Nuprl Lemma : pv11_p1_ldr_state_propose_pred
∀Cmd:ValueAllType. ∀f:pv11_p1_headers_type{i:l}(Cmd). ∀es:EO+(Message(f)). ∀e1,e2:E. ∀ldrs_uid:Id ─→ ℤ.
∀bnum:pv11_p1_Ballot_Num(). ∀active:𝔹. ∀proposals:(ℤ × Cmd) List. ∀s:ℤ. ∀c:Cmd.
((e1 <loc e2)
⇒ <bnum, active, proposals> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e1)
⇒ (header(e1) = [propose] ∈ Name)
⇒ has-es-info-type(es;e1;f;ℤ × Cmd)
⇒ (<s, c> = msgval(e1) ∈ (ℤ × Cmd))
⇒ (¬↑(pv11_p1_in_domain(Cmd) s proposals))
⇒ (∃e:{e:E| ¬↑first(e)}
((e1 = pred(e) ∈ E) ∧ e ≤loc e2 ∧ <bnum, active, pv11_p1_add_if_new() pv11_p1_same_proposal(Cmd) <s, c> proposal\000Cs> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e))))
Proof
Definitions occuring in Statement :
pv11_p1_LeaderState: pv11_p1_LeaderState(Cmd;ldrs_uid;mf),
pv11_p1_add_if_new: pv11_p1_add_if_new(),
pv11_p1_same_proposal: pv11_p1_same_proposal(Cmd),
pv11_p1_in_domain: pv11_p1_in_domain(Cmd),
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),
es-header: header(e),
Message: Message(f),
classrel: v ∈ X(e),
event-ordering+: EO+(Info),
es-le: e ≤loc e' ,
es-locl: (e <loc e'),
es-first: first(e),
es-pred: pred(e),
es-E: E,
Id: Id,
name: Name,
cons: [a / b],
nil: [],
list: T List,
vatype: ValueAllType,
assert: ↑b,
bool: 𝔹,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: 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],
int: ℤ,
token: "$token",
equal: s = t ∈ T
Lemmas :
int_seg_wf,
length_wf,
name_wf,
pv11_p1_headers_wf,
l_all_iff,
l_member_wf,
equal_wf,
pv11_p1_headers_fun_wf,
cons_wf_listp,
nil_wf,
listp_wf,
cons_member,
cons_wf,
equal-wf-base,
iff_weakening_equal,
list_wf,
not_wf,
assert_wf,
pv11_p1_in_domain_wf,
es-info-body_wf,
has-es-info-type_wf,
equal-wf-T-base,
es-header_wf,
classrel_wf,
pv11_p1_LeaderState_wf,
es-locl_wf,
bool_wf,
pv11_p1_Ballot_Num_wf,
Id_wf,
es-E_wf,
event-ordering+_subtype,
event-ordering+_wf,
Message_wf,
subtype_rel_dep_function,
vatype_wf,
pv11_p1_headers_type_wf,
set_wf,
valueall-type_wf,
es-pred-exists-between2,
event-ordering+_cumulative2,
es-pred_wf,
assert_elim,
es-first_wf2,
not_assert_elim,
btrue_neq_bfalse,
es-le_wf,
pv11_p1_add_if_new_wf,
pv11_p1_same_proposal_wf,
pv11_p1_ldr_state_eq2,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
member-base-class,
null_nil_lemma,
btrue_wf,
reduce_tl_cons_lemma,
and_wf,
tl_wf,
null_wf3,
subtype_rel_list,
top_wf,
null_cons_lemma,
bfalse_wf,
base-noloc-classrel,
subtype_rel_weakening,
ext-eq_weakening,
squash_wf,
true_wf,
eclass_wf,
pv11_p1_propose'base_wf,
pv11_p1_on_propose_wf,
es-loc_wf,
exists_wf,
member-base-class_iff
Latex:
\mforall{}Cmd:ValueAllType. \mforall{}f:pv11\_p1\_headers\_type\{i:l\}(Cmd). \mforall{}es:EO+(Message(f)). \mforall{}e1,e2:E.
\mforall{}ldrs$_{uid}$:Id {}\mrightarrow{} \mBbbZ{}. \mforall{}bnum:pv11\_p1\_Ballot\_Num(). \mforall{}active:\mBbbB{}. \mforall{}proposals:(\mBbbZ{} \mtimes{} Cm\000Cd) List. \mforall{}s:\mBbbZ{}. \mforall{}c:Cmd.
((e1 <loc e2)
{}\mRightarrow{} <bnum, active, proposals> \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(e1)
{}\mRightarrow{} (header(e1) = [propose])
{}\mRightarrow{} has-es-info-type(es;e1;f;\mBbbZ{} \mtimes{} Cmd)
{}\mRightarrow{} (<s, c> = msgval(e1))
{}\mRightarrow{} (\mneg{}\muparrow{}(pv11\_p1\_in\_domain(Cmd) s proposals))
{}\mRightarrow{} (\mexists{}e:\{e:E| \mneg{}\muparrow{}first(e)\}
((e1 = pred(e))
\mwedge{} e \mleq{}loc e2
\mwedge{} <bnum, active, pv11\_p1\_add\_if\_new() pv11\_p1\_same\_proposal(Cmd) <s, c> proposals> \mmember{} pv11\_p1\_\000CLeaderState(Cmd;ldrs$_{uid}$;f)(e))))
Date html generated:
2015_07_23-PM-04_48_33
Last ObjectModification:
2015_02_04-AM-08_12_51
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