Nuprl Lemma : new_23_sig_progress-step10
∀[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
∃c':Cmd
((e <loc e')
∧ (<n, c'> ∈ new_23_sig_decided'base(Cmd;notify;propose;f)(e')
∨ (∃r:ℕ+. <<n, r>, c'> ∈ new_23_sig_retry'base(Cmd;notify;propose;f)(e'))))))
Proof
Definitions occuring in Statement :
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_decided'base: new_23_sig_decided'base(Cmd;notify;propose;f),
new_23_sig_retry'base: new_23_sig_retry'base(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),
Message: Message(f),
classrel: v ∈ X(e),
event-ordering+: EO+(Info),
es-locl: (e <loc e'),
es-loc: loc(e),
es-E: E,
Id: Id,
deq: EqDecider(T),
list: T List,
int_upper: {i...},
nat_plus: ℕ+,
nat: ℕ,
vatype: ValueAllType,
assert: ↑b,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
not: ¬A,
squash: ↓T,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
apply: f a,
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 :
new_23_sig_progress-step9-v3,
decidable__assert,
bl-exists_wf,
es-E_wf,
event-ordering+_subtype,
Message_wf,
es-interval_wf,
l_member_wf,
new_23_sig_decided_with_id_wf,
assert-bl-exists,
l_exists_iff,
assert_wf,
new_23_sig_decided_with_id-assert-type,
new_23_sig_decided_with_id-assert-classrel,
es-info-body_wf,
member-es-interval,
new_23_sig_proposal_has_header,
base-noloc-classrel,
cons_wf_listp,
cons_wf,
nil_wf,
listp_wf,
subtype_rel_weakening,
ext-eq_weakening,
and_wf,
equal_wf,
es-header_wf,
length_of_cons_lemma,
length_of_nil_lemma,
false_wf,
le_wf,
not_wf,
less_than_wf,
equal-wf-base,
nat_wf,
assert-name_eq,
exists_wf,
nat_plus_wf,
classrel_wf,
new_23_sig_retry'base_wf,
es-locl_wf,
or_wf,
new_23_sig_decided'base_wf,
mul_bounds_1a,
int_upper_subtype_nat,
decidable__equal_Id,
es-loc_wf,
decidable__es-le,
non_null_iff_length,
filter_wf5,
new_23_sig_vote_with_ballot_first_wf,
eo-forward_wf,
member-eo-forward-E,
Id_wf,
subtype_rel_list,
top_wf,
length-map,
decidable__lt,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
new_23_sig_vote_with_ballot_first-assert-forward,
new_23_sig_vote_with_ballot-assert-type,
poss-maj_wf,
reverse_wf,
mapfilter-wf2,
new_23_sig_vote_with_ballot_first-forward,
valueall-type_wf,
bool_cases_sqequal,
new_23_sig_vote_with_ballot_wf,
subtype_rel_product,
subtype_rel_self,
decidable__equal_int,
es-le_wf,
msg-interface_wf,
new_23_sig_main_wf,
mk-msg-interface_wf,
make-Msg_wf,
list-set-type2,
l_all_iff,
bfalse_wf,
es-le-loc,
bag-member_wf,
new_23_sig-decided,
equal-wf-T-base,
name_wf,
list_subtype_base,
atom_subtype_base,
has-es-info-type_wf,
equal-wf-base-T,
int_subtype_base,
set-sig-member_wf,
new_23_sig_ReplicaStateFun_wf,
no-classrel-in-interval_wf,
new_23_sig_Notify_wf,
new_23_sig_Quorum_wf,
new_23_sig_newvote_wf,
new_23_sig_QuorumStateFun_wf,
length_wf,
list_wf,
bool_wf,
new_23_sig_commands_from_votes_wf,
eo-forward-E-subtype,
new_23_sig_NewRoundsStateFun_wf,
new_23_sig_proposal_classrel2,
subtype_base_sq,
subtype_rel_wf,
sq_stable__no_repeats,
new_23_sig_headers_wf,
squash_wf,
true_wf,
once-classrel,
eclass0_wf,
new_23_sig_decision_wf,
eclass0-classrel,
eq_int_wf,
bool_cases,
bool_subtype_base,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
iff_transitivity,
bnot_wf,
iff_weakening_uiff,
assert_of_bnot,
assert-bl-exists2,
es-le_weakening,
eo-forward-base-classrel,
event-ordering+_cumulative,
member-eclass_wf,
es-info-type_wf,
pi1_wf_top,
subtype_top,
subtype_rel_transitivity,
assert-member-eclass,
assert_of_band,
bag-member-empty-iff,
eclass2-classrel,
eclass1_wf,
bag_wf,
new_23_sig_vote'base_wf,
new_23_sig_when_quorum_wf,
new_23_sig_QuorumState_wf,
eclass1-classrel,
new_23_sig_QuorumState-classrel,
new_23_sig_quorum_state_eq1-forward,
iff_weakening_equal,
map-length,
length-reverse,
filter_append_sq,
filter_cons_lemma,
filter_nil_lemma,
es-ble_wf,
length-append,
es-closed-open-interval_wf,
member-es-closed-open-interval,
assert-es-ble,
es-closed-open-interval-decomp-member,
set_wf,
es-le_transitivity,
length_zero,
assert_of_null,
null_member,
member_filter_2,
es-le-self,
new_23_sig_vote_with_ballot_and_id-if-classrel,
productdeq_reduce_lemma,
intdeq_reduce_lemma,
assert_of_band2,
deq-member_wf,
id-deq_wf,
new_23_sig_vote_with_ballot_and_id-implies,
assert_elim,
assert-bl-all-2,
new_23_sig_vote_with_ballot_and_id_wf,
assert-deq-member,
atom2_subtype_base,
member-reverse,
member-mapfilter,
new_23_sig_vote_with_ballot_and_id-snd,
new_23_sig_vote_with_ballot_and_id-forward,
new_23_sig_vote_with_ballot_first-not2,
es-init_wf,
event_ordering_wf,
es-init-forward,
eo-forward-le,
eo-forward-locl,
es-locl_transitivity2,
new_23_sig_vote_with_ballot-header,
pair-eta,
new_23_sig_vote_with_ballot-assert-classrel,
pi2_wf,
new_23_sig_quorum_inv_vote2_fun,
lelt_wf,
select_wf,
eo-forward-E-member,
new_23_sig_vote_with_ballot_first-assert-forward2,
l_all_fwd,
not_assert_elim,
btrue_neq_bfalse,
length-mapfilter,
new_23_sig_quorum_state_eq1,
es-closed-open-interval-forward,
deq_wf,
mapfilter-append,
mapfilter-singleton,
map_cons_lemma,
map_nil_lemma,
reverse_append_sq,
reverse_singleton,
mapfilter_wf,
new_23_sig_vote_with_ballot_first-assert-type,
list_ind_cons_lemma,
list_ind_nil_lemma,
list-equal-set,
sqequal-tt-to-assert,
sqequal-ff-to-assert,
reverse-cons,
reverse_nil_lemma,
eo-forward-has-es-info-type,
new_23_sig-retry,
make-msg-interface-equal,
es-info_wf,
es-causl_transitivity1,
es-causle_weakening_locl,
base-noloc-classrel-make-Msg2,
msg-authentic_wf
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
\mexists{}c':Cmd
((e <loc e')
\mwedge{} (<n, c'> \mmember{} new\_23\_sig\_decided'base(Cmd;notify;propose;f)(e')
\mvee{} (\mexists{}r:\mBbbN{}\msupplus{}. <<n, r>, c'> \mmember{} new\_23\_sig\_retry'base(Cmd;notify;propose;f)(e'))))))
Date html generated:
2015_07_23-PM-04_09_07
Last ObjectModification:
2015_02_06-PM-03_23_05
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