Nuprl Lemma : pv11_p1_A4_C1
∀Cmd:ValueAllType. ∀f:pv11_p1_headers_type{i:l}(Cmd). ∀es:EO+(Message(f)). ∀e1,e2:E. ∀accpts,ldrs:bag(Id).
∀ldrs_uid:Id ⟶ ℤ. ∀reps:bag(Id). ∀bnum1,bnum2:pv11_p1_Ballot_Num(). ∀accepted1,accepted2:(pv11_p1_Ballot_Num()
× ℤ
× Cmd) List. ∀i1,i2,l:Id.
∀b:pv11_p1_Ballot_Num(). ∀s:ℤ. ∀p1,p2:Cmd.
(Inj(Id;ℤ;ldrs_uid)
⇒ pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)
⇒ ((<bnum1, accepted1> ∈ pv11_p1_AcceptorState(Cmd;ldrs_uid;f)(e1)
∧ <bnum2, accepted2> ∈ pv11_p1_AcceptorState(Cmd;ldrs_uid;f)(e2)
∧ (<b, s, p1> ∈ accepted1)
∧ (<b, s, p2> ∈ accepted2))
∨ (pv11_p1_p2a'send(Cmd;f) i1 <l, b, s, p1> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e1)
∧ pv11_p1_p2a'send(Cmd;f) i2 <l, b, s, p2> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e2)))
⇒ (p1 = p2 ∈ Cmd))
Proof
Definitions occuring in Statement :
pv11_p1_p2a'send: pv11_p1_p2a'send(Cmd;f),
pv11_p1_message-constraint: pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; mf; es),
pv11_p1_main: pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;mf),
pv11_p1_AcceptorState: pv11_p1_AcceptorState(Cmd;ldrs_uid;mf),
pv11_p1_headers_type: pv11_p1_headers_type{i:l}(Cmd),
pv11_p1_Ballot_Num: pv11_p1_Ballot_Num(),
msg-interface: Interface,
Message: Message(f),
classrel: v ∈ X(e),
event-ordering+: EO+(Info),
es-E: E,
Id: Id,
l_member: (x ∈ l),
list: T List,
inject: Inj(A;B;f),
vatype: ValueAllType,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
pair: <a, b>,
product: x:A × B[x],
int: ℤ,
equal: s = t ∈ T,
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),
or: P ∨ Q,
strongwellfounded: SWellFounded(R[x; y]),
exists: ∃x:A. B[x],
nat: ℕ,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
decidable: Dec(P),
pv11_p1_message-constraint: pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; mf; es),
eo-msg-interface-constraint: eo-msg-interface-constraint(es;X;hdrs;f),
pv11_p1_p2a'base: pv11_p1_p2a'base(Cmd;mf),
uiff: uiff(P;Q),
es-header: header(e),
pv11_p1_headers_no_inputs: pv11_p1_headers_no_inputs(),
make-msg-interface: make-msg-interface(i;l;m),
pv11_p1_p2a'send: pv11_p1_p2a'send(Cmd;f),
mk-msg: mk-msg(auth;hdr;val),
make-Msg: make-Msg(hdr;val),
mk-msg-interface: mk-msg-interface(l;m),
sq_type: SQType(T),
pv11_p1_init_active: pv11_p1_init_active(),
pv11_p1_eq_bnums: pv11_p1_eq_bnums(),
pv11_p1_Ballot_Num: pv11_p1_Ballot_Num(),
Id: Id,
pv11_p1_mk_bnum: pv11_p1_mk_bnum(),
outl: outl(x),
isl: isl(x),
assert: ↑b,
pi1: fst(t),
pi2: snd(t),
cand: A c∧ B,
has-es-info-type: has-es-info-type(es;e;f;T),
pv11_p1_propose'base: pv11_p1_propose'base(Cmd;mf),
encodes-msg-type: hdr encodes T,
equal-info-body: v = body(e),
pv11_p1_adopted'base: pv11_p1_adopted'base(Cmd;mf),
es-locl: (e <loc e'),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
single-valued-classrel: single-valued-classrel(es;X;T),
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
strict4: strict4(F),
has-value: (a)↓,
es-le: e ≤loc e' ,
bool: 𝔹,
unit: Unit,
bnot: ¬bb,
pv11_p1_preempted'base: pv11_p1_preempted'base(Cmd;mf),
pv11_p1_when_adopted: pv11_p1_when_adopted(Cmd;ldrs_uid),
spreadn: spread3,
let: let,
pv11_p1_LeaderStateFun: pv11_p1_LeaderStateFun(Cmd;ldrs_uid;mf;es;e),
classfun: X(e),
sv-bag-only: sv-bag-only(b),
hd: hd(l),
pv11_p1_LeaderState: pv11_p1_LeaderState(Cmd;ldrs_uid;mf),
memory-class3: memory-class3(init;tr1;X1;tr2;X2;tr3;X3),
loop-class-memory: loop-class-memory(X;init),
primed-class-opt: Prior(X)?b,
es-local-pred: last(P),
es-first: first(e),
bor: p ∨bq,
es-eq-E: e = e',
es-eq: es-eq(es),
eq_id: a = b,
id-deq: IdDeq,
atom2-deq: Atom2Deq,
eq_atom: eq_atom$n(x;y),
es-loc: loc(e),
record-select: r.x,
label: ...$L... t
Latex:
\mforall{}Cmd:ValueAllType. \mforall{}f:pv11\_p1\_headers\_type\{i:l\}(Cmd). \mforall{}es:EO+(Message(f)). \mforall{}e1,e2:E.
\mforall{}accpts,ldrs:bag(Id). \mforall{}ldrs$_{uid}$:Id {}\mrightarrow{} \mBbbZ{}. \mforall{}reps:bag(Id). \mforall{}bnum1,bnum2:pv11\_p1\000C\_Ballot\_Num().
\mforall{}accepted1,accepted2:(pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List. \mforall{}i1,i2,l:Id. \mforall{}b:pv11\_p1\_Ballot\_Num().
\mforall{}s:\mBbbZ{}. \mforall{}p1,p2:Cmd.
(Inj(Id;\mBbbZ{};ldrs$_{uid}$)
{}\mRightarrow{} pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid\mbackslash{}ff7\000Cd$; reps; f; es)
{}\mRightarrow{} ((<bnum1, accepted1> \mmember{} pv11\_p1\_AcceptorState(Cmd;ldrs$_{uid}$;f)(e1)
\mwedge{} <bnum2, accepted2> \mmember{} pv11\_p1\_AcceptorState(Cmd;ldrs$_{uid}$;f)(e2)
\mwedge{} (<b, s, p1> \mmember{} accepted1)
\mwedge{} (<b, s, p2> \mmember{} accepted2))
\mvee{} (pv11\_p1\_p2a'send(Cmd;f) i1 <l, b, s, p1> \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$_{u\000Cid}$;reps;f)(e1)
\mwedge{} pv11\_p1\_p2a'send(Cmd;f) i2 <l, b, s, p2> \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$_{\000Cuid}$;reps;f)(e2)))
{}\mRightarrow{} (p1 = p2))
Date html generated:
2016_05_17-PM-04_03_48
Last ObjectModification:
2016_01_18-AM-11_27_46
Theory : paxos!synod
Home
Index