Nuprl Lemma : pv11_p1-p2a
∀[Cmd:{T:Type| valueall-type(T)} ]. ∀[accpts,ldrs:bag(Id)]. ∀[ldrs_uid:Id ⟶ ℤ]. ∀[reps:bag(Id)].
∀[mf:pv11_p1_headers_type{i:l}(Cmd)]. ∀[es:EO+(Message(mf))]. ∀[e:E]. ∀[d:ℤ]. ∀[i:Id]. ∀[auth:𝔹]. ∀[i1:Id].
∀[p:pv11_p1_Ballot_Num()]. ∀[k:ℤ]. ∀[v:Cmd].
(<d, i, mk-msg(auth;``pv11_p1 p2a``;<i1, p, k, v>)> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;mf)(e)
⇐⇒ loc(e) ↓∈ ldrs
∧ i ↓∈ accpts
∧ (d = 0 ∈ ℤ)
∧ auth = pv11_p1_init_active()
∧ (i1 = loc(e) ∈ Id)
∧ (↓(((header(e) = [propose] ∈ Name) ∧ has-es-info-type(es;e;mf;ℤ × Cmd))
∧ ((↑(fst(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;mf;es;e)))))
∧ (¬↑(pv11_p1_in_domain(Cmd) (fst(msgval(e))) (snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;mf;es;e)))))))
∧ (p = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;mf;es;e))) ∈ pv11_p1_Ballot_Num())
∧ (<k, v> = msgval(e) ∈ (ℤ × Cmd)))
∨ (((header(e) = ``pv11_p1 adopted`` ∈ Name)
∧ has-es-info-type(es;e;mf;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)))
∧ ((fst(msgval(e))) = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;mf;es;e))) ∈ pv11_p1_Ballot_Num())
∧ ((<k, v> ↓∈ snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;mf;es;e)))
∧ (¬(∃p2:Cmd. (<k, p2> ∈ pv11_p1_pmax(Cmd;ldrs_uid) (snd(msgval(e)))))))
∨ (∃v2:pv11_p1_Ballot_Num()
(<v2, k, v> ↓∈ snd(msgval(e))
∧ (¬(∃z5:pv11_p1_Ballot_Num(). ∃z8:Cmd. ((↑(v2 < z5)) ∧ (<z5, k, z8> ∈ snd(msgval(e)))))))))
∧ (p = (fst(msgval(e))) ∈ pv11_p1_Ballot_Num()))))
Proof
Definitions occuring in Statement :
pv11_p1_main: pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;mf),
pv11_p1_LeaderStateFun: pv11_p1_LeaderStateFun(Cmd;ldrs_uid;mf;es;e),
pv11_p1_init_active: pv11_p1_init_active(),
pv11_p1_in_domain: pv11_p1_in_domain(Cmd),
pv11_p1_pmax: pv11_p1_pmax(Cmd;ldrs_uid),
pv11_p1_headers_type: pv11_p1_headers_type{i:l}(Cmd),
pv11_p1_lt_bnum: pv11_p1_lt_bnum(ldrs_uid),
pv11_p1_Ballot_Num: pv11_p1_Ballot_Num(),
msg-interface: Interface,
mk-msg: mk-msg(auth;hdr;val),
es-info-body: msgval(e),
has-es-info-type: has-es-info-type(es;e;f;T),
es-header: header(e),
Message: Message(f),
classrel: v ∈ X(e),
event-ordering+: EO+(Info),
es-loc: loc(e),
es-E: E,
Id: Id,
name: Name,
l_member: (x ∈ l),
cons: [a / b],
nil: [],
list: T List,
valueall-type: valueall-type(T),
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
squash: ↓T,
or: P ∨ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
pair: <a, b>,
product: x:A × B[x],
natural_number: $n,
int: ℤ,
token: "$token",
universe: Type,
equal: s = t ∈ T,
bag-member: x ↓∈ bs,
bag: bag(T)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
pv11_p1_headers_type: pv11_p1_headers_type{i:l}(Cmd),
subtype_rel: A ⊆r B,
listp: A List+,
name: Name,
prop: ℙ,
and: P ∧ Q,
implies: P ⇒ Q,
sq_stable: SqStable(P),
l_all: (∀x∈L.P[x]),
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
vatype: ValueAllType,
so_apply: x[s],
iff: P ⇐⇒ Q,
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),
bag-member: x ↓∈ bs,
make-msg-interface: make-msg-interface(i;l;m),
classrel: v ∈ X(e),
has-es-info-type: has-es-info-type(es;e;f;T),
top: Top,
pi1: fst(t),
pi2: snd(t),
exists: ∃x:A. B[x],
or: P ∨ Q,
cand: A c∧ B,
nat: ℕ,
pv11_p1_headers_no_rep: pv11_p1_headers_no_rep(),
no_repeats: no_repeats(T;l),
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
pv11_p1_init_active: pv11_p1_init_active()
Latex:
\mforall{}[Cmd:\{T:Type| valueall-type(T)\} ]. \mforall{}[accpts,ldrs:bag(Id)]. \mforall{}[ldrs$_{uid}$:Id {}\mrightarrow{}\000C \mBbbZ{}]. \mforall{}[reps:bag(Id)].
\mforall{}[mf:pv11\_p1\_headers\_type\{i:l\}(Cmd)]. \mforall{}[es:EO+(Message(mf))]. \mforall{}[e:E]. \mforall{}[d:\mBbbZ{}]. \mforall{}[i:Id]. \mforall{}[auth:\mBbbB{}].
\mforall{}[i1:Id]. \mforall{}[p:pv11\_p1\_Ballot\_Num()]. \mforall{}[k:\mBbbZ{}]. \mforall{}[v:Cmd].
(<d, i, mk-msg(auth;``pv11\_p1 p2a``;<i1, p, k, v>)> \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$_\mbackslash{}\000Cff7buid}$;reps;mf)(e)
\mLeftarrow{}{}\mRightarrow{} loc(e) \mdownarrow{}\mmember{} ldrs
\mwedge{} i \mdownarrow{}\mmember{} accpts
\mwedge{} (d = 0)
\mwedge{} auth = pv11\_p1\_init\_active()
\mwedge{} (i1 = loc(e))
\mwedge{} (\mdownarrow{}(((header(e) = [propose]) \mwedge{} has-es-info-type(es;e;mf;\mBbbZ{} \mtimes{} Cmd))
\mwedge{} ((\muparrow{}(fst(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;mf;es;e)))))
\mwedge{} (\mneg{}\muparrow{}(pv11\_p1\_in\_domain(Cmd) (fst(msgval(e)))
(snd(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;mf;es;e))))))\000C)
\mwedge{} (p = (fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;mf;es;e))))
\mwedge{} (<k, v> = msgval(e)))
\mvee{} (((header(e) = ``pv11\_p1 adopted``)
\mwedge{} has-es-info-type(es;e;mf;pv11\_p1\_Ballot\_Num()
\mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List)))
\mwedge{} ((fst(msgval(e))) = (fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;m\000Cf;es;e))))
\mwedge{} ((<k, v> \mdownarrow{}\mmember{} snd(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;mf;es;\000Ce)))
\mwedge{} (\mneg{}(\mexists{}p2:Cmd. (<k, p2> \mmember{} pv11\_p1\_pmax(Cmd;ldrs$_{uid}$) (snd(msgva\000Cl(e)))))))
\mvee{} (\mexists{}v2:pv11\_p1\_Ballot\_Num()
(<v2, k, v> \mdownarrow{}\mmember{} snd(msgval(e))
\mwedge{} (\mneg{}(\mexists{}z5:pv11\_p1\_Ballot\_Num()
\mexists{}z8:Cmd. ((\muparrow{}(v2 < z5)) \mwedge{} (<z5, k, z8> \mmember{} snd(msgval(e)))))))))
\mwedge{} (p = (fst(msgval(e)))))))
Date html generated:
2016_05_17-PM-03_10_05
Last ObjectModification:
2016_04_03-PM-09_39_20
Theory : paxos!synod
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