semanticsPropsScript.sml

(*
  Theorems about the top-level semantics, including totality and determinism.
*)
open preamble
     evaluateTheory
     evaluatePropsTheory
     semanticsTheory lprefix_lubTheory
     typeSoundTheory;

val _ = new_theory"semanticsProps"

Theorem evaluate_prog_events_determ:
   !st env k p k'.
   LENGTH((FST(evaluate_prog_with_clock st env k p)).io_events)
   = LENGTH((FST(evaluate_prog_with_clock st env k' p)).io_events) ==>
   (FST(evaluate_prog_with_clock st env k p)).io_events
    = (FST(evaluate_prog_with_clock st env k' p)).io_events
Proof
  rpt strip_tac
  >> (Cases_on `k <= k'` >| [ALL_TAC,`k' <= k` by simp[]])
  >> fs[evaluate_prog_with_clock_def,ELIM_UNCURRY]
  >> drule evaluate_decs_ffi_mono_clock
  >> disch_then(qspecl_then [`st`,`env`,`p`] assume_tac)
  >> fs[io_events_mono_def,evaluate_prog_with_clock_def,
        ELIM_UNCURRY]
  >> metis_tac[IS_PREFIX_LENGTH_ANTI]
QED

Theorem evaluate_prog_io_events_chain:
   lprefix_chain (IMAGE (λk. fromList (FST (evaluate_prog_with_clock st env k prog)).io_events) UNIV)
Proof
  qho_match_abbrev_tac`lprefix_chain (IMAGE (λk. fromList (g k)) UNIV)` >>
  ONCE_REWRITE_TAC[GSYM o_DEF] >>
  REWRITE_TAC[IMAGE_COMPOSE] >>
  match_mp_tac prefix_chain_lprefix_chain >>
  srw_tac[][prefix_chain_def,Abbr`g`,evaluate_prog_with_clock_def] >> srw_tac[][] >>
  metis_tac[LESS_EQ_CASES,evaluate_decs_ffi_mono_clock,io_events_mono_def,FST]
QED

Theorem semantics_prog_total:
   ∀s e p. ∃b. semantics_prog s e p b
Proof
  srw_tac[][] >>
  Cases_on`∃k. SND(evaluate_prog_with_clock s e k p) = Rerr (Rabort Rtype_error)`
  >- metis_tac[semantics_prog_def] >> full_simp_tac(srw_ss())[] >>
  Cases_on`∃k ffi r.
    evaluate_prog_with_clock s e k p = (ffi,r) ∧
    (r ≠ Rerr (Rabort Rtype_error)) ∧
    (r ≠ Rerr (Rabort Rtimeout_error))`
  >- (fs[semantics_prog_def]
      >> qexists_tac `Terminate
                      (case r of
                      | Rerr(Rabort(Rffi_error outcome)) => FFI_outcome outcome
                      | _ => Success) ffi.io_events`
      >> simp[semantics_prog_def] >> asm_exists_tac
      >> Cases_on `r` >> simp[]
      >> TOP_CASE_TAC >> simp[]
      >> TOP_CASE_TAC >> fs[]) >>
  qexists_tac`Diverge (build_lprefix_lub (IMAGE (λk. fromList (FST (evaluate_prog_with_clock s e k p)).io_events) UNIV))` >>
  simp[semantics_prog_def] >>
  conj_tac >- (
    strip_tac >> fs[] >>
    rpt(first_x_assum(qspec_then`k`mp_tac)) >>
    Cases_on`evaluate_prog_with_clock s e k p`>>simp[]>>
    Cases_on`r`>>simp[]>>
    Cases_on`e'`>>simp[]>>
    Cases_on`a`>>simp[]) >>
  match_mp_tac build_lprefix_lub_thm >>
  MATCH_ACCEPT_TAC evaluate_prog_io_events_chain
QED

Triviality with_clock_ffi:
  (s with clock := x).ffi = s.ffi
Proof
  EVAL_TAC
QED

val tac1 =
    metis_tac[semanticPrimitivesTheory.result_11,evaluate_decs_ffi_mono_clock,io_events_mono_def,
              semanticPrimitivesTheory.error_result_11,option_nchotomy,LESS_EQ_CASES,
              semanticPrimitivesTheory.abort_distinct,pair_CASES,FST,THE_DEF,
              PAIR_EQ,IS_SOME_EXISTS,SOME_11,NOT_SOME_NONE,SND,PAIR,LESS_OR_EQ]

val tac2 = every_case_tac >> rfs[] >> first_x_assum (qspec_then `k` assume_tac) >> rfs[]

Theorem semantics_prog_deterministic:
   ∀s e p b b'.
    semantics_prog s e p b ∧
    semantics_prog s e p b' ⇒
    b = b'
Proof
  rw []
  >> Cases_on `b`
  >> Cases_on `b'`
  >> fs [semantics_prog_def]
  >- metis_tac[unique_lprefix_lub]
  >- tac2
  >- (tac2 >> tac1)
  >- tac2
  >- (
    fs [evaluate_prog_with_clock_def]
    >> pairarg_tac
    >> fs []
    >> pairarg_tac
    >> fs []
    >> rpt var_eq_tac
    >> pop_assum mp_tac
    >> drule evaluate_decs_clock_determ
    >> ntac 2 DISCH_TAC
    >> first_x_assum drule
    >> simp []
    >> every_case_tac
    >> fs [semanticPrimitivesTheory.state_component_equality])
  >- (
    fs [evaluate_prog_with_clock_def]
    >> pairarg_tac
    >> fs []
    >> pairarg_tac
    >> fs []
    >> rpt var_eq_tac
    >> pop_assum mp_tac
    >> drule evaluate_decs_clock_determ
    >> ntac 2 DISCH_TAC
    >> first_x_assum drule
    >> simp []
    >> every_case_tac
    >> fs [semanticPrimitivesTheory.state_component_equality])
  >- tac1
  >- (
    fs [evaluate_prog_with_clock_def]
    >> pairarg_tac
    >> fs []
    >> pairarg_tac
    >> fs []
    >> rpt var_eq_tac
    >> pop_assum mp_tac
    >> drule evaluate_decs_clock_determ
    >> ntac 2 DISCH_TAC
    >> first_x_assum drule
    >> simp []
    >> every_case_tac
    >> fs [semanticPrimitivesTheory.state_component_equality])
QED

Theorem semantics_prog_Terminate_not_Fail:
   semantics_prog s e p (Terminate x y) ⇒
    ¬semantics_prog s e p Fail ∧
    semantics_prog s e p = {Terminate x y}
Proof
  rpt strip_tac
  \\ simp[FUN_EQ_THM]
  \\ imp_res_tac semantics_prog_deterministic \\ fs[]
  \\ metis_tac[semantics_prog_deterministic]
QED

Definition state_invariant_def:
  state_invariant st ⇔
  ?ctMap tenvS.
    FRANGE ((SND ∘ SND) o_f ctMap) ⊆ st.type_ids ∧
    type_sound_invariant st.sem_st st.sem_env ctMap tenvS {} st.tenv
End

Triviality clock_lemmas:
  ((x with clock := c).clock = c) ∧
   (((x with clock := c) with clock := d) = (x with clock := d)) ∧
   (x with clock := x.clock = x)
Proof
  srw_tac[][semanticPrimitivesTheory.state_component_equality]
QED

Theorem semantics_deterministic:
   state_invariant st ⇒
   semantics st prelude inp = Execute bs
   ⇒ ∃b. bs = {b} ∧ b ≠ Fail
Proof
 rw [state_invariant_def, semantics_def]
 >> every_case_tac
 >> fs [can_type_prog_def]
 >> rw []
 >> qspecl_then [`st.sem_st`, `st.sem_env`, `prelude ++ x`] strip_assume_tac semantics_prog_total
 >> imp_res_tac semantics_type_sound
 >> qexists_tac `b`
 >> rw [EXTENSION, IN_DEF]
 >- metis_tac [semantics_prog_deterministic] >>
 `DISJOINT new_tids (FRANGE ((SND ∘ SND) o_f ctMap))`
 by (
   fs [DISJOINT_DEF, EXTENSION, SUBSET_DEF] >>
   rw [] >>
   metis_tac []) >>
 fs [typeSoundInvariantsTheory.type_sound_invariant_def] >>
 rfs [typeSoundInvariantsTheory.consistent_ctMap_def] >>
 metis_tac []
QED

Definition extend_with_resource_limit_def:
  extend_with_resource_limit behaviours =
  behaviours ∪
  { Terminate Resource_limit_hit io_list
    | io_list | ∃t l. Terminate t l ∈ behaviours ∧ io_list ≼ l } ∪
  { Terminate Resource_limit_hit io_list
    | io_list | ∃ll. Diverge ll ∈ behaviours ∧ LPREFIX (fromList io_list) ll }
End

Definition extend_with_resource_limit'_def:
  extend_with_resource_limit' precise behaviours =
    if precise then behaviours else
      extend_with_resource_limit behaviours
End

Definition implements_def:
  implements x y <=>
    (~(Fail IN y) ==> x SUBSET extend_with_resource_limit y)
End

Definition implements'_def:
  implements' precise x y ⇔ Fail ∉ y ⇒ x ⊆ extend_with_resource_limit' precise y
End

Theorem extend_with_resource_limit_not_fail:
   x ∈ extend_with_resource_limit y ∧ Fail ∉ y ⇒ x ≠ Fail
Proof
  rw[extend_with_resource_limit_def] \\ metis_tac[]
QED

Theorem implements_intro:
   (b /\ x <> Fail ==> y = x) ==> b ==> implements {y} {x}
Proof
  full_simp_tac(srw_ss())[implements_def] \\ srw_tac[][] \\ full_simp_tac(srw_ss())[]
  \\ full_simp_tac(srw_ss())[extend_with_resource_limit_def]
QED

Theorem implements_trivial_intro:
   (y = x) ==> implements {y} {x}
Proof
  full_simp_tac(srw_ss())[implements_def] \\ srw_tac[][] \\ full_simp_tac(srw_ss())[]
  \\ full_simp_tac(srw_ss())[extend_with_resource_limit_def]
QED

Theorem implements_intro_ext:
   (b /\ x <> Fail ==> y IN extend_with_resource_limit {x}) ==>
    b ==> implements {y} {x}
Proof
  full_simp_tac(srw_ss())[implements_def] \\ srw_tac[][] \\ full_simp_tac(srw_ss())[]
  \\ full_simp_tac(srw_ss())[extend_with_resource_limit_def]
QED

Triviality isPREFIX_IMP_LPREFIX:
  !xs ys. isPREFIX xs ys ==> LPREFIX (fromList xs) (fromList ys)
Proof
  full_simp_tac(srw_ss())[LPREFIX_def,llistTheory.from_toList]
QED

Theorem implements_trans:
   implements y z ==> implements x y ==> implements x z
Proof
  full_simp_tac(srw_ss())[implements_def] \\ srw_tac[][] \\ full_simp_tac(srw_ss())[]
  \\ full_simp_tac(srw_ss())[extend_with_resource_limit_def]
  \\ Cases_on `Fail IN y` \\ full_simp_tac(srw_ss())[]
  THEN1 (full_simp_tac(srw_ss())[SUBSET_DEF] \\ res_tac \\ full_simp_tac(srw_ss())[])
  \\ full_simp_tac(srw_ss())[SUBSET_DEF] \\ srw_tac[][] \\ rename1 `a IN x`
  \\ reverse (res_tac \\ full_simp_tac(srw_ss())[])
  THEN1 (res_tac \\ full_simp_tac(srw_ss())[] \\ srw_tac[][] \\ metis_tac [])
  \\ res_tac \\ full_simp_tac(srw_ss())[] \\ srw_tac[][]
  \\ imp_res_tac IS_PREFIX_TRANS
  \\ imp_res_tac isPREFIX_IMP_LPREFIX
  \\ imp_res_tac LPREFIX_TRANS
  \\ metis_tac []
QED

Theorem implements'_F:
  implements' F = implements
Proof
  fs [implements'_def,FUN_EQ_THM,extend_with_resource_limit'_def,implements_def]
QED

Theorem implements'_trans:
  !x y z b.
    implements' b y z /\
    implements' b x y ==>
    implements' b x z
Proof
  Cases_on `b` \\ fs [implements'_F]
  THEN1 (fs [implements'_def,extend_with_resource_limit'_def,SUBSET_DEF] \\ metis_tac [])
  \\ metis_tac [implements_trans]
QED

Theorem implements'_strengthen:
  !b b' x y. (b' ==> b) /\ implements' b x y ==> implements' b' x y
Proof
  Cases_on `b` \\ Cases_on `b'` \\ fs [implements'_def]
  \\ fs [extend_with_resource_limit'_def]
  \\ fs [extend_with_resource_limit_def,SUBSET_DEF]
QED

Theorem implements'_IMP_implements:
  implements' b x y ==> implements x y
Proof
  Cases_on `b` \\ fs [implements'_F]
  \\ fs [implements'_def,implements_def,extend_with_resource_limit'_def,
         extend_with_resource_limit_def,SUBSET_DEF]
QED

Theorem extend_with_resource_limit'_SUBSET:
  extend_with_resource_limit' b s SUBSET
  extend_with_resource_limit s
Proof
  Cases_on `b`
  \\ fs [extend_with_resource_limit'_def,extend_with_resource_limit_def,SUBSET_DEF]
QED

val _ = export_theory()