_com_op × 'a) × ('a ann_com_op × 'a)) set"
  and transition :: "(('a com × 'a) × ('a com × 'a)) set"
  and ann_transition' :: "('a ann_com_op × 'a) ⇒ ('a ann_com_op × 'a) ⇒ bool"
    ("_ -1→ _"[81,81] 100)
  and transition' :: "('a com × 'a) ⇒ ('a com × 'a) ⇒ bool"
    ("_ -P1→ _"[81,81] 100)
  and transitions :: "('a com × 'a) ⇒ ('a com × 'a) ⇒ bool"
    ("_ -P*→ _"[81,81] 100)
where
  "con_0 -1→ con_1 ≡ (con_0, con_1) ∈ ann_transition"
| "con_0 -P1→ con_1 ≡ (con_0, con_1) ∈ transition"
| "con_0 -P*→ con_1 ≡ (con_0, con_1) ∈ transition⇧*"

| AnnBasic:  "(Some (AnnBasic r f), s) -1→ (None, f s)"

| AnnSeq1: "(Some c0, s) -1→ (None, t) ⟹
               (Some (AnnSeq c0 c1), s) -1→ (Some c1, t)"
| AnnSeq2: "(Some c0, s) -1→ (Some c2, t) ⟹
               (Some (AnnSeq c0 c1), s) -1→ (Some (AnnSeq c2 c1), t)"

| AnnCond1T: "s ∈ b  ⟹ (Some (AnnCond1 r b c1 c2), s) -1→ (Some c1, s)"
| AnnCond1F: "s ∉ b ⟹ (Some (AnnCond1 r b c1 c2), s) -1→ (Some c2, s)"

| AnnCond2T: "s ∈ b  ⟹ (Some (AnnCond2 r b c), s) -1→ (Some c, s)"
| AnnCond2F: "s ∉ b ⟹ (Some (AnnCond2 r b c), s) -1→ (None, s)"

| AnnWhileF: "s ∉ b ⟹ (Some (AnnWhile r b i c), s) -1→ (None, s)"
| AnnWhileT: "s ∈ b  ⟹ (Some (AnnWhile r b i c), s) -1→
                         (Some (AnnSeq c (AnnWhile i b i c)), s)"

| AnnAwait: "⟦ s ∈ b; atom_com c; (c, s) -P*→ (Parallel [], t) ⟧ ⟹
                   (Some (AnnAwait r b c), s) -1→ (None, t)"

| Parallel: "⟦ i<length Ts; Ts!i = (Some c, q); (Some c, s) -1→ (r, t) ⟧
              ⟹ (Parallel Ts, s) -P1→ (Parallel (Ts [i:=(r, q)]), t)"

| Basic:  "(Basic f, s) -P1→ (Parallel [], f s)"

| Seq1:   "All_None Ts ⟹ (Seq (Parallel Ts) c, s) -P1→ (c, s)"
| Seq2:   "(c0, s) -P1→ (c2, t) ⟹ (Seq c0 c1, s) -P1→ (Seq c2 c1, t)"

| CondT: "s ∈ b ⟹ (Cond b c1 c2, s) -P1→ (c1, s)"
| CondF: "s ∉ b ⟹ (Cond b c1 c2, s) -P1→ (c2, s)"

| WhileF: "s ∉ b ⟹ (While b i c, s) -P1→ (Parallel [], s)"
| WhileT: "s ∈ b ⟹ (While b i c, s) -P1→ (Seq c (While b i c), s)"

monos "rtrancl_mono"

text ‹The corresponding abbreviations are:›

abbreviation
  ann_transition_n :: "('a ann_com_op × 'a) ⇒ nat ⇒ ('a ann_com_op × 'a)
                           ⇒ bool"  ("_ -_→ _"[81,81] 100)  where
  "con_0 -n→ con_1 ≡ (con_0, con_1) ∈ ann_transition ^^ n"

abbreviation
  ann_transitions :: "('a ann_com_op × 'a) ⇒ ('a ann_com_op × 'a) ⇒ bool"
                           ("_ -*→ _"[81,81] 100)  where
  "con_0 -*→ con_1 ≡ (con_0, con_1) ∈ ann_transition⇧*"

abbreviation
  transition_n :: "('a com × 'a) ⇒ nat ⇒ ('a com × 'a) ⇒ bool"
                          ("_ -P_→ _"[81,81,81] 100)  where
  "con_0 -Pn→ con_1 ≡ (con_0, con_1) ∈ transition ^^ n"

subsection ‹Definition of Semantics›

definition ann_sem :: "'a ann_com ⇒ 'a ⇒ 'a set" where
  "ann_sem c ≡ λs. {t. (Some c, s) -*→ (None, t)}"

definition ann_SEM :: "'a ann_com ⇒ 'a set ⇒ 'a set" where
  "ann_SEM c S ≡ ⋃(ann_sem c ` S)"

definition sem :: "'a com ⇒ 'a ⇒ 'a set" where
  "sem c ≡ λs. {t. ∃Ts. (c, s) -P*→ (Parallel Ts, t) ∧ All_None Ts}"

definition SEM :: "'a com ⇒ 'a set ⇒ 'a set" where
  "SEM c S ≡ ⋃(sem c ` S)"

abbreviation Omega :: "'a com"    ("Ω" 63)
  where "Ω ≡ While UNIV UNIV (Basic id)"

primrec fwhile :: "'a bexp ⇒ 'a com ⇒ nat ⇒ 'a com" where
    "fwhile b c 0 = Ω"
  | "fwhile b c (Suc n) = Cond b (Seq c (fwhile b c n)) (Basic id)"

subsubsection ‹Proofs›

declare ann_transition_transition.intros [intro]
inductive_cases transition_cases:
    "(Parallel T,s) -P1→ t"
    "(Basic f, s) -P1→ t"
    "(Seq c1 c2, s) -P1→ t"
    "(Cond b c1 c2, s) -P1→ t"
    "(While b i c, s) -P1→ t"

lemma Parallel_empty_lemma [rule_format (no_asm)]:
  "(Parallel [],s) -Pn→ (Parallel Ts,t) ⟶ Ts=[] ∧ n=0 ∧ s=t"
apply(induct n)
 apply(simp (no_asm))
apply clarify
apply(drule relpow_Suc_D2)
apply(force elim:transition_cases)
done

lemma Parallel_AllNone_lemma [rule_format (no_asm)]:
 "All_None Ss ⟶ (Parallel Ss,s) -Pn→ (Parallel Ts,t) ⟶ Ts=Ss ∧ n=0 ∧ s=t"
apply(induct "n")
 apply(simp (no_asm))
apply clarify
apply(drule relpow_Suc_D2)
apply clarify
apply(erule transition_cases,simp_all)
apply(force dest:nth_mem simp add:All_None_def)
done

lemma Parallel_AllNone: "All_None Ts ⟹ (SEM (Parallel Ts) X) = X"
apply (unfold SEM_def sem_def)
apply auto
apply(drule rtrancl_imp_UN_relpow)
apply clarify
apply(drule Parallel_AllNone_lemma)
apply auto
done

lemma Parallel_empty: "Ts=[] ⟹ (SEM (Parallel Ts) X) = X"
apply(rule Parallel_AllNone)
apply(simp add:All_None_def)
done

text ‹Set of lemmas from Apt and Olderog "Verification of sequential
and concurrent programs", page 63.›

lemma L3_5i: "X⊆Y ⟹ SEM c X ⊆ SEM c Y"
apply (unfold SEM_def)
apply force
done

lemma L3_5ii_lemma1:
 "⟦ (c1, s1) -P*→ (Parallel Ts, s2); All_None Ts;
  (c2, s2) -P*→ (Parallel Ss, s3); All_None Ss ⟧
 ⟹ (Seq c1 c2, s1) -P*→ (Parallel Ss, s3)"
apply(erule converse_rtrancl_induct2)
apply(force intro:converse_rtrancl_into_rtrancl)+
done

lemma L3_5ii_lemma2 [rule_format (no_asm)]:
 "∀c1 c2 s t. (Seq c1 c2, s) -Pn→ (Parallel Ts, t) ⟶
  (All_None Ts) ⟶ (∃y m Rs. (c1,s) -P*→ (Parallel Rs, y) ∧
  (All_None Rs) ∧ (c2, y) -Pm→ (Parallel Ts, t) ∧  m ≤ n)"
apply(induct "n")
 apply(force)
apply(safe dest!: relpow_Suc_D2)
apply(erule transition_cases,simp_all)
 apply (fast intro!: le_SucI)
apply (fast intro!: le_SucI elim!: relpow_imp_rtrancl converse_rtrancl_into_rtrancl)
done

lemma L3_5ii_lemma3:
 "⟦(Seq c1 c2,s) -P*→ (Parallel Ts,t); All_None Ts⟧ ⟹
    (∃y Rs. (c1,s) -P*→ (Parallel Rs,y) ∧ All_None Rs
   ∧ (c2,y) -P*→ (Parallel Ts,t))"
apply(drule rtrancl_imp_UN_relpow)
apply(fast dest: L3_5ii_lemma2 relpow_imp_rtrancl)
done

lemma L3_5ii: "SEM (Seq c1 c2) X = SEM c2 (SEM c1 X)"
apply (unfold SEM_def sem_def)
apply auto
 apply(fast dest: L3_5ii_lemma3)
apply(fast elim: L3_5ii_lemma1)
done

lemma L3_5iii: "SEM (Seq (Seq c1 c2) c3) X = SEM (Seq c1 (Seq c2 c3)) X"
apply (simp (no_asm) add: L3_5ii)
done

lemma L3_5iv:
 "SEM (Cond b c1 c2) X = (SEM c1 (X ∩ b)) Un (SEM c2 (X ∩ (-b)))"
apply (unfold SEM_def sem_def)
apply auto
apply(erule converse_rtranclE)
 prefer 2
 apply (erule transition_cases,simp_all)
  apply(fast intro: converse_rtrancl_into_rtrancl elim: transition_cases)+
done


lemma  L3_5v_lemma1[rule_format]:
 "(S,s) -Pn→ (T,t) ⟶ S=Ω ⟶ (¬(∃Rs. T=(Parallel Rs) ∧ All_None Rs))"
apply (unfold UNIV_def)
apply(rule nat_less_induct)
apply safe
apply(erule relpow_E2)
 apply simp_all
apply(erule transition_cases)
 apply simp_all
apply(erule relpow_E2)
 apply(simp add: Id_def)
apply(erule transition_cases,simp_all)
apply clarify
apply(erule transition_cases,simp_all)
apply(erule relpow_E2,simp)
apply clarify
apply(erule transition_cases)
 apply simp+
    apply clarify
    apply(erule transition_cases)
apply simp_all
done

lemma L3_5v_lemma2: "⟦(Ω, s) -P*→ (Parallel Ts, t); All_None Ts ⟧ ⟹ False"
apply(fast dest: rtrancl_imp_UN_relpow L3_5v_lemma1)
done

lemma L3_5v_lemma3: "SEM (Ω) S = {}"
apply (unfold SEM_def sem_def)
apply(fast dest: L3_5v_lemma2)
done

lemma L3_5v_lemma4 [rule_format]:
 "∀s. (While b i c, s) -Pn→ (Parallel Ts, t) ⟶ All_None Ts ⟶
  (∃k. (fwhile b c k, s) -P*→ (Parallel Ts, t))"
apply(rule nat_less_induct)
apply safe
apply(erule relpow_E2)
 apply safe
apply(erule transition_cases,simp_all)
 apply (rule_tac x = "1" in exI)
 apply(force dest: Parallel_empty_lemma intro: converse_rtrancl_into_rtrancl simp add: Id_def)
apply safe
apply(drule L3_5ii_lemma2)
 apply safe
apply(drule le_imp_less_Suc)
apply (erule allE , erule impE,assumption)
apply (erule allE , erule impE, assumption)
apply safe
apply (rule_tac x = "k+1" in exI)
apply(simp (no_asm))
apply(rule converse_rtrancl_into_rtrancl)
 apply fast
apply(fast elim: L3_5ii_lemma1)
done

lemma L3_5v_lemma5 [rule_format]:
 "∀s. (fwhile b c k, s) -P*→ (Parallel Ts, t) ⟶ All_None Ts ⟶
  (While b i c, s) -P*→ (Parallel Ts,t)"
apply(induct "k")
 apply(force dest: L3_5v_lemma2)
apply safe
apply(erule converse_rtranclE)
 apply simp_all
apply(erule transition_cases,simp_all)
 apply(rule converse_rtrancl_into_rtrancl)
  apply(fast)
 apply(fast elim!: L3_5ii_lemma1 dest: L3_5ii_lemma3)
apply(drule rtrancl_imp_UN_relpow)
apply clarify
apply(erule relpow_E2)
 apply simp_all
apply(erule transition_cases,simp_all)
apply(fast dest: Parallel_empty_lemma)
done

lemma L3_5v: "SEM (While b i c) = (λx. (⋃k. SEM (fwhile b c k) x))"
apply(rule ext)
apply (simp add: SEM_def sem_def)
apply safe
 apply(drule rtrancl_imp_UN_relpow,simp)
 apply clarify
 apply(fast dest:L3_5v_lemma4)
apply(fast intro: L3_5v_lemma5)
done

section ‹Validity of Correctness Formulas›

definition com_validity :: "'a assn ⇒ 'a com ⇒ 'a assn ⇒ bool" ("(3∥= _// _//_)" [90,55,90] 50) where
  "∥= p c q ≡ SEM c p ⊆ q"

definition ann_com_validity :: "'a ann_com ⇒ 'a assn ⇒ bool" ("⊨ _ _" [60,90] 45) where
  "⊨ c q ≡ ann_SEM c (pre c) ⊆ q"

end