Nuprl Lemma : pv11_p1-decision
∀[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]. ∀[k:ℤ]. ∀[v:Cmd].
(<d, i, mk-msg(auth;[decision];<k, v>)> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;mf)(e)
⇐⇒ loc(e) ↓∈ ldrs
∧ (↓∃e':{e':E| e' ≤loc e }
((header(e) = ``pv11_p1 p2b`` ∈ Name)
∧ has-es-info-type(es;e;mf;Id × pv11_p1_Ballot_Num() × ℤ × pv11_p1_Ballot_Num())
∧ (d = 0 ∈ ℤ)
∧ ((fst(snd(msgval(e)))) = (snd(snd(snd(msgval(e))))) ∈ pv11_p1_Ballot_Num())
∧ #(pv11_p1_CommanderStateFun(Cmd;accpts;mf;fst(snd(msgval(e)));fst(snd(snd(msgval(e))));es.e';e)) < ...
∧ i ↓∈ reps
∧ auth = pv11_p1_init_active()
∧ (k = (fst(snd(snd(msgval(e))))) ∈ ℤ)
∧ ((((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')))))))
∧ ((fst(snd(msgval(e)))) = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;mf;es;e'))) ∈ pv11_p1_Ballot_Num())
∧ (<fst(snd(snd(msgval(e)))), 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())
∧ ((<fst(snd(snd(msgval(e)))), v> ↓∈ snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;mf;es;e')))
∧ (¬(∃p2:Cmd. (<fst(snd(snd(msgval(e)))), p2> ∈ pv11_p1_pmax(Cmd;ldrs_uid) (snd(msgval(e')))))))
∨ (∃v2:pv11_p1_Ballot_Num()
(<v2, fst(snd(snd(msgval(e)))), v> ↓∈ snd(msgval(e'))
∧ (¬(∃z5:pv11_p1_Ballot_Num()
∃z8:Cmd. ((↑(v2 < z5)) ∧ (<z5, fst(snd(snd(msgval(e)))), z8> ∈ snd(msgval(e')))))))))
∧ ((fst(snd(msgval(e)))) = (fst(msgval(e'))) ∈ pv11_p1_Ballot_Num())))
∧ (no ((pv11_p1_commander_output(Cmd;accpts;reps;mf) <fst(snd(msgval(e))), fst(snd(snd(msgval(e)))), v> o pv1\000C1_p1_p2b'base(Cmd;mf)) o
pv11_p1_CommanderState(Cmd;accpts;mf) (fst(snd(msgval(e)))) (fst(snd(snd(msgval(e)))))) between e' and
e))))
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_commander_output: pv11_p1_commander_output(Cmd;accpts;reps;mf),
pv11_p1_CommanderStateFun: pv11_p1_CommanderStateFun(Cmd;accpts;mf;x;zzs;es;e),
pv11_p1_CommanderState: pv11_p1_CommanderState(Cmd;accpts;mf),
pv11_p1_init_active: pv11_p1_init_active(),
pv11_p1_p2b'base: pv11_p1_p2b'base(Cmd;mf),
pv11_p1_threshold: pv11_p1_threshold(accpts),
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),
eclass2: (X o Y),
eclass1: (f o X),
no-classrel-in-interval: (no X between start and e),
classrel: v ∈ X(e),
eo-forward: eo.e,
event-ordering+: EO+(Info),
es-le: e ≤loc e' ,
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,
less_than: a < 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-size: #(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),
cand: A c∧ B,
has-es-info-type: has-es-info-type(es;e;f;T),
pi1: fst(t),
pi2: snd(t),
nat: ℕ,
top: Top,
exists: ∃x:A. B[x],
or: P ∨ Q,
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),
hint: hint(t),
uiff: uiff(P;Q),
assert: ↑b,
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{}[k:\mBbbZ{}]. \mforall{}[v:Cmd].
(<d, i, mk-msg(auth;[decision];<k, v>)> \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$_{uid}\000C$;reps;mf)(e)
\mLeftarrow{}{}\mRightarrow{} loc(e) \mdownarrow{}\mmember{} ldrs
\mwedge{} (\mdownarrow{}\mexists{}e':\{e':E| e' \mleq{}loc e \}
((header(e) = ``pv11\_p1 p2b``)
\mwedge{} has-es-info-type(es;e;mf;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} pv11\_p1\_Ballot\_Num())
\mwedge{} (d = 0)
\mwedge{} ((fst(snd(msgval(e)))) = (snd(snd(snd(msgval(e))))))
\mwedge{} \#(pv11\_p1\_CommanderStateFun(Cmd;accpts;mf;fst(snd(msgval(e)));fst(...);es.e';e)) < ...
\mwedge{} i \mdownarrow{}\mmember{} reps
\mwedge{} auth = pv11\_p1\_init\_active()
\mwedge{} (k = (fst(snd(snd(msgval(e))))))
\mwedge{} ((((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{} ((fst(snd(msgval(e)))) = (fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}\mbackslash{}\000Cff24;mf;es;e'))))
\mwedge{} (<fst(snd(snd(msgval(e)))), v> = msgval(e')))
\mvee{} (((header(e') = ``pv11\_p1 adopted``)
\mwedge{} has-es-info-type(es;e';mf;pv11\_p1\_Ballot\_Num()
\mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List)))
\mwedge{} ((fst(msgval(e'))) = (fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}\mbackslash{}ff\000C24;mf;es;e'))))
\mwedge{} ((<fst(snd(snd(msgval(e))))
, v
> \mdownarrow{}\mmember{} snd(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;mf;es;e'\000C)))
\mwedge{} (\mneg{}(\mexists{}p2:Cmd
(<fst(snd(snd(msgval(e)))), p2> \mmember{} pv11\_p1\_pmax(Cmd;ldrs$_{uid\mbackslash{}ff7\000Cd$)
(snd(msgval(e')))))))
\mvee{} (\mexists{}v2:pv11\_p1\_Ballot\_Num()
(<v2, fst(snd(snd(msgval(e)))), v> \mdownarrow{}\mmember{} snd(msgval(e'))
\mwedge{} (\mneg{}(\mexists{}z5:pv11\_p1\_Ballot\_Num()
\mexists{}z8:Cmd
((\muparrow{}(v2 < z5))
\mwedge{} (<z5, fst(snd(snd(msgval(e)))), z8> \mmember{} snd(msgval(e')))))))))
\mwedge{} ((fst(snd(msgval(e)))) = (fst(msgval(e'))))))
\mwedge{} (no ((pv11\_p1\_commander\_output(Cmd;accpts;reps;mf)
<fst(snd(msgval(e))), fst(snd(snd(msgval(e)))), v> o pv11\_p1\_p2b'base(Cmd;mf)) o
pv11\_p1\_CommanderState(Cmd;accpts;mf) (fst(snd(msgval(e))))
(fst(snd(snd(msgval(e)))))) between e' and e))))
Date html generated:
2016_05_17-PM-03_12_14
Last ObjectModification:
2016_04_03-PM-09_39_32
Theory : paxos!synod
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