Nuprl Lemma : pv11_p1_ord_comm
∀Cmd:ValueAllType. ∀accpts:bag(Id). ∀mf:pv11_p1_headers_type{i:l}(Cmd). ∀es:EO+(Message(mf)). ∀e1,e2:E.
∀b:pv11_p1_Ballot_Num(). ∀s:ℤ. ∀waitfor1,waitfor2:bag(Id).
((e1 <loc e2)
⇒ waitfor1 ∈ pv11_p1_CommanderState(Cmd;accpts;mf) b s(e1)
⇒ waitfor2 ∈ pv11_p1_CommanderState(Cmd;accpts;mf) b s(e2)
⇒ sub-bag(Id;waitfor2;waitfor1))
Proof
Definitions occuring in Statement :
pv11_p1_CommanderState: pv11_p1_CommanderState(Cmd;accpts;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),
event-ordering+: EO+(Info),
es-locl: (e <loc e'),
es-E: E,
Id: Id,
vatype: ValueAllType,
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
int: ℤ,
sub-bag: sub-bag(T;as;bs),
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),
pv11_p1_CommanderState: pv11_p1_CommanderState(Cmd;accpts;mf),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
State1: State1(init;tr;X),
refl: Refl(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
pv11_p1_on_p2b: pv11_p1_on_p2b(),
spreadn: spread4,
bool: 𝔹,
unit: Unit,
uiff: uiff(P;Q),
top: Top,
or: P ∨ Q,
pv11_p1_p2b'base: pv11_p1_p2b'base(Cmd;mf),
encodes-msg-type: hdr encodes T
Latex:
\mforall{}Cmd:ValueAllType. \mforall{}accpts:bag(Id). \mforall{}mf:pv11\_p1\_headers\_type\{i:l\}(Cmd). \mforall{}es:EO+(Message(mf)).
\mforall{}e1,e2:E. \mforall{}b:pv11\_p1\_Ballot\_Num(). \mforall{}s:\mBbbZ{}. \mforall{}waitfor1,waitfor2:bag(Id).
((e1 <loc e2)
{}\mRightarrow{} waitfor1 \mmember{} pv11\_p1\_CommanderState(Cmd;accpts;mf) b s(e1)
{}\mRightarrow{} waitfor2 \mmember{} pv11\_p1\_CommanderState(Cmd;accpts;mf) b s(e2)
{}\mRightarrow{} sub-bag(Id;waitfor2;waitfor1))
Date html generated:
2016_05_17-PM-02_53_48
Last ObjectModification:
2016_01_18-AM-11_22_50
Theory : paxos!synod
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