Nuprl Lemma : pv11_p1_append_news_iff
∀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,
implies: P ⇒ Q,
or: P ∨ Q,
apply: f a,
function: x:A ⟶ B[x],
product: x:A × B[x],
int: ℤ,
bag: bag(T)
Definitions unfolded in proof :
vatype: ValueAllType,
all: ∀x:A. B[x],
implies: P ⇒ Q,
pv11_p1_headers_type: pv11_p1_headers_type{i:l}(Cmd),
l_all: (∀x∈L.P[x]),
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
prop: ℙ,
so_apply: x[s],
iff: P ⇐⇒ Q,
listp: A List+,
name: Name,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
less_than: a < b,
squash: ↓T,
length: ||as||,
list_ind: list_ind,
pv11_p1_headers: pv11_p1_headers(),
cons: [a / b],
nil: [],
it: ⋅,
true: True,
select: L[n],
subtract: n - m,
uimplies: b supposing a,
guard: {T},
rev_implies: P ⇐ Q,
pv11_p1_headers_fun: pv11_p1_headers_fun(Cmd),
name_eq: name_eq(x;y),
name-deq: NameDeq,
list-deq: list-deq(eq),
band: p ∧b q,
ifthenelse: if b then t else f fi ,
atom-deq: AtomDeq,
eq_atom: x =a y,
bfalse: ff,
btrue: tt,
null: null(as),
exists: ∃x:A. B[x],
pv11_p1_append_news: pv11_p1_append_news(Cmd),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
or: P ∨ Q,
top: Top,
pv11_p1_add_if_new: pv11_p1_add_if_new(),
sq_type: SQType(T),
uiff: uiff(P;Q),
bool: 𝔹,
unit: Unit,
bnot: ¬bb,
assert: ↑b,
pv11_p1_same_pvalue: pv11_p1_same_pvalue(Cmd),
pv11_p1_same_proposal: pv11_p1_same_proposal(Cmd),
pv11_p1_Ballot_Num: pv11_p1_Ballot_Num(),
Id: Id,
pi1: fst(t),
pi2: snd(t)
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.\mexists{}e:E. (x \mmember{} snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;e\000C))))
{}\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:
2016_05_17-PM-04_04_26
Last ObjectModification:
2016_01_18-AM-11_17_29
Theory : paxos!synod
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