Nuprl Lemma : pv11_p1_ldr_mem_propose2
∀Cmd:ValueAllType. ∀f:pv11_p1_headers_type{i:l}(Cmd). ∀es:EO+(Message(f)). ∀e1,e2:E. ∀ldrs_uid:Id ⟶ ℤ.
∀bnum1,bnum2:pv11_p1_Ballot_Num(). ∀active1,active2:𝔹. ∀proposals1,proposals2:(ℤ × Cmd) List. ∀s:ℤ. ∀c:Cmd.
((e1 <loc e2)
⇒ <s, c> ∈ pv11_p1_propose'base(Cmd;f)(e1)
⇒ <bnum1, active1, proposals1> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e1)
⇒ <bnum2, active2, proposals2> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e2)
⇒ (¬↑(pv11_p1_in_domain(Cmd) s proposals1))
⇒ (↑(pv11_p1_in_domain(Cmd) s proposals2)))
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_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),
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,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
pair: <a, b>,
product: x:A × B[x],
int: ℤ
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,
pv11_p1_headers: pv11_p1_headers(),
rev_implies: P ⇐ Q,
guard: {T},
or: P ∨ Q,
uimplies: b supposing a,
pv11_p1_headers_fun: pv11_p1_headers_fun(Cmd),
name_eq: name_eq(x;y),
name-deq: NameDeq,
list-deq: list-deq(eq),
list_ind: list_ind,
cons: [a / b],
band: p ∧b q,
ifthenelse: if b then t else f fi ,
atom-deq: AtomDeq,
eq_atom: x =a y,
bfalse: ff,
btrue: tt,
nil: [],
it: ⋅,
null: null(as),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
isl: isl(x),
outl: outl(x),
outr: outr(x),
assert: ↑b,
cand: A c∧ B,
true: True,
not: ¬A,
false: False,
spreadn: spread3
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{}bnum1,bnum2:pv11\_p1\_Ballot\_Num(). \mforall{}active1,active2:\mBbbB{}.
\mforall{}proposals1,proposals2:(\mBbbZ{} \mtimes{} Cmd) List. \mforall{}s:\mBbbZ{}. \mforall{}c:Cmd.
((e1 <loc e2)
{}\mRightarrow{} <s, c> \mmember{} pv11\_p1\_propose'base(Cmd;f)(e1)
{}\mRightarrow{} <bnum1, active1, proposals1> \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(e1)
{}\mRightarrow{} <bnum2, active2, proposals2> \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(e2)
{}\mRightarrow{} (\mneg{}\muparrow{}(pv11\_p1\_in\_domain(Cmd) s proposals1))
{}\mRightarrow{} (\muparrow{}(pv11\_p1\_in\_domain(Cmd) s proposals2)))
Date html generated:
2016_05_17-PM-03_18_19
Last ObjectModification:
2015_12_29-PM-11_19_49
Theory : paxos!synod
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