Nuprl Lemma : pv11_p1_ldr_state_eq
∀[Cmd:ValueAllType]. ∀[f:pv11_p1_headers_type{i:l}(Cmd)]. ∀[es:EO+(Message(f))]. ∀[e:E]. ∀[ldrs_uid:Id ─→ ℤ].
∀[v:pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List)].
uiff(v ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e);↓if first(e)
then v
= (pv11_p1_init_leader(Cmd) loc(e))
∈ (pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List))
if pred(e) ∈b pv11_p1_propose'base(Cmd;f)
then ∃x:ℤ × Cmd
∃s:pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List)
(x ∈ pv11_p1_propose'base(Cmd;f)(pred(e))
∧ s ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(pred(e))
∧ (v
= (pv11_p1_on_propose(Cmd) loc(e) x s)
∈ (pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List))))
if pred(e) ∈b pv11_p1_adopted'base(Cmd;f)
then ∃x:pv11_p1_Ballot_Num()
× ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
∃s:pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List)
(x ∈ pv11_p1_adopted'base(Cmd;f)(pred(e))
∧ s ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(pred(e))
∧ (v
= (pv11_p1_when_adopted(Cmd;ldrs_uid) loc(e) x s)
∈ (pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List))))
if pred(e) ∈b pv11_p1_preempted'base(Cmd;f)
then ∃x:pv11_p1_Ballot_Num()
∃s:pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List)
(x ∈ pv11_p1_preempted'base(Cmd;f)(pred(e))
∧ s ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(pred(e))
∧ (v
= (pv11_p1_when_preempted(Cmd;ldrs_uid) loc(e) x s)
∈ (pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List))))
else v ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(pred(e))
fi )
Proof
Definitions occuring in Statement :
pv11_p1_LeaderState: pv11_p1_LeaderState(Cmd;ldrs_uid;mf),
pv11_p1_when_preempted: pv11_p1_when_preempted(Cmd;ldrs_uid),
pv11_p1_when_adopted: pv11_p1_when_adopted(Cmd;ldrs_uid),
pv11_p1_on_propose: pv11_p1_on_propose(Cmd),
pv11_p1_init_leader: pv11_p1_init_leader(Cmd),
pv11_p1_propose'base: pv11_p1_propose'base(Cmd;mf),
pv11_p1_adopted'base: pv11_p1_adopted'base(Cmd;mf),
pv11_p1_preempted'base: pv11_p1_preempted'base(Cmd;mf),
pv11_p1_headers_type: pv11_p1_headers_type{i:l}(Cmd),
pv11_p1_Ballot_Num: pv11_p1_Ballot_Num(),
Message: Message(f),
classrel: v ∈ X(e),
member-eclass: e ∈b X,
event-ordering+: EO+(Info),
es-first: first(e),
es-pred: pred(e),
es-loc: loc(e),
es-E: E,
Id: Id,
list: T List,
vatype: ValueAllType,
ifthenelse: if b then t else f fi ,
bool: 𝔹,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
and: P ∧ Q,
apply: f a,
function: x:A ─→ B[x],
product: x:A × B[x],
int: ℤ,
equal: s = t ∈ T
Definitions :
assert: ↑b,
ifthenelse: if b then t else f fi ,
name_eq: name_eq(x;y),
name-deq: NameDeq,
list-deq: list-deq(eq),
cons: [a / b],
band: p ∧b q,
atom-deq: AtomDeq,
eq_atom: x =a y,
bfalse: ff,
false: False,
btrue: tt
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,
eqtt_to_assert,
es-loc_wf,
eqff_to_assert,
bool_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
member-eclass_wf,
es-pred_wf,
iff_weakening_uiff,
classrel_wf,
memory-class3_wf,
pv11_p1_init_leader_wf,
pv11_p1_on_propose_wf,
pv11_p1_propose'base_wf,
pv11_p1_when_adopted_wf,
pv11_p1_adopted'base_wf,
pv11_p1_when_preempted_wf,
pv11_p1_preempted'base_wf,
es-first_wf2,
event-ordering+_subtype,
exists_wf,
bor_wf,
iff_transitivity,
assert_wf,
or_wf,
assert_of_bor,
memory-class3-classrel,
pv11_p1_LeaderState_wf,
uiff_wf,
pv11_p1_Ballot_Num_wf,
Id_wf,
es-E_wf,
event-ordering+_wf,
Message_wf,
subtype_rel_dep_function,
pv11_p1_headers_type_wf,
set_wf,
valueall-type_wf,
base-noloc-classrel,
subtype_rel_weakening,
ext-eq_weakening,
member-base-class,
assert-name_eq,
and_wf
Latex:
\mforall{}[Cmd:ValueAllType]. \mforall{}[f:pv11\_p1\_headers\_type\{i:l\}(Cmd)]. \mforall{}[es:EO+(Message(f))]. \mforall{}[e:E].
\mforall{}[ldrs$_{uid}$:Id {}\mrightarrow{} \mBbbZ{}]. \mforall{}[v:pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbB{} \mtimes{} ((\mBbbZ{} \mtimes{} Cmd) List)].
uiff(v \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(e);\mdownarrow{}if first(e)
then v = (pv11\_p1\_init\_leader(Cmd) loc(e))
if pred(e) \mmember{}\msubb{} pv11\_p1\_propose'base(Cmd;f)
then \mexists{}x:\mBbbZ{} \mtimes{} Cmd
\mexists{}s:pv11\_p1\_Ballot\_Num()
\mtimes{} \mBbbB{}
\mtimes{} ((\mBbbZ{} \mtimes{} Cmd) List)
(x \mmember{} pv11\_p1\_propose'base(Cmd;f)(pred(e))
\mwedge{} s \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$\mbackslash{}f\000Cf5f{uid}$;f)(
pred(e))
\mwedge{} (v
= (pv11\_p1\_on\_propose(Cmd) loc(e) x
s)))
if pred(e) \mmember{}\msubb{} pv11\_p1\_adopted'base(Cmd;f)
then \mexists{}x:pv11\_p1\_Ballot\_Num()
\mtimes{} ((pv11\_p1\_Ballot\_Num()
\mtimes{} \mBbbZ{}
\mtimes{} Cmd) List)
\mexists{}s:pv11\_p1\_Ballot\_Num()
\mtimes{} \mBbbB{}
\mtimes{} ((\mBbbZ{} \mtimes{} Cmd) List)
(x \mmember{} pv11\_p1\_adopted'base(Cmd;f)(pred(e))
\mwedge{} s \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$\mbackslash{}f\000Cf5f{uid}$;f)(
pred(e))
\mwedge{} (v
= (pv11\_p1\_when\_adopted(Cmd;ldrs$\mbackslash{}f\000Cf5f{uid}$)
loc(e)
x
s)))
if pred(e) \mmember{}\msubb{} pv11\_p1\_preempted'base(Cmd;f)
then \mexists{}x:pv11\_p1\_Ballot\_Num()
\mexists{}s:pv11\_p1\_Ballot\_Num()
\mtimes{} \mBbbB{}
\mtimes{} ((\mBbbZ{} \mtimes{} Cmd) List)
(x \mmember{} pv11\_p1\_preempted'base(Cmd;f)(
pred(e))
\mwedge{} s \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$\mbackslash{}f\000Cf5f{uid}$;f)(
pred(e))
\mwedge{} (v
= (pv11\_p1\_when\_preempted(Cmd;ldrs$\000C_{uid}$)
loc(e)
x
s)))
else v \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_\mbackslash{}ff\000C7buid}$;f)(
pred(e))
fi )
Date html generated:
2015_07_23-PM-04_46_47
Last ObjectModification:
2015_02_04-AM-08_25_37
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