Nuprl Lemma : pv11_p1_inc_acc
∀Cmd:ValueAllType. ∀ldrs_uid:Id ⟶ ℤ. ∀mf:pv11_p1_headers_type{i:l}(Cmd). ∀es:EO+(Message(mf)). ∀e1,e2:E.
∀zh,z:pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List).
((e1 <loc e2)
⇒ zh ∈ pv11_p1_AcceptorState(Cmd;ldrs_uid;mf)(e1)
⇒ z ∈ pv11_p1_AcceptorState(Cmd;ldrs_uid;mf)(e2)
⇒ let ballot_num1,accepted1 = zh
in let ballot_num2,accepted2 = z
in (↑(pv11_p1_leq_bnum(ldrs_uid) ballot_num1 ballot_num2)) ∧ accepted1 ≤ accepted2)
Proof
Definitions occuring in Statement :
pv11_p1_AcceptorState: pv11_p1_AcceptorState(Cmd;ldrs_uid;mf),
pv11_p1_headers_type: pv11_p1_headers_type{i:l}(Cmd),
pv11_p1_leq_bnum: pv11_p1_leq_bnum(ldrs_uid),
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,
iseg: l1 ≤ l2,
list: T List,
vatype: ValueAllType,
assert: ↑b,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
spread: spread def,
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,
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_AcceptorState: pv11_p1_AcceptorState(Cmd;ldrs_uid;mf),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
State2: State2(init;tr1;X1;tr2;X2),
refl: Refl(T;x,y.E[x; y]),
pv11_p1_leq_bnum: pv11_p1_leq_bnum(ldrs_uid),
pv11_p1_Ballot_Num: pv11_p1_Ballot_Num(),
pv11_p1_leq_bnum': pv11_p1_leq_bnum'(ldrs_uid),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
or: P ∨ Q,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
assert: ↑b,
trans: Trans(T;x,y.E[x; y]),
cand: A c∧ B,
pv11_p1_on_p1a: pv11_p1_on_p1a(Cmd;ldrs_uid),
pv11_p1_max_bnum: pv11_p1_max_bnum(ldrs_uid),
bool: 𝔹,
unit: Unit,
sq_type: SQType(T),
bnot: ¬bb,
pv11_p1_on_p2a: pv11_p1_on_p2a(Cmd;ldrs_uid),
let: let,
pv11_p1_add_if_new: pv11_p1_add_if_new(),
pv11_p1_p1a'base: pv11_p1_p1a'base(Cmd;mf),
encodes-msg-type: hdr encodes T,
pv11_p1_p2a'base: pv11_p1_p2a'base(Cmd;mf),
sq_stable: SqStable(P)
Latex:
\mforall{}Cmd:ValueAllType. \mforall{}ldrs$_{uid}$:Id {}\mrightarrow{} \mBbbZ{}. \mforall{}mf:pv11\_p1\_headers\_type\{i:l\}(Cmd). \mforall{}e\000Cs:EO+(Message(mf)).
\mforall{}e1,e2:E. \mforall{}zh,z:pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List).
((e1 <loc e2)
{}\mRightarrow{} zh \mmember{} pv11\_p1\_AcceptorState(Cmd;ldrs$_{uid}$;mf)(e1)
{}\mRightarrow{} z \mmember{} pv11\_p1\_AcceptorState(Cmd;ldrs$_{uid}$;mf)(e2)
{}\mRightarrow{} let ballot$_{num1}$,accepted1 = zh
in let ballot$_{num2}$,accepted2 = z
in (\muparrow{}(pv11\_p1\_leq\_bnum(ldrs$_{uid}$) ballot$_{num1}$\000C ballot$_{num2}$)) \mwedge{} accepted1 \mleq{} accepted2)
Date html generated:
2016_05_17-PM-02_52_07
Last ObjectModification:
2016_01_18-AM-11_24_58
Theory : paxos!synod
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