Nuprl Lemma : pv11_p1_acc_state_fun_eq
∀[Cmd:ValueAllType]. ∀[f:pv11_p1_headers_type{i:l}(Cmd)]. ∀[es:EO+(Message(f))]. ∀[e:E]. ∀[ldrs_uid:Id ⟶ ℤ].
(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e)
= if e ∈b pv11_p1_p1a'base(Cmd;f) (+) pv11_p1_p2a'base(Cmd;f)
then if first(e)
then pv11_p1_on_p1a(Cmd;ldrs_uid) + pv11_p1_on_p2a(Cmd;ldrs_uid) loc(e)
pv11_p1_p1a'base(Cmd;f) (+) pv11_p1_p2a'base(Cmd;f)@e
pv11_p1_init_acceptor(Cmd)
else pv11_p1_on_p1a(Cmd;ldrs_uid) + pv11_p1_on_p2a(Cmd;ldrs_uid) loc(e)
pv11_p1_p1a'base(Cmd;f) (+) pv11_p1_p2a'base(Cmd;f)@e
pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e))
fi
if first(e) then pv11_p1_init_acceptor(Cmd)
else pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e))
fi
∈ (pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)))
Proof
Definitions occuring in Statement :
pv11_p1_AcceptorStateFun: pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;mf;es;e),
pv11_p1_on_p2a: pv11_p1_on_p2a(Cmd;ldrs_uid),
pv11_p1_on_p1a: pv11_p1_on_p1a(Cmd;ldrs_uid),
pv11_p1_init_acceptor: pv11_p1_init_acceptor(Cmd),
pv11_p1_p2a'base: pv11_p1_p2a'base(Cmd;mf),
pv11_p1_p1a'base: pv11_p1_p1a'base(Cmd;mf),
pv11_p1_headers_type: pv11_p1_headers_type{i:l}(Cmd),
pv11_p1_Ballot_Num: pv11_p1_Ballot_Num(),
disjoint-union-tr: tr1 + tr2,
Message: Message(f),
disjoint-union-comb: X (+) Y,
classfun-res: 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 ,
uall: ∀[x:A]. B[x],
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,
uall: ∀[x:A]. B[x],
pv11_p1_headers_type: pv11_p1_headers_type{i:l}(Cmd),
l_all: (∀x∈L.P[x]),
all: ∀x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
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),
pv11_p1_AcceptorStateFun: pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;mf;es;e),
pv11_p1_AcceptorState: pv11_p1_AcceptorState(Cmd;ldrs_uid;mf),
state-class2: state-class2(init;tr1;X1;tr2;X2),
top: Top,
pv11_p1_p2a'base: pv11_p1_p2a'base(Cmd;mf),
pv11_p1_p1a'base: pv11_p1_p1a'base(Cmd;mf),
uiff: uiff(P;Q),
bnot: ¬bb,
assert: ↑b,
sq_stable: SqStable(P),
encodes-msg-type: hdr encodes T,
bool: 𝔹,
unit: Unit,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bor: p ∨bq,
rev_uimplies: rev_uimplies(P;Q)
Latex:
\mforall{}[Cmd:ValueAllType]. \mforall{}[f:pv11\_p1\_headers\_type\{i:l\}(Cmd)]. \mforall{}[es:EO+(Message(f))]. \mforall{}[e:E].
\mforall{}[ldrs$_{uid}$:Id {}\mrightarrow{} \mBbbZ{}].
(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;e)
= if e \mmember{}\msubb{} pv11\_p1\_p1a'base(Cmd;f) (+) pv11\_p1\_p2a'base(Cmd;f)
then if first(e)
then pv11\_p1\_on\_p1a(Cmd;ldrs$_{uid}$) + pv11\_p1\_on\_p2a(Cmd;ldrs$\mbackslash{}\000Cff5f{uid}$) loc(e)
pv11\_p1\_p1a'base(Cmd;f) (+) pv11\_p1\_p2a'base(Cmd;f)@e
pv11\_p1\_init\_acceptor(Cmd)
else pv11\_p1\_on\_p1a(Cmd;ldrs$_{uid}$) + pv11\_p1\_on\_p2a(Cmd;ldrs$\mbackslash{}\000Cff5f{uid}$) loc(e)
pv11\_p1\_p1a'base(Cmd;f) (+) pv11\_p1\_p2a'base(Cmd;f)@e
pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;pred(e))
fi
if first(e) then pv11\_p1\_init\_acceptor(Cmd)
else pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;pred(e))
fi )
Date html generated:
2016_05_17-PM-03_33_37
Last ObjectModification:
2016_01_18-AM-11_18_08
Theory : paxos!synod
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