Nuprl Lemma : pv11_p1_append_news_iff2
∀Cmd:ValueAllType. ∀f:pv11_p1_headers_type{i:l}(Cmd). ∀es:EO+(Message(f)). ∀accpts,ldrs:bag(Id). ∀ldrs_uid:Id ─→ ℤ.
∀reps:bag(Id). ∀pv:pv11_p1_Ballot_Num() × ℤ × Cmd. ∀L1,L2:(pv11_p1_Ballot_Num() × ℤ × Cmd) List.
(Inj(Id;ℤ;ldrs_uid)
⇒ pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)
⇒ (∀x∈L1 @ L2.↓∃e:E. (x ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e))))
⇒ ((pv ∈ pv11_p1_append_news(Cmd) pv11_p1_same_pvalue(Cmd) L1 L2) ⇐⇒ (pv ∈ L1) ∨ (pv ∈ L2)))
Proof
Definitions occuring in Statement :
pv11_p1_message-constraint: pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; mf; es),
pv11_p1_AcceptorStateFun: pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;mf;es;e),
pv11_p1_append_news: pv11_p1_append_news(Cmd),
pv11_p1_same_pvalue: pv11_p1_same_pvalue(Cmd),
pv11_p1_headers_type: pv11_p1_headers_type{i:l}(Cmd),
pv11_p1_Ballot_Num: pv11_p1_Ballot_Num(),
Message: Message(f),
event-ordering+: EO+(Info),
es-E: E,
Id: Id,
l_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
append: as @ bs,
list: T List,
inject: Inj(A;B;f),
vatype: ValueAllType,
pi2: snd(t),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
squash: ↓T,
implies: P ⇒ Q,
or: P ∨ Q,
apply: f a,
function: x:A ─→ B[x],
product: x:A × B[x],
int: ℤ,
bag: bag(T)
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,
l_all_wf2,
append_wf,
squash_wf,
exists_wf,
es-E_wf,
event-ordering+_subtype,
pv11_p1_AcceptorStateFun_wf,
pv11_p1_message-constraint_wf,
inject_wf,
list_wf,
pv11_p1_Ballot_Num_wf,
bag_wf,
Id_wf,
event-ordering+_wf,
Message_wf,
subtype_rel_dep_function,
vatype_wf,
pv11_p1_headers_type_wf,
set_wf,
valueall-type_wf,
list_induction,
all_wf,
iff_wf,
list_accum_wf,
pv11_p1_add_if_new_wf,
bool_wf,
pv11_p1_same_pvalue_wf,
or_wf,
list_accum_nil_lemma,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
btrue_neq_bfalse,
list_accum_cons_lemma,
member_append,
pv11_p1_add_if_new_iff2,
bl-exists_wf,
assert_wf,
bnot_wf,
not_wf,
l_exists_wf,
bool_cases,
subtype_base_sq,
bool_subtype_base,
eqtt_to_assert,
assert-bl-exists,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
false_wf,
bool_cases_sqequal,
assert-bnot,
l_exists_iff,
assert_of_band2,
pv11_p1_eq_bnums_wf,
pv11_p1_eq_bnums-assert,
pv11_p1_same_proposal_wf,
assert_of_eq_int,
assert-union-deq,
unit_wf2,
product-deq_wf,
int-deq_wf,
id-deq_wf,
unit-deq_wf,
pv11_p1_A4_C1_funA,
and_wf,
pi2_wf
Latex:
\mforall{}Cmd:ValueAllType. \mforall{}f:pv11\_p1\_headers\_type\{i:l\}(Cmd). \mforall{}es:EO+(Message(f)). \mforall{}accpts,ldrs:bag(Id).
\mforall{}ldrs$_{uid}$:Id {}\mrightarrow{} \mBbbZ{}. \mforall{}reps:bag(Id). \mforall{}pv:pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd. \mforall{}L1,L2\000C:(pv11\_p1\_Ballot\_Num()
\mtimes{} \mBbbZ{}
\mtimes{} Cmd) List.
(Inj(Id;\mBbbZ{};ldrs$_{uid}$)
{}\mRightarrow{} pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid\mbackslash{}ff7\000Cd$; reps; f; es)
{}\mRightarrow{} (\mforall{}x\mmember{}L1 @ L2.\mdownarrow{}\mexists{}e:E. (x \mmember{} snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;\000Ce))))
{}\mRightarrow{} ((pv \mmember{} pv11\_p1\_append\_news(Cmd) pv11\_p1\_same\_pvalue(Cmd) L1 L2) \mLeftarrow{}{}\mRightarrow{} (pv \mmember{} L1) \mvee{} (pv \mmember{} L2)))
Date html generated:
2015_07_23-PM-04_54_29
Last ObjectModification:
2015_02_04-AM-07_58_22
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