Nuprl Lemma : new_23_sig_progress-step7
∀[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)))
⇒ (↓∃bs:(Id × E × Cmd) List
((bag-map(λi.(fst(i));bs) = [x∈reps|¬bbag-deq-member(IdDeq;x;faulty)] ∈ bag(Id))
∧ (∀x∈bs.(loc(fst(snd(x))) = loc(e) ∈ Id)
∧ <<<n, 0>, snd(snd(x))>, fst(x)> ∈ new_23_sig_vote'base(Cmd;notify;propose;f)(fst(snd(x))))
∧ l-ordered(Id × E × Cmd;x,y.(fst(snd(x)) <loc fst(snd(y)));bs)
∧ (∀x∈bs.e ≤loc fst(snd(x)) )
∧ (((coeff * flrs) + 1) ≤ ||bs||))))
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_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),
Message: Message(f),
classrel: v ∈ X(e),
event-ordering+: EO+(Info),
es-le: e ≤loc e' ,
es-locl: (e <loc e'),
es-loc: loc(e),
es-E: E,
id-deq: IdDeq,
Id: Id,
deq: EqDecider(T),
l_all: (∀x∈L.P[x]),
length: ||as||,
list: T List,
int_upper: {i...},
nat: ℕ,
vatype: ValueAllType,
bnot: ¬bb,
assert: ↑b,
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
le: A ≤ B,
exists: ∃x:A. B[x],
not: ¬A,
squash: ↓T,
implies: P ⇒ Q,
and: 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-deq-member: bag-deq-member(eq;x;b),
l-ordered: l-ordered(T;x,y.R[x; y];L),
bag-member: x ↓∈ bs,
bag-no-repeats: bag-no-repeats(T;bs),
bag-size: #(bs),
bag-filter: [x∈b|p[x]],
bag-map: bag-map(f;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-step6,
bag-map_wf,
list-subtype-bag,
subtype_rel_self,
l_all_wf2,
new_23_sig_vote'base_wf,
l-ordered_wf,
es-locl_wf,
es-le_wf,
le_wf,
l_all_fwd,
es-locl-trichotomy,
es-le_weakening,
es-le-self,
and_wf,
new_23_sig_replica_state_mem_fun,
new_23_sig_proposal_classrel,
eclass0-classrel,
new_23_sig_vote2prop_wf,
bag-member-single,
new_23_sig_propose'base_wf,
map_length,
bnot_wf,
bag-deq-member_wf,
id-deq_wf,
bag-filter_wf,
bag-filter-split,
bag-size-append,
bag-size-filter-member-bound,
le_transitivity
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{}bs:(Id \mtimes{} E \mtimes{} Cmd) List
((bag-map(\mlambda{}i.(fst(i));bs) = [x\mmember{}reps|\mneg{}\msubb{}bag-deq-member(IdDeq;x;faulty)])
\mwedge{} (\mforall{}x\mmember{}bs.(loc(fst(snd(x))) = loc(e))
\mwedge{} <<<n, 0>, snd(snd(x))>, fst(x)> \mmember{} new\_23\_sig\_vote'base(Cmd;notify;propose;f)(
fst(snd(x))))
\mwedge{} l-ordered(Id \mtimes{} E \mtimes{} Cmd;x,y.(fst(snd(x)) <loc fst(snd(y)));bs)
\mwedge{} (\mforall{}x\mmember{}bs.e \mleq{}loc fst(snd(x)) )
\mwedge{} (((coeff * flrs) + 1) \mleq{} ||bs||))))
Date html generated:
2015_07_23-PM-04_04_06
Last ObjectModification:
2015_02_04-PM-03_53_12
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