Nuprl Lemma : new_23_sig_retry_property
∀[Cmd:ValueAllType]. ∀[eq:EqDecider(Cmd)]. ∀[reps,clients:bag(Id)]. ∀[coeff,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))].
(new_23_sig_message-constraint{i:l}(Cmd;clients;eq;coeff;flrs;notify;propose;reps;slots;f)
⇒ (∀e:E. ∀c:Cmd. ∀n,r:ℤ.
(<<n, r + 1>, c> ∈ new_23_sig_retry'base(Cmd;notify;propose;f)(e)
⇒ (↓∃x:Id
∃cs:Cmd List+
∃L:Id List
((no_repeats(Id;L) ∧ (||L|| = ((coeff * flrs) + 1) ∈ ℤ) ∧ (||cs|| = ||L|| ∈ ℤ))
∧ (∀i:ℕ||L||. ((↓∃e':E. mk-msg-interface(x;make-Msg(``new_23_sig vote``;<<<n, r>, cs[i]>, L[i]>)) ∈ new_\000C23_sig_main()(e')) ∧ L[i] ↓∈ reps))
∧ (c = (snd(poss-maj(eq;cs;hd(cs)))) ∈ Cmd))))))
Proof
Definitions occuring in Statement :
new_23_sig_message-constraint: new_23_sig_message-constraint{i:l}(Cmd;clients;cmdeq;coeff;flrs;notify;propose;reps;slots;f),
new_23_sig_main: new_23_sig_main(),
new_23_sig_retry'base: new_23_sig_retry'base(Cmd;notify;propose;f),
new_23_sig_headers_type: new_23_sig_headers_type{i:l}(Cmd;notify;propose),
mk-msg-interface: mk-msg-interface(l;m),
msg-interface: Interface,
make-Msg: make-Msg(hdr;val),
Message: Message(f),
classrel: v ∈ X(e),
event-ordering+: EO+(Info),
es-E: E,
Id: Id,
poss-maj: poss-maj(eq;L;x),
deq: EqDecider(T),
listp: A List+,
no_repeats: no_repeats(T;l),
select: L[n],
hd: hd(l),
length: ||as||,
cons: [a / b],
nil: [],
list: T List,
int_seg: {i..j-},
nat: ℕ,
vatype: ValueAllType,
uall: ∀[x:A]. B[x],
pi2: snd(t),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
and: P ∧ Q,
pair: <a, b>,
product: x:A × B[x],
multiply: n * m,
add: n + m,
natural_number: $n,
int: ℤ,
token: "$token",
atom: Atom,
equal: s = t ∈ T,
bag-member: x ↓∈ bs,
bag: bag(T),
set-sig: set-sig{i:l}(Item)
Definitions :
assert: ↑b,
ifthenelse: if b then t else f fi ,
deq-member: x ∈b L),
reduce: reduce(f;k;as),
new_23_sig_headers_internal: new_23_sig_headers_internal(),
cons: [a / b],
bor: p ∨bq,
list-deq: list-deq(eq),
band: p ∧b q,
atom-deq: AtomDeq,
eq_atom: x =a y,
btrue: tt,
nil: [],
it: ⋅,
null: null(as),
true: True,
member: t ∈ T
Lemmas :
int_seg_wf,
length_wf,
name_wf,
new_23_sig_headers_wf,
l_all_iff,
l_member_wf,
equal_wf,
new_23_sig_headers_fun_wf,
cons_member,
cons_wf,
cons_wf_listp,
listp_wf,
nil_wf,
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,
squash_wf,
true_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
base-noloc-classrel-make-Msg2,
hdrmkmsg_lemma,
msg-header_wf,
new_23_sig_headers_internal_wf,
exists_wf,
es-causl_wf,
classrel_wf,
msg-interface_wf,
new_23_sig_main_wf,
make-msg-interface_wf,
es-loc_wf,
Message_wf,
subtype_rel_dep_function,
vatype_wf,
new_23_sig_retry'base_wf,
es-E_wf,
event-ordering+_subtype,
new_23_sig_message-constraint_wf,
event-ordering+_wf,
new_23_sig_headers_type_wf,
set-sig_wf,
list_wf,
nat_wf,
bag_wf,
Id_wf,
deq_wf,
valueall-type_wf,
msg-authentic_wf,
es-info_wf,
new_23_sig_assert_newvote,
es-info-body_wf,
new_23_sig_QuorumStateFun_wf,
eo-forward_wf,
member-eo-forward-E,
mul_bounds_1a,
eo-forward-E-subtype,
new_23_sig-retry,
new_23_sig_quorum_invariant_fun,
new_23_sig_quorum_inv_vote_fun,
length_of_cons_lemma,
reduce_hd_cons_lemma,
no_repeats_cons,
select-cons,
le_int_wf,
assert_of_le_int,
le_wf,
no_repeats_wf,
all_wf,
mk-msg-interface_wf,
make-Msg_wf,
select_wf,
sq_stable__le,
length_cons,
non_neg_length,
length_wf_nat,
poss-maj_wf,
subtype_rel_weakening,
ext-eq_weakening,
bag-member_wf,
less_than_transitivity1,
le_weakening,
hd_wf,
decidable__le,
false_wf,
not-ge-2,
le_antisymmetry_iff,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
add-commutes,
add-associates,
zero-add,
add_functionality_wrt_le,
le-add-cancel2,
assert-deq-member,
es-info-mk-msg,
event-ordering+_cumulative,
es-info-auth_wf,
pair-eta,
subtype_rel_product,
top_wf,
subtype_top,
mk-msg_wf,
eclass_wf,
new_23_sig-vote,
decidable__equal_int,
not-equal-2,
int_subtype_base,
subtract_wf,
not-le-2,
minus-zero,
add-zero,
minus-minus,
le-add-cancel,
decidable__lt,
less-iff-le,
lelt_wf,
and_wf,
pi1_wf_top,
add_functionality_wrt_eq,
pi2_wf,
eo-forward-base-classrel,
eo-forward-locl,
atom2_subtype_base,
sq_stable__bag-member
Latex:
\mforall{}[Cmd:ValueAllType]. \mforall{}[eq:EqDecider(Cmd)]. \mforall{}[reps,clients:bag(Id)]. \mforall{}[coeff,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))].
(new\_23\_sig\_message-constraint\{i:l\}(Cmd;clients;eq;coeff;flrs;notify;propose;reps;slots;f)
{}\mRightarrow{} (\mforall{}e:E. \mforall{}c:Cmd. \mforall{}n,r:\mBbbZ{}.
(<<n, r + 1>, c> \mmember{} new\_23\_sig\_retry'base(Cmd;notify;propose;f)(e)
{}\mRightarrow{} (\mdownarrow{}\mexists{}x:Id
\mexists{}cs:Cmd List\msupplus{}
\mexists{}L:Id List
((no\_repeats(Id;L) \mwedge{} (||L|| = ((coeff * flrs) + 1)) \mwedge{} (||cs|| = ||L||))
\mwedge{} (\mforall{}i:\mBbbN{}||L||
((\mdownarrow{}\mexists{}e':E
mk-msg-interface(x;make-Msg(``new\_23\_sig vote``;<<<n, r>, cs[i]>, L[i]>)) \mmember{}\000C new\_23\_sig\_main()(e'))
\mwedge{} L[i] \mdownarrow{}\mmember{} reps))
\mwedge{} (c = (snd(poss-maj(eq;cs;hd(cs))))))))))
Date html generated:
2015_07_23-PM-03_58_47
Last ObjectModification:
2015_02_04-PM-01_47_26
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