Nuprl Lemma : new_23_sig_progress-step9-v2
∀[Cmd:ValueAllType]. ∀[eq:EqDecider(Cmd)]. ∀[reps,clients:bag(Id)]. ∀[coeff:{2...}]. ∀[flrs:ℕ].
∀[propose,notify:Atom List]. ∀[slots:set-sig{i:l}(ℤ)]. ∀[f:new_23_sig_headers_type{i:l}(Cmd;notify;propose)].
∀[es:EO+(Message(f))]. ∀[e:E]. ∀[n:ℤ]. ∀[c:Cmd]. ∀[faulty:bag(Id)].
(msgs-interface-delivered-with-omissions(f;es;new_23_sig_main();faulty;flrs;reps)
⇒ bag-no-repeats(Id;reps)
⇒ (#(reps) = ((coeff * flrs) + flrs + 1) ∈ ℤ)
⇒ loc(e) ↓∈ reps
⇒ (¬loc(e) ↓∈ faulty)
⇒ <n, c> ∈ new_23_sig_Proposal(Cmd;notify;propose;f)(e)
⇒ (¬↑(set-sig-member(slots) n new_23_sig_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e)))
⇒ (↓∃e':E
(((coeff * flrs) + 1)
= ||mapfilter(λe.<snd(msgval(e)), e, snd(fst(msgval(e)))>;
λe'.new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es.e;e';n;0);
[e, e'])||
∈ ℤ)))
Proof
Definitions occuring in Statement :
new_23_sig_vote_with_ballot_first: new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es;e;n;r),
new_23_sig_main: new_23_sig_main(),
new_23_sig_ReplicaStateFun: new_23_sig_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e),
new_23_sig_Proposal: new_23_sig_Proposal(Cmd;notify;propose;f),
new_23_sig_headers_type: new_23_sig_headers_type{i:l}(Cmd;notify;propose),
msgs-interface-delivered-with-omissions: msgs-interface-delivered-with-omissions(f;es;X;faulty;failures;ids),
es-info-body: msgval(e),
Message: Message(f),
classrel: v ∈ X(e),
eo-forward: eo.e,
event-ordering+: EO+(Info),
es-interval: [e, e'],
es-loc: loc(e),
es-E: E,
Id: Id,
deq: EqDecider(T),
mapfilter: mapfilter(f;P;L),
length: ||as||,
list: T List,
int_upper: {i...},
nat: ℕ,
vatype: ValueAllType,
assert: ↑b,
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
exists: ∃x:A. B[x],
not: ¬A,
squash: ↓T,
implies: P ⇒ Q,
apply: f a,
lambda: λx.A[x],
pair: <a, b>,
product: x:A × B[x],
multiply: n * m,
add: n + m,
natural_number: $n,
int: ℤ,
atom: Atom,
equal: s = t ∈ T,
bag-member: x ↓∈ bs,
bag-no-repeats: bag-no-repeats(T;bs),
bag-size: #(bs),
bag: bag(T),
set-sig-member: set-sig-member(s),
set-sig: set-sig{i:l}(Item)
Lemmas :
sq_stable__and,
equal_wf,
vatype_wf,
cons_wf_listp,
cons_wf,
nil_wf,
listp_wf,
sq_stable__equal,
squash_wf,
int_seg_wf,
length_wf,
name_wf,
new_23_sig_headers_wf,
l_all_iff,
l_member_wf,
new_23_sig_headers_fun_wf,
cons_member,
equal-wf-base,
list_subtype_base,
atom_subtype_base,
iff_weakening_equal,
name_eq_wf,
bool_wf,
eqtt_to_assert,
assert-name_eq,
sq_stable__no_repeats,
true_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
not_wf,
assert_wf,
set-sig-member_wf,
new_23_sig_ReplicaStateFun_wf,
classrel_wf,
new_23_sig_Proposal_wf,
bag-member_wf,
es-loc_wf,
bag-size_wf,
nat_wf,
bag-no-repeats_wf,
msgs-interface-delivered-with-omissions_wf,
new_23_sig_main_wf,
es-E_wf,
event-ordering+_subtype,
event-ordering+_wf,
Message_wf,
subtype_rel_dep_function,
new_23_sig_headers_type_wf,
set-sig_wf,
list_wf,
int_upper_wf,
bag_wf,
Id_wf,
deq_wf,
set_wf,
valueall-type_wf,
new_23_sig_progress-step8-v3,
mul_bounds_1a,
int_upper_subtype_nat,
false_wf,
le_wf,
bag-size-append,
bag-filter-split,
bag-deq-member_wf,
id-deq_wf,
bag-size-filter-member-bound,
bnot_wf,
bag-filter_wf,
sub-bag-list-if-bag-no-repeats-sq,
subtype_rel_bag,
mapfilter_wf,
es-interval_wf,
new_23_sig_vote_with_ballot_wf,
new_23_sig_vote_with_ballot-assert-type,
es-info-body_wf,
bag-filter-no-repeats,
sub-bag-remove-repeats-if-no-repeats,
list-subtype-bag,
subtype_rel_self,
bag-no-repeats-filter,
remove-repeats-mapfilter-with-fun,
es-blocl_wf,
es-locl_transitivity2,
es-le_weakening,
es-locl_irreflexivity,
es-locl_wf,
es-blocl-iff,
all_wf,
select_wf,
es-interval-sorted-by,
es-locl-antireflexive,
sub-bag_wf,
list-to-set_wf,
sub-bag-size,
atom2_subtype_base,
le_transitivity,
bag-remove-repeats-eq-remove-repeats,
map-length,
filter-vote-with-ballot-lemma,
list-decomp-nat-iseg,
filter_wf2,
new_23_sig_vote_with_ballot_first_wf,
member-eo-forward-E,
member-es-interval,
subtype_rel_list,
iseg_filter2,
eo-forward_wf,
length-filter,
iseg_member,
list-subtype,
non_null_iff_length,
top_wf,
last_wf,
filter_wf5,
iseg-es-interval
Latex:
\mforall{}[Cmd:ValueAllType]. \mforall{}[eq:EqDecider(Cmd)]. \mforall{}[reps,clients:bag(Id)]. \mforall{}[coeff:\{2...\}]. \mforall{}[flrs:\mBbbN{}].
\mforall{}[propose,notify:Atom List]. \mforall{}[slots:set-sig\{i:l\}(\mBbbZ{})].
\mforall{}[f:new\_23\_sig\_headers\_type\{i:l\}(Cmd;notify;propose)]. \mforall{}[es:EO+(Message(f))]. \mforall{}[e:E]. \mforall{}[n:\mBbbZ{}].
\mforall{}[c:Cmd]. \mforall{}[faulty:bag(Id)].
(msgs-interface-delivered-with-omissions(f;es;new\_23\_sig\_main();faulty;flrs;reps)
{}\mRightarrow{} bag-no-repeats(Id;reps)
{}\mRightarrow{} (\#(reps) = ((coeff * flrs) + flrs + 1))
{}\mRightarrow{} loc(e) \mdownarrow{}\mmember{} reps
{}\mRightarrow{} (\mneg{}loc(e) \mdownarrow{}\mmember{} faulty)
{}\mRightarrow{} <n, c> \mmember{} new\_23\_sig\_Proposal(Cmd;notify;propose;f)(e)
{}\mRightarrow{} (\mneg{}\muparrow{}(set-sig-member(slots) n new\_23\_sig\_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e)))
{}\mRightarrow{} (\mdownarrow{}\mexists{}e':E
(((coeff * flrs) + 1)
= ||mapfilter(\mlambda{}e.<snd(msgval(e)), e, snd(fst(msgval(e)))>
\mlambda{}e'.new\_23\_sig\_vote\_with\_ballot\_first(Cmd;notify;propose;f;es.e;e';n;0);
[e, e'])||)))
Date html generated:
2015_07_23-PM-04_06_59
Last ObjectModification:
2015_02_04-PM-03_50_29
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