Nuprl Lemma : new_23_sig-decided
∀[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]. ∀[auth:𝔹].
∀[k:ℤ]. ∀[v:Cmd].
(<d, i, mk-msg(auth;``new_23_sig decided``;<k, v>)> ∈ 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))\000C)
∨ ((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)
∧ (((no new_23_sig_Quorum(Cmd;cmdeq;coeff;flrs;notify;propose;reps;f) <z1, 0> between e' and 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, 0> msgval(e) new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.\000Ce';e)))
∧ (||fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e';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, 0>\000C;es.e';e)))];snd(fst(msgval(e)))))) = ((coeff * flrs) + 1) ∈ ℤ)
∧ i ↓∈ reps
∧ auth = ff
∧ (k = (fst(fst(fst(msgval(e))))) ∈ ℤ)
∧ (v
= (snd(poss-maj(cmdeq;[snd(fst(msgval(e))) /
(fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e';e)))];snd(fst(\000Cmsgval(e))))))
∈ Cmd))
∨ (∃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))))
∧ (no new_23_sig_Quorum(Cmd;cmdeq;coeff;flrs;notify;propose;reps;f) <z1, z6> between e1 and 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,\000C z6>;es.e1;e)))
∧ (||fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, z6>;es.e1;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>;es.e1;e)))];sn\000Cd(fst(msgval(e))))))
= ((coeff * flrs) + 1)
∈ ℤ)
∧ i ↓∈ reps
∧ auth = ff
∧ (k = (fst(fst(fst(msgval(e))))) ∈ ℤ)
∧ (v
= (snd(poss-maj(cmdeq;[snd(fst(msgval(e))) /
(fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, z6>;es.e1;e)))];s\000Cnd(fst(msgval(e))))))
∈ 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_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_headers_type: new_23_sig_headers_type{i:l}(Cmd;notify;propose),
msg-interface: Interface,
mk-msg: mk-msg(auth;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),
no-classrel-in-interval: (no X between start and 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,
bfalse: ff,
bool: 𝔹,
less_than: a < b,
uall: ∀[x:A]. B[x],
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,
eqtt_to_assert,
assert-name_eq,
sq_stable__no_repeats,
true_wf,
eqff_to_assert,
bool_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
classrel_wf,
msg-interface_wf,
new_23_sig_main_wf,
make-msg-interface_wf,
mk-msg_wf,
subtype_rel_weakening,
ext-eq_weakening,
bag-member_wf,
es-loc_wf,
event-ordering+_subtype,
exists_wf,
es-E_wf,
es-le_wf,
or_wf,
equal-wf-T-base,
es-header_wf,
has-es-info-type_wf,
es-info-body_wf,
equal-wf-base-T,
int_subtype_base,
not_wf,
assert_wf,
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,
eo-forward_wf,
member-eo-forward-E,
list_wf,
poss-maj_wf,
nat_wf,
es-info-type_wf,
pi1_wf_top,
top_wf,
subtype_rel_product,
subtype_top,
subtype_rel_transitivity,
eo-forward-E-subtype,
less_than_wf,
new_23_sig_NewRoundsStateFun_wf,
event-ordering+_wf,
Message_wf,
subtype_rel_dep_function,
new_23_sig_headers_type_wf,
set-sig_wf,
deq_wf,
bag_wf,
Id_wf,
set_wf,
valueall-type_wf,
new_23_sig-ilf,
eo-forward-header,
eo-forward-has-es-info-type,
make-Msg_wf,
eo-forward-info-body,
eclass0_wf,
new_23_sig_decided'base_wf,
new_23_sig_decision_wf,
msg-header_wf,
hdrmkmsg_lemma,
hdrmakeMsg_lemma,
false_wf,
and_wf,
make-Msg-as-mk-msg,
mk-msg-equal,
bfalse_wf,
subtype_rel_set,
equal_functionality_wrt_subtype_rel2
Latex:
\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{}[auth:\mBbbB{}]. \mforall{}[k:\mBbbZ{}]. \mforall{}[v:Cmd].
(<d, i, mk-msg(auth;``new\_23\_sig decided``;<k, v>)> \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{} (((no new\_23\_sig\_Quorum(Cmd;cmdeq;coeff;flrs;notify;propose;reps;f) <z1, 0> between e\000C' and e)
\mwedge{} ((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, 0> msgval(e) new\_23\_sig\_QuorumStateFun(Cmd;notify;p\000Cropose;f;<z1, 0>es.e';e)))
\mwedge{} (||fst(new\_23\_sig\_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>es.e';e))|| = (coeff \000C* flrs))
\mwedge{} (d = 0)
\mwedge{} ((fst(poss-maj(cmdeq;[snd(fst(msgval(e))) /
(fst(new\_23\_sig\_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>e\000Cs.e';e)))];snd(fst(msgval(e))))))
= ((coeff * flrs) + 1))
\mwedge{} i \mdownarrow{}\mmember{} reps
\mwedge{} auth = ff
\mwedge{} (k = (fst(fst(fst(msgval(e))))))
\mwedge{} (v
= (snd(poss-maj(cmdeq;[snd(fst(msgval(e))) /
(fst(new\_23\_sig\_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>\000Ces.e';e)))];snd(fst(msgval(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{} (no new\_23\_sig\_Quorum(Cmd;cmdeq;coeff;flrs;notify;propose;reps;f) <z1, z6> be\000Ctween e1 and e)
\mwedge{} ((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;n\000Cotify;propose;f;<z1, z6>es.e1;e)))
\mwedge{} (||fst(new\_23\_sig\_QuorumStateFun(Cmd;notify;propose;f;<z1, z6>es.e1;e))|| = \000C(coeff * flrs))
\mwedge{} (d = 0)
\mwedge{} ((fst(poss-maj(cmdeq;[snd(fst(msgval(e))) /
(fst(new\_23\_sig\_QuorumStateFun(Cmd;notify;...;f;<z1
, z6
>es.e1\000C;e)))];snd(fst(msgval(e))))))
= ((coeff * flrs) + 1))
\mwedge{} i \mdownarrow{}\mmember{} reps
\mwedge{} auth = ff
\mwedge{} (k = (fst(fst(fst(msgval(e))))))
\mwedge{} (v
= (snd(poss-maj(cmdeq;[snd(fst(msgval(e))) /
(fst(new\_23\_sig\_QuorumStateFun(Cmd;notify;...;f;<z1
, z6
>es.e\000C1;e)))];snd(fst(msgval(e)))))))))))))
Date html generated:
2015_07_23-PM-03_56_43
Last ObjectModification:
2015_02_04-PM-03_28_17
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