Nuprl Lemma : pv11_p1_validity-lemma
∀[Cmd:{T:Type| valueall-type(T)} ]
∀f:pv11_p1_headers_type{i:l}(Cmd). ∀es:EO+(Message(f)). ∀accpts,ldrs:bag(Id). ∀ldrs_uid:Id ⟶ ℤ. ∀reps:bag(Id).
(pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)
⇒ (∀e:E
(pv11_p1_valid-AcceptorState(Cmd;ldrs_uid;es;e;f)
∧ pv11_p1_valid-LeaderState(Cmd;ldrs_uid;es;e;f)
∧ (∀b:pv11_p1_Ballot_Num(). ∀start:E. (start ≤loc e ⇒ pv11_p1_valid-ScoutState(Cmd;accpts;b;start;es;e;f)))
∧ ((header(e) = ``pv11_p1 adopted`` ∈ Name)
⇒ has-es-info-type(es;e;f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
⇒ pv11_p1_valid-adopted-message(Cmd;es;e;f))
∧ ((header(e) = ``pv11_p1 p1b`` ∈ Name)
⇒ has-es-info-type(es;e;f;Id
× pv11_p1_Ballot_Num()
× pv11_p1_Ballot_Num()
× ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
⇒ pv11_p1_valid-p1b-message(Cmd;es;e;f))
∧ ((header(e) = ``pv11_p1 p2a`` ∈ Name)
⇒ has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)
⇒ pv11_p1_valid-p2a-message(Cmd;es;e;f)))))
Proof
Definitions occuring in Statement :
pv11_p1_valid-p2a-message: pv11_p1_valid-p2a-message(Cmd;es;e;f),
pv11_p1_valid-p1b-message: pv11_p1_valid-p1b-message(Cmd;es;e;f),
pv11_p1_valid-adopted-message: pv11_p1_valid-adopted-message(Cmd;es;e;f),
pv11_p1_valid-ScoutState: pv11_p1_valid-ScoutState(Cmd;accpts;b;start;es;e;f),
pv11_p1_valid-AcceptorState: pv11_p1_valid-AcceptorState(Cmd;ldrs_uid;es;e;f),
pv11_p1_valid-LeaderState: pv11_p1_valid-LeaderState(Cmd;ldrs_uid;es;e;f),
pv11_p1_message-constraint: pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; mf; es),
pv11_p1_headers_type: pv11_p1_headers_type{i:l}(Cmd),
pv11_p1_Ballot_Num: pv11_p1_Ballot_Num(),
has-es-info-type: has-es-info-type(es;e;f;T),
es-header: header(e),
Message: Message(f),
event-ordering+: EO+(Info),
es-le: e ≤loc e' ,
es-E: E,
Id: Id,
name: Name,
cons: [a / b],
nil: [],
list: T List,
valueall-type: valueall-type(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
product: x:A × B[x],
int: ℤ,
token: "$token",
universe: Type,
equal: s = t ∈ T,
bag: bag(T)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
pv11_p1_headers_type: pv11_p1_headers_type{i:l}(Cmd),
subtype_rel: A ⊆r B,
listp: A List+,
name: Name,
vatype: ValueAllType,
prop: ℙ,
and: P ∧ Q,
sq_stable: SqStable(P),
l_all: (∀x∈L.P[x]),
so_lambda: λ2x.t[x],
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),
pv11_p1_valid-AcceptorState: pv11_p1_valid-AcceptorState(Cmd;ldrs_uid;es;e;f),
pv11_p1_valid-proposals: pv11_p1_valid-proposals(Cmd;es;e;L;f),
pv11_p1_valid-proposal: pv11_p1_valid-proposal(Cmd;es;e;p;f),
pi2: snd(t),
pv11_p1_valid-LeaderState: pv11_p1_valid-LeaderState(Cmd;ldrs_uid;es;e;f),
pv11_p1_valid-ScoutState: pv11_p1_valid-ScoutState(Cmd;accpts;b;start;es;e;f),
pv11_p1_valid-adopted-message: pv11_p1_valid-adopted-message(Cmd;es;e;f),
has-es-info-type: has-es-info-type(es;e;f;T),
pv11_p1_valid-p1b-message: pv11_p1_valid-p1b-message(Cmd;es;e;f),
pv11_p1_valid-p2a-message: pv11_p1_valid-p2a-message(Cmd;es;e;f),
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),
or: P ∨ Q,
cand: A c∧ B,
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),
sq_type: SQType(T),
pv11_p1_headers_no_inputs: pv11_p1_headers_no_inputs(),
es-info-body: msgval(e),
es-info-auth: es-info-auth(es;e),
make-msg-interface: make-msg-interface(i;l;m),
pi1: fst(t),
pv11_p1_p2a'base: pv11_p1_p2a'base(Cmd;mf),
encodes-msg-type: hdr encodes T,
uiff: uiff(P;Q),
es-le: e ≤loc e' ,
es-info-type: es-info-type(es;e;f),
msg-type: msg-type(msg;f),
es-header: header(e),
equal-info-body: v = body(e),
pv11_p1_propose'base: pv11_p1_propose'base(Cmd;mf),
rev_uimplies: rev_uimplies(P;Q),
pv11_p1_adopted'base: pv11_p1_adopted'base(Cmd;mf),
pv11_p1_p1b'base: pv11_p1_p1b'base(Cmd;mf)
Latex:
\mforall{}[Cmd:\{T:Type| valueall-type(T)\} ]
\mforall{}f:pv11\_p1\_headers\_type\{i:l\}(Cmd). \mforall{}es:EO+(Message(f)). \mforall{}accpts,ldrs:bag(Id). \mforall{}ldrs$_{\000Cuid}$:Id {}\mrightarrow{} \mBbbZ{}.
\mforall{}reps:bag(Id).
(pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid\mbackslash{}ff7\000Cd$; reps; f; es)
{}\mRightarrow{} (\mforall{}e:E
(pv11\_p1\_valid-AcceptorState(Cmd;ldrs$_{uid}$;es;e;f)
\mwedge{} pv11\_p1\_valid-LeaderState(Cmd;ldrs$_{uid}$;es;e;f)
\mwedge{} (\mforall{}b:pv11\_p1\_Ballot\_Num(). \mforall{}start:E.
(start \mleq{}loc e {}\mRightarrow{} pv11\_p1\_valid-ScoutState(Cmd;accpts;b;start;es;e;f)))
\mwedge{} ((header(e) = ``pv11\_p1 adopted``)
{}\mRightarrow{} has-es-info-type(es;e;f;pv11\_p1\_Ballot\_Num()
\mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List))
{}\mRightarrow{} pv11\_p1\_valid-adopted-message(Cmd;es;e;f))
\mwedge{} ((header(e) = ``pv11\_p1 p1b``)
{}\mRightarrow{} has-es-info-type(es;e;f;Id
\mtimes{} pv11\_p1\_Ballot\_Num()
\mtimes{} pv11\_p1\_Ballot\_Num()
\mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List))
{}\mRightarrow{} pv11\_p1\_valid-p1b-message(Cmd;es;e;f))
\mwedge{} ((header(e) = ``pv11\_p1 p2a``)
{}\mRightarrow{} has-es-info-type(es;e;f;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd)
{}\mRightarrow{} pv11\_p1\_valid-p2a-message(Cmd;es;e;f)))))
Date html generated:
2016_05_17-PM-04_15_13
Last ObjectModification:
2016_01_18-AM-11_28_37
Theory : paxos!synod
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