Nuprl Lemma : new_23_sig_progress-step8-v2
∀[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
((((coeff * flrs) + 1) ≤ ||bs||)
∧ (∀x∈bs.(loc(fst(snd(x))) = loc(e) ∈ Id)
∧ e ≤loc fst(snd(x))
∧ <<<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)
∧ (∀e'∈[e, fst(snd(last(bs)))].(↑new_23_sig_vote_with_ballot(Cmd;notify;propose;f;es;e';n;0))
⇒ (e' ∈ map(λx.(fst(snd(x)));bs)))
∧ b_all(Id;[x∈reps|¬bbag-deq-member(IdDeq;x;faulty)];i.(i ∈ map(λi.(fst(i));bs)))
∧ (bs
= mapfilter(λe.<snd(msgval(e)), e, snd(fst(msgval(e)))>;
λe.new_23_sig_vote_with_ballot(Cmd;notify;propose;f;es;e;n;0);
[e, fst(snd(last(bs)))])
∈ ((Id × E × Cmd) List)))))
Proof
Definitions occuring in Statement :
new_23_sig_vote_with_ballot: new_23_sig_vote_with_ballot(Cmd;notify;propose;f;es;e;n;r),
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),
es-info-body: msgval(e),
Message: Message(f),
classrel: v ∈ X(e),
event-ordering+: EO+(Info),
es-interval: [e, e'],
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),
mapfilter: mapfilter(f;P;L),
l_all: (∀x∈L.P[x]),
last: last(L),
l_member: (x ∈ l),
map: map(f;as),
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),
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-filter: [x∈b|p[x]],
bag: bag(T),
set-sig-member: set-sig-member(s),
set-sig: set-sig{i:l}(Item)
Lemmas :
new_23_sig_progress-step7,
es-info-body_wf,
new_23_sig_vote_with_ballot-assert-type,
assert_wf,
new_23_sig_vote_with_ballot_wf,
subtype_top,
subtype_rel_product,
le-add-cancel,
zero-add,
add-associates,
add_functionality_wrt_le,
add-commutes,
add-swap,
minus-one-mul,
minus-add,
condition-implies-le,
vatype_wf,
name_wf,
subtype_rel_dep_function,
Id_wf,
length_wf,
decidable__lt,
le_wf,
false_wf,
int_upper_subtype_nat,
mul_bounds_1a,
top_wf,
subtype_rel_list,
non_null_iff_length,
last_wf,
pi2_wf,
pi1_wf_top,
es-interval_wf,
Message_wf,
event-ordering+_subtype,
es-E_wf,
mapfilter_wf,
es-causl_wf,
true_wf,
squash_wf,
es-loc_wf,
equal_wf,
and_wf,
l_before_wf,
es-causl_weakening,
not_wf,
es-blocl-iff,
es-locl_wf,
es-locl_irreflexivity,
es-le_weakening,
es-locl_transitivity2,
es-blocl_wf,
sublist-if-no_repeats,
es-locl-antireflexive,
l-ordered-no_repeats,
l_before-es-interval,
l_before_mapfilter,
member-mapfilter,
l_member_wf,
l_all_iff,
subtract_wf,
decidable__equal_int,
le-add-cancel2,
add-zero,
less-iff-le,
sq_stable__le,
not-le-2,
decidable__le,
list_wf,
equal-wf-T-base,
assert_of_null,
member-es-interval,
es-le_wf,
l_all_fwd,
iff_weakening_equal,
less_than_wf,
select_wf,
es-le-self,
minus-minus,
le_antisymmetry_iff,
le-add-cancel-alt,
not-equal-2,
lelt_wf,
subtract-is-less,
l-ordered-inst,
assert_of_band,
iff_weakening_uiff,
iff_transitivity,
assert-member-eclass,
int_subtype_base,
subtype_rel_transitivity,
es-info-type_wf,
assert_of_eq_int,
eq_int_wf,
eqtt_to_assert,
bool_wf,
member-eclass_wf,
ext-eq_weakening,
subtype_rel_weakening,
listp_wf,
nil_wf,
cons_wf,
cons_wf_listp,
base-noloc-classrel,
new_23_sig_vote'base_wf,
classrel_wf,
le_transitivity,
length_sublist,
new_23_sig_vote_with_ballot-assert-classrel,
es-le-loc,
valueall-type_wf,
l_all-mapfilter,
map_wf,
member-map,
es-interval-last,
last_member,
last-mapfilter3,
bag-member-map,
list-subtype-bag,
subtype_rel_self,
sq_stable__l_member,
decidable__equal_Id,
bag-member-sq-list-member,
bag-member_wf,
b_all_wf,
new_23_sig_vote_with_ballot-if-classrel,
filter_wf5,
null_wf3,
null-filter,
null-mapfilter,
set_wf,
l-ordered_wf,
l_all_wf2,
null_cons_lemma,
product_subtype_list,
length_of_nil_lemma,
null_nil_lemma,
list-cases
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
((((coeff * flrs) + 1) \mleq{} ||bs||)
\mwedge{} (\mforall{}x\mmember{}bs.(loc(fst(snd(x))) = loc(e))
\mwedge{} e \mleq{}loc fst(snd(x))
\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{}e'\mmember{}[e, fst(snd(last(bs)))].
(\muparrow{}new\_23\_sig\_vote\_with\_ballot(Cmd;notify;propose;f;es;e';n;0))
{}\mRightarrow{} (e' \mmember{} map(\mlambda{}x.(fst(snd(x)));bs)))
\mwedge{} b\_all(Id;[x\mmember{}reps|\mneg{}\msubb{}bag-deq-member(IdDeq;x;faulty)];i.(i \mmember{} map(\mlambda{}i.(fst(i));bs)))
\mwedge{} (bs
= mapfilter(\mlambda{}e.<snd(msgval(e)), e, snd(fst(msgval(e)))>
\mlambda{}e.new\_23\_sig\_vote\_with\_ballot(Cmd;notify;propose;f;es;e;n;0);
[e, fst(snd(last(bs)))])))))
Date html generated:
2015_07_23-PM-04_05_08
Last ObjectModification:
2015_07_16-AM-09_32_17
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