Nuprl Lemma : pv11_p1_decision_from_p2a_acc
∀[Cmd:ValueAllType]. ∀[f:pv11_p1_headers_type{i:l}(Cmd)]. ∀[es:EO+(Message(f))]. ∀[accpts,ldrs:bag(Id)].
∀[ldrs_uid:Id ─→ ℤ]. ∀[reps:bag(Id)]. ∀[e:E]. ∀[s:ℤ]. ∀[c:Cmd]. ∀[a:Id].
(Inj(Id;ℤ;ldrs_uid)
⇒ pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)
⇒ pv11_p1_decision{i:l}(Cmd;f;accpts;ldrs;ldrs_uid;reps;es;e;s;c)
⇒ has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × pv11_p1_Ballot_Num())
⇒ a ↓∈ accpts
⇒ (↓∃e':E
((e' < e)
∧ loc(e') ↓∈ accpts
∧ (s = (fst(snd(snd(msgval(e))))) ∈ ℤ)
∧ ((fst(snd(msgval(e)))) = (snd(snd(snd(msgval(e))))) ∈ pv11_p1_Ballot_Num())
∧ <loc(e), snd(snd(snd(msgval(e)))), s, c> ∈ pv11_p1_p2a'base(Cmd;f)(e'))))
Proof
Definitions occuring in Statement :
pv11_p1_decision: pv11_p1_decision{i:l}(Cmd;f;accpts;ldrs;ldrs_uid;reps;es;e;k;v),
pv11_p1_message-constraint: pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; mf; es),
pv11_p1_p2a'base: pv11_p1_p2a'base(Cmd;mf),
pv11_p1_headers_type: pv11_p1_headers_type{i:l}(Cmd),
pv11_p1_Ballot_Num: pv11_p1_Ballot_Num(),
es-info-body: msgval(e),
has-es-info-type: has-es-info-type(es;e;f;T),
Message: Message(f),
classrel: v ∈ X(e),
event-ordering+: EO+(Info),
es-causl: (e < e'),
es-loc: loc(e),
es-E: E,
Id: Id,
inject: Inj(A;B;f),
vatype: ValueAllType,
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
exists: ∃x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ─→ B[x],
pair: <a, b>,
product: x:A × B[x],
int: ℤ,
equal: s = t ∈ T,
bag-member: x ↓∈ bs,
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 :
sq_stable__and,
equal_wf,
cons_wf_listp,
nil_wf,
listp_wf,
vatype_wf,
cons_wf,
list_wf,
equal-wf-T-base,
sq_stable__equal,
squash_wf,
int_seg_wf,
length_wf,
name_wf,
pv11_p1_headers_wf,
l_all_iff,
l_member_wf,
pv11_p1_headers_fun_wf,
cons_member,
equal-wf-base,
iff_weakening_equal,
bag-member_wf,
has-es-info-type_wf,
pv11_p1_Ballot_Num_wf,
pv11_p1_decision_wf,
pv11_p1_message-constraint_wf,
inject_wf,
es-E_wf,
event-ordering+_subtype,
bag_wf,
Id_wf,
event-ordering+_wf,
Message_wf,
subtype_rel_dep_function,
pv11_p1_headers_type_wf,
set_wf,
valueall-type_wf,
make-Msg-as-mk-msg,
bfalse_wf,
subtype_base_sq,
list_subtype_base,
atom_subtype_base,
assert-deq-member,
pv11_p1_headers_no_inputs_wf,
es-info-mk-msg,
event-ordering+_cumulative,
es-info-auth_wf,
es-info-body_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,
es-loc_wf,
pair-eta,
subtype_rel_product,
top_wf,
subtype_top,
mk-msg_wf,
pi1_wf_top,
es-causl_wf,
equal-wf-base-T,
int_subtype_base,
pv11_p1_p2a'base_wf,
pv11_p1-decision,
pv11_p1-p2b,
base-classrel-equal,
es-le-loc,
and_wf,
pv11_p1-p2a,
assert-name_eq,
bool_wf,
bool_subtype_base,
pv11_p1_p2a'send_wf,
pv11_p1_A4_C1_funC
Latex:
\mforall{}[Cmd:ValueAllType]. \mforall{}[f:pv11\_p1\_headers\_type\{i:l\}(Cmd)]. \mforall{}[es:EO+(Message(f))].
\mforall{}[accpts,ldrs:bag(Id)]. \mforall{}[ldrs$_{uid}$:Id {}\mrightarrow{} \mBbbZ{}]. \mforall{}[reps:bag(Id)]. \mforall{}[e:E]. \mforall{}[s:\mBbbZ{}]\000C. \mforall{}[c:Cmd]. \mforall{}[a:Id].
(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{} pv11\_p1\_decision\{i:l\}(Cmd;f;accpts;ldrs;ldrs$_{uid}$;reps;es;e;s;c)
{}\mRightarrow{} has-es-info-type(es;e;f;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} pv11\_p1\_Ballot\_Num())
{}\mRightarrow{} a \mdownarrow{}\mmember{} accpts
{}\mRightarrow{} (\mdownarrow{}\mexists{}e':E
((e' < e)
\mwedge{} loc(e') \mdownarrow{}\mmember{} accpts
\mwedge{} (s = (fst(snd(snd(msgval(e))))))
\mwedge{} ((fst(snd(msgval(e)))) = (snd(snd(snd(msgval(e))))))
\mwedge{} <loc(e), snd(snd(snd(msgval(e)))), s, c> \mmember{} pv11\_p1\_p2a'base(Cmd;f)(e'))))
Date html generated:
2015_07_23-PM-05_05_35
Last ObjectModification:
2015_02_04-AM-07_50_38
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