Nuprl Lemma : new_23_sig_progress2-step2
∀[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,r:ℤ]. ∀[c:Cmd]. ∀[faulty:bag(Id)]. ∀[e':E]. ∀[c':Cmd].
(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)
⇒ (e' ≤loc e
∧ <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')))
∧ (no new_23_sig_Notify(Cmd;clients;notify;propose;f) n between e' and e))
⇒ <<n, r>, c> ∈ new_23_sig_RoundInfo(Cmd;notify;propose;f)(e)
⇒ new_23_sig_NewRoundsStateFun(Cmd;notify;propose;f;n;es.e';e) < r
⇒ b_all(Id;reps;i.↓∃e1:E
((e < e1)
∧ (loc(e1) = i ∈ Id)
∧ <<<n, r>, c>, loc(e)> ∈ new_23_sig_vote'base(Cmd;notify;propose;f)(e1))))
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_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_RoundInfo: new_23_sig_RoundInfo(Cmd;notify;propose;f),
new_23_sig_vote'base: new_23_sig_vote'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),
msg-interface: Interface,
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-causl: (e < e'),
es-loc: loc(e),
es-E: E,
Id: Id,
deq: EqDecider(T),
list: T List,
int_upper: {i...},
nat: ℕ,
vatype: ValueAllType,
assert: ↑b,
less_than: a < b,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
not: ¬A,
squash: ↓T,
implies: 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,
b_all: b_all(T;b;x.P[x]),
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,
bag-member_wf,
less_than_wf,
new_23_sig_NewRoundsStateFun_wf,
eo-forward_wf,
member-eo-forward-E,
es-loc_wf,
classrel_wf,
new_23_sig_RoundInfo_wf,
es-le_wf,
new_23_sig_Proposal_wf,
not_wf,
assert_wf,
set-sig-member_wf,
new_23_sig_ReplicaStateFun_wf,
no-classrel-in-interval_wf,
msg-interface_wf,
new_23_sig_Notify_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_progress2-step1,
mk-msg-interface_wf,
make-Msg_wf,
subtype_rel_weakening,
ext-eq_weakening,
make-msg-interface-equal,
es-info_wf,
base-noloc-classrel-make-Msg2,
es-causl_wf,
new_23_sig_vote'base_wf,
bfalse_wf,
equal-wf-T-base,
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,r:\mBbbZ{}].
\mforall{}[c:Cmd]. \mforall{}[faulty:bag(Id)]. \mforall{}[e':E]. \mforall{}[c':Cmd].
(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{} (e' \mleq{}loc e
\mwedge{} <n, c'> \mmember{} new\_23\_sig\_Proposal(Cmd;notify;propose;f)(e')
\mwedge{} (\mneg{}\muparrow{}(set-sig-member(slots) n new\_23\_sig\_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e')))
\mwedge{} (no new\_23\_sig\_Notify(Cmd;clients;notify;propose;f) n between e' and e))
{}\mRightarrow{} <<n, r>, c> \mmember{} new\_23\_sig\_RoundInfo(Cmd;notify;propose;f)(e)
{}\mRightarrow{} new\_23\_sig\_NewRoundsStateFun(Cmd;notify;propose;f;n;es.e';e) < r
{}\mRightarrow{} b\_all(Id;reps;i.\mdownarrow{}\mexists{}e1:E
((e < e1)
\mwedge{} (loc(e1) = i)
\mwedge{} <<<n, r>, c>, loc(e)> \mmember{} new\_23\_sig\_vote'base(Cmd;notify;propose;f)(e1))))
Date html generated:
2015_07_23-PM-04_09_21
Last ObjectModification:
2015_02_04-PM-03_44_50
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