Nuprl Lemma : new_23_sig_progress-step9
∀[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|| ∈ ℤ)
∧ (∀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)))
∧ (∀e'∈[e;fst(snd(x))).¬↑new_23_sig_vote_with_ballot_and_id(Cmd;notify;propose;f;es;e';n;0;fst(x))))
∧ l-ordered(Id × E × Cmd;x,y.(fst(snd(x)) <loc fst(snd(y)));bs)
∧ (∀e'∈[e, fst(snd(last(bs)))].
∀i:Id
((↑new_23_sig_vote_with_ballot_and_id(Cmd;notify;propose;f;es;e';n;0;i))
⇒ (∀e''∈[e;e').¬↑new_23_sig_vote_with_ballot_and_id(Cmd;notify;propose;f;es;e'';n;0;i))
⇒ (e' ∈ map(λx.(fst(snd(x)));bs))))
∧ no_repeats(Id;map(λx.(fst(x));bs)))))
Proof
Definitions occuring in Statement :
new_23_sig_vote_with_ballot_and_id: new_23_sig_vote_with_ballot_and_id(Cmd;notify;propose;f;es;e;n;r;i),
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-closed-open-interval: [e;e'),
es-interval: [e, e'],
es-le: e ≤loc e' ,
es-locl: (e <loc e'),
es-loc: loc(e),
es-E: E,
Id: Id,
deq: EqDecider(T),
l_all: (∀x∈L.P[x]),
no_repeats: no_repeats(T;l),
last: last(L),
l_member: (x ∈ l),
map: map(f;as),
length: ||as||,
list: T List,
int_upper: {i...},
nat: ℕ,
vatype: ValueAllType,
assert: ↑b,
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
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,
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: bag(T),
set-sig-member: set-sig-member(s),
set-sig: set-sig{i:l}(Item)
Lemmas :
new_23_sig_progress-step8,
mul_bounds_1a,
int_upper_subtype_nat,
false_wf,
le_wf,
squash_wf,
true_wf,
remove-repeats-fun-length-as-remove-repeats-map,
Id_wf,
es-E_wf,
event-ordering+_subtype,
Message_wf,
id-deq_wf,
iff_weakening_equal,
sub-bag-list-if-bag-no-repeats-sq,
bag-filter_wf,
bnot_wf,
bag-deq-member_wf,
subtype_rel_bag,
assert_wf,
map_wf,
bag-filter-no-repeats,
sq_stable__le,
length_wf,
remove-repeats_wf,
bag-size_wf,
bag-remove-repeats_wf,
bag-size-append,
nat_wf,
bag-remove-repeats-filter,
bag-filter-split,
equal_wf,
bag-size-filter-member-bound,
le_transitivity,
bag-remove-repeats-eq-remove-repeats,
bag-remove-repeats-append,
add_functionality_wrt_eq,
bool_wf,
bag-remove-repeats-if-no-repeats,
list-decomp-nat,
remove-repeats-fun_wf,
decidable__le,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
decidable__lt,
subtype_rel_dep_function,
name_wf,
vatype_wf,
le-add-cancel2,
lelt_wf,
non_null_iff_length,
subtype_rel_list,
top_wf,
le_antisymmetry_iff,
remove-repeats-fun-sublist,
sublist_append1,
sublist_wf,
sublist_transitivity,
l_all_wf2,
l_member_wf,
es-loc_wf,
es-le_wf,
classrel_wf,
new_23_sig_vote'base_wf,
es-closed-open-interval_wf,
not_wf,
new_23_sig_vote_with_ballot_and_id_wf,
l-ordered_wf,
es-locl_wf,
es-interval_wf,
last_wf,
list-cases,
null_nil_lemma,
length_of_nil_lemma,
product_subtype_list,
null_cons_lemma,
all_wf,
no_repeats_wf,
l_all_sublist,
l_all_iff,
new_23_sig_vote_with_ballot_and_id-implies,
member-es-closed-open-interval,
member-es-interval,
member_sublist,
decidable__equal_int,
subtract_wf,
l-ordered-inst,
es-le_weakening,
es-locl_transitivity2,
minus-minus,
es-le-self,
and_wf,
pi2_wf,
pi1_wf_top,
subtype_rel_product,
subtype_top,
select_wf,
less_than_wf,
list_wf,
l_all_fwd,
new_23_sig_vote_with_ballot_wf,
remove-repeats-fun-member,
member-map,
es-le-loc,
int_trichot,
es-locl_irreflexivity,
es-causl_wf,
select_member,
new_23_sig_vote_with_ballot_and_id-assert-classrel,
base-noloc-classrel,
cons_wf_listp,
cons_wf,
nil_wf,
listp_wf,
subtype_rel_weakening,
ext-eq_weakening,
es-info-body_wf,
subtype_base_sq,
list_subtype_base,
atom_subtype_base,
l-ordered-sublist,
last_member,
subtract-is-less,
not-equal-2,
le-add-cancel-alt,
less-iff-le,
es-le_transitivity,
less_than_transitivity2,
le_weakening2,
int_seg_wf,
new_23_sig_vote_with_ballot_and_id-if-classrel,
event-ordering+_wf,
eclass_wf,
map_append_sq,
member_append,
l-ordered-remove-repeats-fun,
l-ordered-append,
sublist_map,
remove-repeats-fun-as-remove-repeats-map,
remove-repeats-no_repeats,
no_repeats-sublist
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) = ||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{} (\mforall{}e'\mmember{}[e;fst(snd(x))).
\mneg{}\muparrow{}new\_23\_sig\_vote\_with\_ballot\_and\_id(Cmd;notify;propose;f;es;e';n;0;fst(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)))].
\mforall{}i:Id
((\muparrow{}new\_23\_sig\_vote\_with\_ballot\_and\_id(Cmd;notify;propose;f;es;e';n;0;i))
{}\mRightarrow{} (\mforall{}e''\mmember{}[e;e').
\mneg{}\muparrow{}new\_23\_sig\_vote\_with\_ballot\_and\_id(Cmd;notify;propose;f;es;e'';n;0;i))
{}\mRightarrow{} (e' \mmember{} map(\mlambda{}x.(fst(snd(x)));bs))))
\mwedge{} no\_repeats(Id;map(\mlambda{}x.(fst(x));bs)))))
Date html generated:
2015_07_23-PM-04_06_50
Last ObjectModification:
2015_02_04-PM-03_57_26
Home
Index