Nuprl Lemma : pv11_p1_ldr_state_eq2
∀Cmd:ValueAllType. ∀f:pv11_p1_headers_type{i:l}(Cmd). ∀es:EO+(Message(f)). ∀e:E. ∀ldrs_uid:Id ⟶ ℤ.
∀v:pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List).
uiff(v ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e);if first(e)
then v
= (pv11_p1_init_leader(Cmd) loc(e))
∈ (pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List))
if pred(e) ∈b pv11_p1_propose'base(Cmd;f)
then ∃x:ℤ × Cmd
∃s:pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List)
(x ∈ pv11_p1_propose'base(Cmd;f)(pred(e))
∧ s ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(pred(e))
∧ (v = (pv11_p1_on_propose(Cmd) loc(e) x s) ∈ (pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List))))
if pred(e) ∈b pv11_p1_adopted'base(Cmd;f)
then ∃x:pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
∃s:pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List)
(x ∈ pv11_p1_adopted'base(Cmd;f)(pred(e))
∧ s ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(pred(e))
∧ (v = (pv11_p1_when_adopted(Cmd;ldrs_uid) loc(e) x s) ∈ (pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List))))
if pred(e) ∈b pv11_p1_preempted'base(Cmd;f)
then ∃x:pv11_p1_Ballot_Num()
∃s:pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List)
(x ∈ pv11_p1_preempted'base(Cmd;f)(pred(e))
∧ s ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(pred(e))
∧ (v = (pv11_p1_when_preempted(Cmd;ldrs_uid) loc(e) x s) ∈ (pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List))))
else v ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(pred(e))
fi )
Proof
Definitions occuring in Statement :
pv11_p1_LeaderState: pv11_p1_LeaderState(Cmd;ldrs_uid;mf),
pv11_p1_when_preempted: pv11_p1_when_preempted(Cmd;ldrs_uid),
pv11_p1_when_adopted: pv11_p1_when_adopted(Cmd;ldrs_uid),
pv11_p1_on_propose: pv11_p1_on_propose(Cmd),
pv11_p1_init_leader: pv11_p1_init_leader(Cmd),
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_headers_type: pv11_p1_headers_type{i:l}(Cmd),
pv11_p1_Ballot_Num: pv11_p1_Ballot_Num(),
Message: Message(f),
classrel: v ∈ X(e),
member-eclass: e ∈b X,
event-ordering+: EO+(Info),
es-first: first(e),
es-pred: pred(e),
es-loc: loc(e),
es-E: E,
Id: Id,
list: T List,
vatype: ValueAllType,
ifthenelse: if b then t else f fi ,
bool: 𝔹,
uiff: uiff(P;Q),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
product: x:A × B[x],
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
vatype: ValueAllType,
all: ∀x:A. B[x],
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,
implies: 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),
bool: 𝔹,
unit: Unit,
uiff: uiff(P;Q),
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
pv11_p1_propose'base: pv11_p1_propose'base(Cmd;mf),
encodes-msg-type: hdr encodes T,
pv11_p1_adopted'base: pv11_p1_adopted'base(Cmd;mf),
pv11_p1_preempted'base: pv11_p1_preempted'base(Cmd;mf),
classrel: v ∈ X(e),
bag-member: x ↓∈ bs,
single-valued-classrel: single-valued-classrel(es;X;T),
cand: A c∧ B
Latex:
\mforall{}Cmd:ValueAllType. \mforall{}f:pv11\_p1\_headers\_type\{i:l\}(Cmd). \mforall{}es:EO+(Message(f)). \mforall{}e:E. \mforall{}ldrs$_\mbackslash{}ff7\000Cbuid}$:Id {}\mrightarrow{} \mBbbZ{}.
\mforall{}v:pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbB{} \mtimes{} ((\mBbbZ{} \mtimes{} Cmd) List).
uiff(v \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(e);if first(e)
then v = (pv11\_p1\_init\_leader(Cmd) loc(e))
if pred(e) \mmember{}\msubb{} pv11\_p1\_propose'base(Cmd;f)
then \mexists{}x:\mBbbZ{} \mtimes{} Cmd
\mexists{}s:pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbB{} \mtimes{} ((\mBbbZ{} \mtimes{} Cmd) List)
(x \mmember{} pv11\_p1\_propose'base(Cmd;f)(pred(e))
\mwedge{} s \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(pred(e))
\mwedge{} (v = (pv11\_p1\_on\_propose(Cmd) loc(e) x s)))
if pred(e) \mmember{}\msubb{} pv11\_p1\_adopted'base(Cmd;f)
then \mexists{}x:pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List)
\mexists{}s:pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbB{} \mtimes{} ((\mBbbZ{} \mtimes{} Cmd) List)
(x \mmember{} pv11\_p1\_adopted'base(Cmd;f)(pred(e))
\mwedge{} s \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(pred(e))
\mwedge{} (v = (pv11\_p1\_when\_adopted(Cmd;ldrs$_{uid}$) loc(e) x s)))
if pred(e) \mmember{}\msubb{} pv11\_p1\_preempted'base(Cmd;f)
then \mexists{}x:pv11\_p1\_Ballot\_Num()
\mexists{}s:pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbB{} \mtimes{} ((\mBbbZ{} \mtimes{} Cmd) List)
(x \mmember{} pv11\_p1\_preempted'base(Cmd;f)(pred(e))
\mwedge{} s \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(pred(e))
\mwedge{} (v = (pv11\_p1\_when\_preempted(Cmd;ldrs$_{uid}$) loc(e) x s)))
else v \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(pred(e))
fi )
Date html generated:
2016_05_17-PM-03_35_10
Last ObjectModification:
2016_01_18-AM-11_17_41
Theory : paxos!synod
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