Nuprl Lemma : pv11_p1_ldr_mem_propose

∀Cmd:ValueAllType. ∀f:pv11_p1_headers_type{i:l}(Cmd). ∀es:EO+(Message(f)). ∀e1,e2:E. ∀ldrs_uid:Id ─→ ℤ.
∀v1,v2:pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List).
∀p:ℤ × Cmd + pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List) + pv11_p1_Ballot_Num().
  ((e1 <loc e2)
  ⇒ p ∈ pv11_p1_propose'base(Cmd;f) (+) pv11_p1_adopted'base(Cmd;f) (+) pv11_p1_preempted'base(Cmd;f)(e1)
  ⇒ v1 ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e1)
  ⇒ v2 ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e2)
  ⇒ case p
      of inl(x) =>
      let s,c = x 
      in let bnum1,active1,proposals1 = v1 in 
         let bnum2,active2,proposals2 = v2 in 
         (¬↑(pv11_p1_in_domain(Cmd) s proposals1)) ⇒ (↑(pv11_p1_in_domain(Cmd) s proposals2))
      | inr(x) =>
      True)


Proof



Definitions occuring in Statement :  pv11_p1_LeaderState: pv11_p1_LeaderState(Cmd;ldrs_uid;mf),  pv11_p1_propose'base: pv11_p1_propose'base(Cmd;mf),  pv11_p1_adopted'base: pv11_p1_adopted'base(Cmd;mf),  pv11_p1_preempted'base: pv11_p1_preempted'base(Cmd;mf),  pv11_p1_in_domain: pv11_p1_in_domain(Cmd),  pv11_p1_headers_type: pv11_p1_headers_type{i:l}(Cmd),  pv11_p1_Ballot_Num: pv11_p1_Ballot_Num(),  Message: Message(f),  disjoint-union-comb: X (+) Y,  classrel: v ∈ X(e),  event-ordering+: EO+(Info),  es-locl: (e <loc e'),  es-E: E,  Id: Id,  list: T List,  vatype: ValueAllType,  assert: ↑b,  bool: 𝔹,  spreadn: spread3,  all: ∀x:A. B[x],  not: ¬A,  implies: P ⇒ Q,  true: True,  apply: f a,  function: x:A ─→ B[x],  spread: spread def,  product: x:A × B[x],  decide: case b of inl(x) => s[x] | inr(y) => t[y],  union: left + right,  int: ℤ
Lemmas :  int_seg_wf,  length_wf,  name_wf,  pv11_p1_headers_wf,  l_all_iff,  l_member_wf,  equal_wf,  pv11_p1_headers_fun_wf,  cons_wf_listp,  nil_wf,  listp_wf,  cons_member,  cons_wf,  equal-wf-base,  iff_weakening_equal,  classrel_wf,  squash_wf,  true_wf,  es-E_wf,  event-ordering+_subtype,  event-ordering+_wf,  eclass_wf,  pv11_p1_Ballot_Num_wf,  bool_wf,  list_wf,  Memory3-memory-class3,  pv11_p1_init_leader_wf,  pv11_p1_on_propose_wf,  pv11_p1_propose'base_wf,  pv11_p1_when_adopted_wf,  pv11_p1_adopted'base_wf,  pv11_p1_when_preempted_wf,  pv11_p1_preempted'base_wf,  Memory-loc-class-mem,  not_wf,  assert_wf,  pv11_p1_in_domain_wf,  disjoint-union-tr_wf,  single-bag_wf,  Id_wf,  disjoint-union-comb_wf,  disj_un_tr_ap_inl_lemma,  assert-deq-member,  int-deq_wf,  map_wf,  deq-member_wf,  eqff_to_assert,  bool_cases_sqequal,  subtype_base_sq,  bool_subtype_base,  assert-bnot,  pv11_p1_add_if_new_wf,  pv11_p1_same_proposal_wf,  member_map,  eqtt_to_assert,  disj_un_tr_ap_inr_lemma,  pv11_p1_eq_bnums_wf,  pv11_p1_eq_bnums-assert,  pv11_p1_update_proposals_wf,  pv11_p1_pmax_wf,  pv11_p1_is_bnum_wf,  pv11_p1_lt_bnum_wf,  disjoint-union-comb-single-val,  base-headers-msg-val-single-val,  subtype_rel_weakening,  ext-eq_weakening,  base-disjoint-classrel,  iff_transitivity,  name_eq_wf,  bnot_wf,  assert-name_eq,  iff_weakening_uiff,  assert_of_bnot,  sq_stable__encodes-msg-type,  l_all_fwd,  valueall-type_wf,  equal-wf-T-base,  disjoint-classrel-symm,  disjoint-union-comb-disjoint-classrel,  single-valued-bag-single,  es-loc_wf,  pv11_p1_LeaderState_wf,  es-locl_wf,  Message_wf,  subtype_rel_dep_function,  vatype_wf,  pv11_p1_headers_type_wf,  set_wf,  bl-exists_wf,  l_exists_wf,  assert-bl-exists,  l_exists_iff,  assert_of_eq_int,  equal-wf-base-T,  int_subtype_base,  member_append,  pv11_p1_upd_left
Latex:
\mforall{}Cmd:ValueAllType.  \mforall{}f:pv11\_p1\_headers\_type\{i:l\}(Cmd).  \mforall{}es:EO+(Message(f)).  \mforall{}e1,e2:E.
\mforall{}ldrs$_{uid}$:Id  {}\mrightarrow{}  \mBbbZ{}.  \mforall{}v1,v2:pv11\_p1\_Ballot\_Num()  \mtimes{}  \mBbbB{}  \mtimes{}  ((\mBbbZ{}  \mtimes{}  Cmd)  List).
\mforall{}p:\mBbbZ{}  \mtimes{}  Cmd  +  pv11\_p1\_Ballot\_Num()  \mtimes{}  ((pv11\_p1\_Ballot\_Num()  \mtimes{}  \mBbbZ{}  \mtimes{}  Cmd)  List)  +  pv11\_p1\_Ballot\_Num().
    ((e1  <loc  e2)
    {}\mRightarrow{}  p  \mmember{}
            pv11\_p1\_propose'base(Cmd;f)  (+)  pv11\_p1\_adopted'base(Cmd;f)  (+)  pv11\_p1\_preempted'base(Cmd;f)(
            e1)
    {}\mRightarrow{}  v1  \mmember{}  pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(e1)
    {}\mRightarrow{}  v2  \mmember{}  pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(e2)
    {}\mRightarrow{}  case  p
            of  inl(x)  =>
            let  s,c  =  x 
            in  let  bnum1,active1,proposals1  =  v1  in 
                  let  bnum2,active2,proposals2  =  v2  in 
                  (\mneg{}\muparrow{}(pv11\_p1\_in\_domain(Cmd)  s  proposals1))  {}\mRightarrow{}  (\muparrow{}(pv11\_p1\_in\_domain(Cmd)  s  proposals2))
            |  inr(x)  =>
            True)


Date html generated: 2015_07_23-PM-04_45_22
Last ObjectModification: 2015_02_04-AM-07_35_15

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