Nuprl Lemma : new_23_sig-ilf
(∀[z1:ℤ]. ∀[propose,notify:Atom List]. ∀[clients:bag(Id)]. ∀[Cmd:{T:Type| valueall-type(T)} ].
∀[f:new_23_sig_headers_type{i:l}(Cmd;notify;propose)]. ∀[es:EO+(Message(f))]. ∀[e:E]. ∀[d:ℤ]. ∀[i:Id]. ∀[m:Message(f)].
{<d, i, m> ∈ new_23_sig_Notify(Cmd;clients;notify;propose;f) z1(e)
⇐⇒ (no (new_23_sig_decision(Cmd;clients;notify;propose;f)
z1 o new_23_sig_decided'base(Cmd;notify;propose;f)) prior to e)
∧ <d, i, m> ∈
{(new_23_sig_decision(Cmd;clients;notify;propose;f) z1 o new_23_sig_decided'base(Cmd;notify;propose;f))}(e)})
∧ (∀[z6,z1:ℤ]. ∀[reps:bag(Id)]. ∀[propose,notify:Atom List]. ∀[flrs,coeff:ℤ]. ∀[Cmd:{T:Type| valueall-type(T)} ].
∀[cmdeq:EqDecider(Cmd)]. ∀[f:new_23_sig_headers_type{i:l}(Cmd;notify;propose)]. ∀[es:EO+(Message(f))]. ∀[e:E].
∀[d:ℤ]. ∀[i:Id]. ∀[m:Message(f)].
{<d, i, m> ∈ new_23_sig_Quorum(Cmd;cmdeq;coeff;flrs;notify;propose;reps;f) <z1, z6>(e)
⇐⇒ ((header(e) = ``new_23_sig vote`` ∈ Name) ∧ has-es-info-type(es;e;f;ℤ × ℤ × Cmd × Id))
∧ (↑(new_23_sig_newvote(Cmd) <z1, z6> msgval(e) new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, z6>;es;e)))
∧ (||fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, z6>;es;e))|| = (coeff * flrs) ∈ ℤ)
∧ (d = 0 ∈ ℤ)
∧ (↓(((fst(poss-maj(cmdeq;[snd(fst(msgval(e))) / (fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, z6>;e\000Cs;e)))];snd(fst(msgval(e)))))) = ((coeff * flrs) + 1) ∈ ℤ)
∧ i ↓∈ reps
∧ (m
= make-Msg(``new_23_sig decided``;<fst(fst(fst(msgval(e))))
, snd(poss-maj(cmdeq;[snd(fst(msgval(e))) /
(fst(new_23_sig_QuorumStateFun(Cmd;...;...;f;<z1
, z6
>;es\000C;e)))];snd(fst(msgval(e)))))
>)
∈ Message(f)))
∨ ((¬((fst(poss-maj(cmdeq;[snd(fst(msgval(e))) /
(fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, z6>;es;e)))];snd(fst(ms\000Cgval(e))))))
= ((coeff * flrs) + 1)
∈ ℤ))
∧ (i = loc(e) ∈ Id)
∧ (m
= make-Msg(``new_23_sig retry``;<<fst(fst(fst(msgval(e)))), (snd(fst(fst(msgval(e))))) + 1>
, snd(poss-maj(cmdeq;[snd(fst(msgval(e))) /
(fst(new_23_sig_QuorumStateFun(Cmd;...;...;f;<z1
, z6
>;es\000C;e)))];snd(fst(msgval(e)))))
>)
∈ Message(f))))})
∧ (∀[z1:ℤ]. ∀[propose,notify:Atom List]. ∀[clients:bag(Id)]. ∀[Cmd:{T:Type| valueall-type(T)} ].
∀[f:new_23_sig_headers_type{i:l}(Cmd;notify;propose)]. ∀[es:EO+(Message(f))]. ∀[e:E]. ∀[d:ℤ]. ∀[i:Id].
∀[m:Message(f)].
{<d, i, m> ∈
(new_23_sig_decision(Cmd;clients;notify;propose;f) z1 o new_23_sig_decided'base(Cmd;notify;propose;f))(e)
⇐⇒ (header(e) = ``new_23_sig decided`` ∈ Name)
∧ has-es-info-type(es;e;f;ℤ × Cmd)
∧ ((fst(msgval(e))) = z1 ∈ ℤ)
∧ i ↓∈ clients
∧ (d = 0 ∈ ℤ)
∧ (m = make-Msg(notify;msgval(e)) ∈ Message(f))})
∧ (∀[Cmd:{T:Type| valueall-type(T)} ]. ∀[clients:bag(Id)]. ∀[cmdeq:EqDecider(Cmd)]. ∀[coeff,flrs:ℤ].
∀[notify,propose:Atom List]. ∀[reps:bag(Id)]. ∀[slots:set-sig{i:l}(ℤ)].
∀[f:new_23_sig_headers_type{i:l}(Cmd;notify;propose)]. ∀[es:EO+(Message(f))]. ∀[e:E]. ∀[d:ℤ]. ∀[i:Id].
∀[m:Message(f)].
{<d, i, m> ∈ new_23_sig_main()(e)
⇐⇒ loc(e) ↓∈ reps
∧ (↓∃e':{e':E| e' ≤loc e }
∃z1:ℤ
∃z2:Cmd
(((((header(e') = propose ∈ Name)
∧ has-es-info-type(es;e';f;ℤ × Cmd)
∧ (<z1, z2> = msgval(e') ∈ (ℤ × Cmd)))
∨ ((header(e') = ``new_23_sig vote`` ∈ Name)
∧ has-es-info-type(es;e';f;ℤ × ℤ × Cmd × Id)
∧ (z1 = (fst(fst(fst(msgval(e'))))) ∈ ℤ)
∧ (z2 = (snd(fst(msgval(e')))) ∈ Cmd)))
∧ (¬↑(set-sig-member(slots) z1 new_23_sig_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e'))))
∧ (((no new_23_sig_Notify(Cmd;clients;notify;propose;f) z1 between e' and e)
∧ ((((i ↓∈ reps ∧ (d = 0 ∈ ℤ) ∧ (m = make-Msg(``new_23_sig vote``;<<<z1, 0>, z2>, loc(e)>) ∈ Message(f\000C))) ∧ (e = e' ∈ E))
∨ ((no new_23_sig_Quorum(Cmd;cmdeq;coeff;flrs;notify;propose;reps;f) <z1, 0> between e' and e)
∧ <d, i, m> ∈ {new_23_sig_Quorum(Cmd;cmdeq;coeff;flrs;notify;propose;reps;f) <z1, 0>}(e)))
∨ (∃e1:{e1:E| e1 ≤loc e }
∃z6:ℤ
∃z7:Cmd
(new_23_sig_NewRoundsStateFun(Cmd;notify;propose;f;z1;es.e';e1) < z6
∧ (((header(e1) = ``new_23_sig retry`` ∈ Name)
∧ has-es-info-type(es;e1;f;ℤ × ℤ × Cmd)
∧ (<<z1, z6>, z7> = msgval(e1) ∈ (ℤ × ℤ × Cmd)))
∨ ((header(e1) = ``new_23_sig vote`` ∈ Name)
∧ has-es-info-type(es;e1;f;ℤ × ℤ × Cmd × Id)
∧ (<<z1, z6>, z7> = (fst(msgval(e1))) ∈ (ℤ × ℤ × Cmd))))
∧ (((... ↓∈ ... ∧ ...) ∧ ...) ∨ ...)))))
∨ ...)))})
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_Notify: new_23_sig_Notify(Cmd;clients;notify;propose;f),
new_23_sig_decision: new_23_sig_decision(Cmd;clients;notify;propose;f),
new_23_sig_NewRoundsStateFun: new_23_sig_NewRoundsStateFun(Cmd;notify;propose;f;x;es;e),
new_23_sig_Quorum: new_23_sig_Quorum(Cmd;cmdeq;coeff;flrs;notify;propose;reps;f),
new_23_sig_QuorumStateFun: new_23_sig_QuorumStateFun(Cmd;notify;propose;f;x;es;e),
new_23_sig_newvote: new_23_sig_newvote(Cmd),
new_23_sig_decided'base: new_23_sig_decided'base(Cmd;notify;propose;f),
new_23_sig_headers_type: new_23_sig_headers_type{i:l}(Cmd;notify;propose),
msg-interface: Interface,
make-Msg: make-Msg(hdr;val),
es-info-body: msgval(e),
has-es-info-type: has-es-info-type(es;e;f;T),
es-header: header(e),
Message: Message(f),
eclass0: (f o X),
no-classrel-in-interval: (no X between start and e),
no-prior-classrel: (no X prior to e),
classrel: v ∈ X(e),
eo-forward: eo.e,
event-ordering+: EO+(Info),
es-le: e ≤loc e' ,
es-loc: loc(e),
es-E: E,
Id: Id,
name: Name,
poss-maj: poss-maj(eq;L;x),
deq: EqDecider(T),
length: ||as||,
cons: [a / b],
nil: [],
list: T List,
valueall-type: valueall-type(T),
assert: ↑b,
less_than: a < b,
uall: ∀[x:A]. B[x],
guard: {T},
pi1: fst(t),
pi2: snd(t),
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
squash: ↓T,
or: P ∨ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
pair: <a, b>,
product: x:A × B[x],
multiply: n * m,
add: n + m,
natural_number: $n,
int: ℤ,
token: "$token",
atom: Atom,
universe: Type,
equal: s = t ∈ T,
bag-member: x ↓∈ 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,
classrel_wf,
msg-interface_wf,
eclass0_wf,
new_23_sig_decided'base_wf,
new_23_sig_decision_wf,
es-locl_wf,
new_23_sig_Notify_wf,
make-msg-interface_wf,
no-prior-classrel_wf,
es-E_wf,
event-ordering+_subtype,
event-ordering+_wf,
Message_wf,
subtype_rel_dep_function,
new_23_sig_headers_type_wf,
valueall-type_wf,
bag_wf,
Id_wf,
list_wf,
once-classrel-weak,
assert_witness,
es-info-body_wf,
new_23_sig_newvote_wf,
new_23_sig_QuorumStateFun_wf,
new_23_sig_Quorum_wf,
deq_wf,
iff_transitivity,
iff_weakening_uiff,
eclass2-eclass1-classrel,
new_23_sig_when_quorum_wf,
new_23_sig_vote'base_wf,
new_23_sig_QuorumState_wf,
base-classrel-equal,
subtype_rel_weakening,
ext-eq_weakening,
new_23_sig_QuorumState-classrel,
has-es-info-type_wf,
exists_wf,
bag-member_wf,
es-loc_wf,
equal-wf-T-base,
es-header_wf,
bag-member-ifthenelse,
empty-bag_wf,
ifthenelse_functionality_wrt_iff,
false_wf,
bag-member-empty-weak,
bag-member-spread-to-pi,
eq_int_wf,
nat_wf,
bag-member-single-weak,
bag-member-map,
ifthenelse-false-right,
new_23_sig_roundout_wf,
assert_wf,
poss-maj_wf,
assert_of_eq_int,
mk-msg-interface_wf,
make-Msg_wf,
neg_assert_of_eq_int,
new_23_sig_retry'send_wf,
new_23_sig_decided'broadcast_wf,
single-bag_wf,
subtype_rel_product,
top_wf,
sq_stable__bag-member,
sq_stable__has-es-info-type,
squash-assert,
member_wf,
int_subtype_base,
bool_cases,
not_wf,
or_wf,
eclass0-base-classrel,
new_23_sig_notify'broadcast_wf,
pair-eta,
subtype_rel_transitivity,
es-info-type_wf,
pi1_wf_top,
new_23_sig_main_wf,
set-sig_wf,
classrel-at,
new_23_sig_Replica_wf,
bind-class-rel-weak,
new_23_sig_NewVoters_wf,
new_23_sig_Voter_wf,
set-sig-set_wf,
new_23_sig_when_new_proposal_wf,
new_23_sig_Proposal_wf,
new_23_sig_ReplicaState_wf,
parallel-classrel-weak,
new_23_sig_propose'base_wf,
new_23_sig_vote2prop_wf,
new_23_sig_ReplicaState-classrel,
new_23_sig_ReplicaStateFun_wf,
eo-forward_wf,
until-classrel,
new_23_sig_Rounds_wf,
new_23_sig_Round_wf,
bind-class_wf,
new_23_sig_NewRounds_wf,
once-class_wf,
return-loc-bag-class-classrel,
new_23_sig_when_new_round_wf,
new_23_sig_RoundInfo_wf,
new_23_sig_NewRoundsState_wf,
new_23_sig_retry'base_wf,
new_23_sig_vote2retry_wf,
new_23_sig_NewRoundsState-classrel,
equal-wf-base-T,
new_23_sig_NewRoundsStateFun_wf,
return-loc-bag-class_wf,
new_23_sig_vote'broadcast_wf,
sq_or_wf,
until-class_wf,
es-interface-subtype_rel2,
subtype_top,
member-eo-forward-E,
es-first_wf2,
eo-forward-header,
eo-forward-has-es-info-type,
eo-forward-info-body,
eo-forward-E-subtype,
es-le_wf,
set-sig-member_wf,
ifthenelse-false-left,
lt_int_wf,
assert_of_lt_int,
less_than_wf,
assert_of_band,
and_wf,
eo-forward-no-prior-classrel,
eo-forward-loc,
eo-forward-first2,
no-classrel-in-interval_wf,
eo-forward-le,
uiff_transitivity,
equal-eo-forward-E,
eo-forward-no-classrel-in-interval,
es-le-loc,
decidable__es-le,
eo-forward-le2,
subtype_rel_sets,
eo-forward-forward2
Latex:
(\mforall{}[z1:\mBbbZ{}]. \mforall{}[propose,notify:Atom List]. \mforall{}[clients:bag(Id)]. \mforall{}[Cmd:\{T:Type| valueall-type(T)\} ].
\mforall{}[f:new\_23\_sig\_headers\_type\{i:l\}(Cmd;notify;propose)]. \mforall{}[es:EO+(Message(f))]. \mforall{}[e:E]. \mforall{}[d:\mBbbZ{}].
\mforall{}[i:Id]. \mforall{}[m:Message(f)].
\{<d, i, m> \mmember{} new\_23\_sig\_Notify(Cmd;clients;notify;propose;f) z1(e)
\mLeftarrow{}{}\mRightarrow{} (no (new\_23\_sig\_decision(Cmd;clients;notify;propose;f)
z1 o new\_23\_sig\_decided'base(Cmd;notify;propose;f)) prior to e)
\mwedge{} <d, i, m> \mmember{}
\{(new\_23\_sig\_decision(Cmd;clients;notify;propose;f)
z1 o new\_23\_sig\_decided'base(Cmd;notify;propose;f))\}(e)\})
\mwedge{} (\mforall{}[z6,z1:\mBbbZ{}]. \mforall{}[reps:bag(Id)]. \mforall{}[propose,notify:Atom List]. \mforall{}[flrs,coeff:\mBbbZ{}].
\mforall{}[Cmd:\{T:Type| valueall-type(T)\} ]. \mforall{}[cmdeq:EqDecider(Cmd)].
\mforall{}[f:new\_23\_sig\_headers\_type\{i:l\}(Cmd;notify;propose)]. \mforall{}[es:EO+(Message(f))]. \mforall{}[e:E]. \mforall{}[d:\mBbbZ{}].
\mforall{}[i:Id]. \mforall{}[m:Message(f)].
\{<d, i, m> \mmember{} new\_23\_sig\_Quorum(Cmd;cmdeq;coeff;flrs;notify;propose;reps;f) <z1, z6>(e)
\mLeftarrow{}{}\mRightarrow{} ((header(e) = ``new\_23\_sig vote``) \mwedge{} has-es-info-type(es;e;f;\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd \mtimes{} Id))
\mwedge{} (\muparrow{}(new\_23\_sig\_newvote(Cmd) <z1, z6> msgval(e) new\_23\_sig\_QuorumStateFun(Cmd;notify;propos\000Ce;f;<z1, z6>es;e)))
\mwedge{} (||fst(new\_23\_sig\_QuorumStateFun(Cmd;notify;propose;f;<z1, z6>es;e))|| = (coeff * flrs))
\mwedge{} (d = 0)
\mwedge{} (\mdownarrow{}(((fst(poss-maj(cmdeq;[snd(fst(msgval(e))) /
(fst(new\_23\_sig\_QuorumStateFun(Cmd;notify;propose;f;<z1, z6>es;\000Ce)))];snd(fst(msgval(e))))))
= ((coeff * flrs) + 1))
\mwedge{} i \mdownarrow{}\mmember{} reps
\mwedge{} (m
= make-Msg(``new\_23\_sig decided``;<fst(fst(fst(msgval(e))))
, snd(poss-maj(cmdeq;[snd(fst(msgval(e))) /
(fst(...))];snd(fst(...))))
>)))
\mvee{} ((\mneg{}((fst(poss-maj(cmdeq;[snd(fst(msgval(e))) /
(fst(new\_23\_sig\_QuorumStateFun(Cmd;notify;propose;f;<z1, z6>\000C;es;e)))];snd(fst(msgval(e))))))
= ((coeff * flrs) + 1)))
\mwedge{} (i = loc(e))
\mwedge{} (m
= make-Msg(``new\_23\_sig retry``;<<fst(fst(fst(msgval(e))))
, (snd(fst(fst(msgval(e))))) + 1
>
, snd(poss-maj(cmdeq;[snd(fst(msgval(e))) /
(fst(...))];snd(fst(...))))
>))))\})
\mwedge{} (\mforall{}[z1:\mBbbZ{}]. \mforall{}[propose,notify:Atom List]. \mforall{}[clients:bag(Id)]. \mforall{}[Cmd:\{T:Type| valueall-type(T)\} ].
\mforall{}[f:new\_23\_sig\_headers\_type\{i:l\}(Cmd;notify;propose)]. \mforall{}[es:EO+(Message(f))]. \mforall{}[e:E]. \mforall{}[d:\mBbbZ{}].
\mforall{}[i:Id]. \mforall{}[m:Message(f)].
\{<d, i, m> \mmember{}
(new\_23\_sig\_decision(Cmd;clients;notify;propose;f)
z1 o new\_23\_sig\_decided'base(Cmd;notify;propose;f))(e)
\mLeftarrow{}{}\mRightarrow{} (header(e) = ``new\_23\_sig decided``)
\mwedge{} has-es-info-type(es;e;f;\mBbbZ{} \mtimes{} Cmd)
\mwedge{} ((fst(msgval(e))) = z1)
\mwedge{} i \mdownarrow{}\mmember{} clients
\mwedge{} (d = 0)
\mwedge{} (m = make-Msg(notify;msgval(e)))\})
\mwedge{} (\mforall{}[Cmd:\{T:Type| valueall-type(T)\} ]. \mforall{}[clients:bag(Id)]. \mforall{}[cmdeq:EqDecider(Cmd)]. \mforall{}[coeff,flrs:\mBbbZ{}].
\mforall{}[notify,propose:Atom List]. \mforall{}[reps:bag(Id)]. \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{}[d:\mBbbZ{}].
\mforall{}[i:Id]. \mforall{}[m:Message(f)].
\{<d, i, m> \mmember{} new\_23\_sig\_main()(e)
\mLeftarrow{}{}\mRightarrow{} loc(e) \mdownarrow{}\mmember{} reps
\mwedge{} (\mdownarrow{}\mexists{}e':\{e':E| e' \mleq{}loc e \}
\mexists{}z1:\mBbbZ{}
\mexists{}z2:Cmd
(((((header(e') = propose)
\mwedge{} has-es-info-type(es;e';f;\mBbbZ{} \mtimes{} Cmd)
\mwedge{} (<z1, z2> = msgval(e')))
\mvee{} ((header(e') = ``new\_23\_sig vote``)
\mwedge{} has-es-info-type(es;e';f;\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd \mtimes{} Id)
\mwedge{} (z1 = (fst(fst(fst(msgval(e'))))))
\mwedge{} (z2 = (snd(fst(msgval(e')))))))
\mwedge{} (\mneg{}\muparrow{}(set-sig-member(slots) z1
new\_23\_sig\_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e'))))
\mwedge{} (((no new\_23\_sig\_Notify(Cmd;clients;notify;propose;f) z1 between e' and e)
\mwedge{} ((((i \mdownarrow{}\mmember{} reps
\mwedge{} (d = 0)
\mwedge{} (m = make-Msg(``new\_23\_sig vote``;<<<z1, 0>, z2>, loc(e)>)))
\mwedge{} (e = e'))
\mvee{} ((no new\_23\_sig\_Quorum(Cmd;cmdeq;coeff;flrs;notify;propose;reps;f) <z1, 0> bet\000Cween e' and e)
\mwedge{} <d, i, m> \mmember{} \{new\_23\_sig\_Quorum(Cmd;cmdeq;coeff;flrs;notify;propose;reps;f)
<z1, 0>\}(e)))
\mvee{} (\mexists{}e1:\{e1:E| e1 \mleq{}loc e \}
\mexists{}z6:\mBbbZ{}
\mexists{}z7:Cmd
(new\_23\_sig\_NewRoundsStateFun(Cmd;notify;propose;f;z1;es.e';e1) < z6
\mwedge{} (((header(e1) = ``new\_23\_sig retry``)
\mwedge{} has-es-info-type(es;e1;f;\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd)
\mwedge{} (<<z1, z6>, z7> = msgval(e1)))
\mvee{} ((header(e1) = ``new\_23\_sig vote``)
\mwedge{} has-es-info-type(es;e1;f;\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd \mtimes{} Id)
\mwedge{} (<<z1, z6>, z7> = (fst(msgval(e1))))))
\mwedge{} (((i \mdownarrow{}\mmember{} reps
\mwedge{} (d = 0)
\mwedge{} (m = make-Msg(``new\_23\_sig vote``;<<<z1, z6>, z7>, loc(e)>)))
\mwedge{} (e = e1))
\mvee{} ((no new\_23\_sig\_Quorum(Cmd;cmdeq;coeff;flrs;notify;propose;reps;f)
<z1, z6> between e1 and e)
\mwedge{} <d, i, m> \mmember{}
\{new\_23\_sig\_Quorum(Cmd;cmdeq;coeff;flrs;...;...;...;...) ...\}(
...)))))))
\mvee{} ...)))\})
Date html generated:
2015_07_23-PM-03_55_13
Last ObjectModification:
2015_02_04-PM-03_35_14
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