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 :
assert: ↑b,
ifthenelse: if b then t else f fi ,
deq-member: x ∈b L),
reduce: reduce(f;k;as),
pv11_p1_headers_no_inputs: pv11_p1_headers_no_inputs(),
cons: [a / b],
bor: p ∨bq,
list-deq: list-deq(eq),
band: p ∧b q,
atom-deq: AtomDeq,
eq_atom: x =a y,
btrue: tt,
bfalse: ff,
nil: [],
it: ⋅,
null: null(as),
true: True,
member: t ∈ T
Lemmas :
es-causl-swellfnd,
event-ordering+_subtype,
Message_wf,
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
int_seg_wf,
length_wf,
map_wf,
pv11_p1_AcceptorStateFun_wf,
pv11_p1_LeaderStateFun_wf,
bool_wf,
list_wf,
pv11_p1_ScoutStateFun_wf,
eo-forward_wf,
member-eo-forward-E,
equal_wf,
Id_wf,
es-loc_wf,
bag_wf,
es-le_wf,
es-E_wf,
es-info-body_wf,
has-es-info-type_wf,
equal-wf-T-base,
name_wf,
es-header_wf,
int_seg_subtype-nat,
decidable__le,
subtract_wf,
false_wf,
not-ge-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
decidable__equal_int,
subtype_rel-int_seg,
le_weakening,
int_seg_properties,
le_wf,
nat_wf,
zero-le-nat,
lelt_wf,
es-causl_wf,
l_all_iff,
l_member_wf,
pv11_p1_valid-proposal_wf,
subtype_base_sq,
list_subtype_base,
atom_subtype_base,
assert-deq-member,
pv11_p1_headers_no_inputs_wf,
cons_wf,
nil_wf,
es-info-mk-msg,
event-ordering+_cumulative,
es-info-auth_wf,
subtype_rel_transitivity,
es-info-type_wf,
subtype_rel_weakening,
ext-eq_weakening,
classrel_wf,
msg-interface_wf,
pv11_p1_main_wf,
make-msg-interface_wf,
pair-eta,
subtype_rel_product,
top_wf,
subtype_top,
mk-msg_wf,
decidable__lt,
not-equal-2,
le-add-cancel-alt,
not-le-2,
sq_stable__le,
add-mul-special,
zero-mul,
pi1_wf_top,
pv11_p1-p2a,
member_map,
pv11_p1_acc_fun_p2a_pv,
base-classrel-equal,
cons_wf_listp,
listp_wf,
es-causl_weakening,
subtype_rel_wf,
pv11_p1_valid-p2a-message_wf,
and_wf,
member_wf,
es-causle_weakening,
es-causle_wf,
pv11_p1_propose'base_wf,
pv11_p1_valid-proposal-transitivity,
bag-member-sq-list-member,
member-map,
subtype_rel_list,
l_member_subtype,
es-causle_weakening_locl,
pi2_wf,
pv11_p1_ldr_fun_proposal3,
es-le_weakening,
pv11_p1_valid-proposals_wf,
pv11_p1_scout_fun_from_acc,
eo-forward-header,
eo-forward-has-es-info-type,
eo-forward-info-body,
eo-forward-le,
eo-forward-E-subtype,
squash_wf,
true_wf,
event-ordering+_wf,
iff_weakening_equal,
subtype_rel_self,
pv11_p1_valid-p1b-message_wf,
pv11_p1-p1b,
pv11_p1-adopted,
es-init_wf,
es-init-le,
es-first-init,
assert_of_tt,
map_functionality,
pv11_p1_Ballot_Num_wf,
subtype_rel_dep_function,
vatype_wf,
pv11_p1_headers_type_wf,
eo-forward-trivial,
decidable__es-le,
pv11_p1_upd_bnum_wf
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:
2015_07_23-PM-05_04_25
Last ObjectModification:
2015_02_04-AM-07_59_58
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