Nuprl Lemma : pv11_p1_A4_C1
∀Cmd:ValueAllType. ∀f:pv11_p1_headers_type{i:l}(Cmd). ∀es:EO+(Message(f)). ∀e1,e2:E. ∀accpts,ldrs:bag(Id).
∀ldrs_uid:Id ─→ ℤ. ∀reps:bag(Id). ∀bnum1,bnum2:pv11_p1_Ballot_Num(). ∀accepted1,accepted2:(pv11_p1_Ballot_Num()
× ℤ
× Cmd) List. ∀i1,i2,l:Id.
∀b:pv11_p1_Ballot_Num(). ∀s:ℤ. ∀p1,p2:Cmd.
(Inj(Id;ℤ;ldrs_uid)
⇒ pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)
⇒ ((<bnum1, accepted1> ∈ pv11_p1_AcceptorState(Cmd;ldrs_uid;f)(e1)
∧ <bnum2, accepted2> ∈ pv11_p1_AcceptorState(Cmd;ldrs_uid;f)(e2)
∧ (<b, s, p1> ∈ accepted1)
∧ (<b, s, p2> ∈ accepted2))
∨ (pv11_p1_p2a'send(Cmd;f) i1 <l, b, s, p1> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e1)
∧ pv11_p1_p2a'send(Cmd;f) i2 <l, b, s, p2> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e2)))
⇒ (p1 = p2 ∈ Cmd))
Proof
Definitions occuring in Statement :
pv11_p1_p2a'send: pv11_p1_p2a'send(Cmd;f),
pv11_p1_message-constraint: pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; mf; es),
pv11_p1_main: pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;mf),
pv11_p1_AcceptorState: pv11_p1_AcceptorState(Cmd;ldrs_uid;mf),
pv11_p1_headers_type: pv11_p1_headers_type{i:l}(Cmd),
pv11_p1_Ballot_Num: pv11_p1_Ballot_Num(),
msg-interface: Interface,
Message: Message(f),
classrel: v ∈ X(e),
event-ordering+: EO+(Info),
es-E: E,
Id: Id,
l_member: (x ∈ l),
list: T List,
inject: Inj(A;B;f),
vatype: ValueAllType,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
apply: f a,
function: x:A ─→ B[x],
pair: <a, b>,
product: x:A × B[x],
int: ℤ,
equal: s = t ∈ T,
bag: bag(T)
Definitions :
assert: ↑b,
ifthenelse: if b then t else f fi ,
name_eq: name_eq(x;y),
name-deq: NameDeq,
list-deq: list-deq(eq),
cons: [a / b],
band: p ∧b q,
atom-deq: AtomDeq,
eq_atom: x =a y,
bfalse: ff,
false: False,
btrue: tt
Lemmas :
es-causl-swellfnd,
event-ordering+_subtype,
Message_wf,
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
or_wf,
classrel_wf,
pv11_p1_AcceptorState_wf,
l_member_wf,
msg-interface_wf,
pv11_p1_main_wf,
pv11_p1_p2a'send_wf,
Id_wf,
list_wf,
es-E_wf,
int_seg_wf,
int_seg_subtype-nat,
decidable__le,
subtract_wf,
false_wf,
not-ge-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
decidable__equal_int,
subtype_rel-int_seg,
le_weakening,
int_seg_properties,
le_wf,
nat_wf,
zero-le-nat,
lelt_wf,
es-causl_wf,
equal_wf,
decidable__lt,
not-equal-2,
le-add-cancel-alt,
not-le-2,
sq_stable__le,
add-mul-special,
zero-mul,
pv11_p1_acc_p2a,
base-noloc-classrel-make-Msg2,
cons_wf_listp,
cons_wf,
nil_wf,
listp_wf,
hdrmkmsg_lemma,
cons_member,
name_wf,
equal-wf-base,
msg-header_wf,
pv11_p1_headers_no_inputs_wf,
squash_wf,
exists_wf,
make-msg-interface_wf,
es-loc_wf,
msg-authentic_wf,
es-info_wf,
subtype_base_sq,
int_subtype_base,
bool_wf,
bool_subtype_base,
pv11_p1-p2a,
pv11_p1_bnum_p2a,
assert-union-deq,
unit_wf2,
product-deq_wf,
int-deq_wf,
id-deq_wf,
unit-deq_wf,
pv11_p1_mk_bnum_wf,
pv11_p1_Ballot_Num_wf,
union_subtype_base,
product_subtype_base,
atom2_subtype_base,
unit_subtype_base,
btrue_wf,
and_wf,
isl_wf,
bfalse_wf,
btrue_neq_bfalse,
outl_wf,
es-causl_transitivity2,
es-causle_weakening_locl,
pv11_p1_bnum_p2a_loc,
es-locl-trichotomy,
es-info-body_wf,
true_wf,
event-ordering+_wf,
pi2_wf,
pv11_p1_ldr_fun_mem_propose2,
base-classrel-equal,
subtype_rel_weakening,
ext-eq_weakening,
pi1_wf_top,
subtype_rel_product,
top_wf,
subtype_top,
pv11_p1_in_domain_wf,
assert_elim,
pv11_p1_LeaderStateFun_wf,
not_assert_elim,
assert_wf,
null_nil_lemma,
reduce_tl_cons_lemma,
es-header_wf,
tl_wf,
null_wf3,
subtype_rel_list,
null_cons_lemma,
pv11_p1_ldr_fun_mem_adopted,
iff_weakening_equal,
member_wf,
bag-member_wf,
list-subtype-bag,
not_wf,
pv11_p1_pmax_wf,
pv11_p1_lt_bnum_wf,
l_member-bag-member,
pv11_p1_ldr_active_fun,
pv11_p1_ldr_fun_proposal3,
es-info-type_wf,
subtype_rel_transitivity,
pair-eta,
pv11_p1_propose'base_wf,
pv11_p1_ldr_fun_ord,
pv11_p1_leq_bnum_wf,
pv11_p1_leq_bnum_implies_eq_or_lt,
pv11_p1_unique_adopted_fun2,
eclass_wf,
pv11_p1_adopted'base_wf,
es-locl_transitivity2,
es-le_weakening,
pv11_p1_lt_bnum_irrefl2,
pv11_p1_lt_bnum_trans2,
subtype_rel_dep_function,
vatype_wf,
pv11_p1_headers_type_wf,
pv11_p1_unique_adopted_fun,
bag_wf,
pv11_p1_pmax_desc_iff,
all_wf,
base-headers-msg-val-single-val,
base-noloc-classrel,
list_subtype_base,
atom_subtype_base,
assert-name_eq,
decidable__es-locl,
es-le-not-locl,
pv11_p1_leq_bnum_implies_eq,
has-es-info-type_wf,
es-pred-exists-between,
event-ordering+_cumulative2,
es-first_wf2,
es-pred_wf,
valueall-type_wf,
es-locl_wf,
event_ordering_wf,
es-pred-locl-implies-le,
es-pred-locl,
es-le_weakening_eq,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
assert-bnot,
member-eclass_wf,
member-base-class_iff,
pv11_p1_ldr_state_fun_eq,
classfun-res-base-classrel,
base-headers-msg-val_wf,
pv11_p1_eq_bnums_wf,
pv11_p1_eq_bnums-assert,
pv11_p1_when_adopted_wf,
pv11_p1_adopted_prior,
pv11_p1_LeaderState-classrel,
pv11_p1_adopted_from_acceptor,
es-causle_weakening_eq,
es-causle_wf,
equal-wf-T-base,
decidable__assert,
not-pv11_p1_leq_bnum
Latex:
\mforall{}Cmd:ValueAllType. \mforall{}f:pv11\_p1\_headers\_type\{i:l\}(Cmd). \mforall{}es:EO+(Message(f)). \mforall{}e1,e2:E.
\mforall{}accpts,ldrs:bag(Id). \mforall{}ldrs$_{uid}$:Id {}\mrightarrow{} \mBbbZ{}. \mforall{}reps:bag(Id). \mforall{}bnum1,bnum2:pv11\_p1\000C\_Ballot\_Num().
\mforall{}accepted1,accepted2:(pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List. \mforall{}i1,i2,l:Id. \mforall{}b:pv11\_p1\_Ballot\_Num().
\mforall{}s:\mBbbZ{}. \mforall{}p1,p2:Cmd.
(Inj(Id;\mBbbZ{};ldrs$_{uid}$)
{}\mRightarrow{} pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid\mbackslash{}ff7\000Cd$; reps; f; es)
{}\mRightarrow{} ((<bnum1, accepted1> \mmember{} pv11\_p1\_AcceptorState(Cmd;ldrs$_{uid}$;f)(e1)
\mwedge{} <bnum2, accepted2> \mmember{} pv11\_p1\_AcceptorState(Cmd;ldrs$_{uid}$;f)(e2)
\mwedge{} (<b, s, p1> \mmember{} accepted1)
\mwedge{} (<b, s, p2> \mmember{} accepted2))
\mvee{} (pv11\_p1\_p2a'send(Cmd;f) i1 <l, b, s, p1> \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$_{u\000Cid}$;reps;f)(e1)
\mwedge{} pv11\_p1\_p2a'send(Cmd;f) i2 <l, b, s, p2> \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$_{\000Cuid}$;reps;f)(e2)))
{}\mRightarrow{} (p1 = p2))
Date html generated:
2015_07_23-PM-04_54_00
Last ObjectModification:
2015_02_04-AM-08_20_36
Home
Index