Nuprl Lemma : new_23_sig_rounds_strict_inc
∀Cmd:ValueAllType. ∀notify,propose:Atom List. ∀f:new_23_sig_headers_type{i:l}(Cmd;notify;propose). ∀es:EO+(Message(f)).
∀e1,e2:E. ∀n,round1,round2:ℤ.
((∃z:ℤ × ℤ × Cmd
∃round:ℤ
∃e:E
(e1 ≤loc e
∧ (e <loc e2)
∧ z ∈ new_23_sig_RoundInfo(Cmd;notify;propose;f)(e)
∧ round ∈ new_23_sig_NewRoundsState(Cmd;notify;propose;f) n(e)
∧ let z,cmd = z
in let n',round' = z
in (n' = n ∈ ℤ) ∧ round < round'))
⇒ (e1 <loc e2)
⇒ round1 ∈ new_23_sig_NewRoundsState(Cmd;notify;propose;f) n(e1)
⇒ round2 ∈ new_23_sig_NewRoundsState(Cmd;notify;propose;f) n(e2)
⇒ round1 < round2)
Proof
Definitions occuring in Statement :
new_23_sig_NewRoundsState: new_23_sig_NewRoundsState(Cmd;notify;propose;f),
new_23_sig_RoundInfo: new_23_sig_RoundInfo(Cmd;notify;propose;f),
new_23_sig_headers_type: new_23_sig_headers_type{i:l}(Cmd;notify;propose),
Message: Message(f),
classrel: v ∈ X(e),
event-ordering+: EO+(Info),
es-le: e ≤loc e' ,
es-locl: (e <loc e'),
es-E: E,
list: T List,
vatype: ValueAllType,
less_than: a < b,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
spread: spread def,
product: x:A × B[x],
int: ℤ,
atom: Atom,
equal: s = t ∈ T
Definitions unfolded in proof :
vatype: ValueAllType,
all: ∀x:A. B[x],
implies: P ⇒ Q,
new_23_sig_NewRoundsState: new_23_sig_NewRoundsState(Cmd;notify;propose;f),
member: t ∈ T,
squash: ↓T,
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
Memory1: Memory1(init;tr;X),
so_lambda: λ2x.t[x],
exists: ∃x:A. B[x],
cand: A c∧ B,
trans: Trans(T;x,y.E[x; y]),
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
new_23_sig_update_round: new_23_sig_update_round(Cmd),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
band: p ∧b q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
new_23_sig_RoundInfo: new_23_sig_RoundInfo(Cmd;notify;propose;f),
new_23_sig_vote'base: new_23_sig_vote'base(Cmd;notify;propose;f),
new_23_sig_retry'base: new_23_sig_retry'base(Cmd;notify;propose;f),
listp: A List+,
name: Name,
rev_implies: P ⇐ Q,
name_eq: name_eq(x;y),
name-deq: NameDeq,
list-deq: list-deq(eq),
list_ind: list_ind,
cons: [a / b],
atom-deq: AtomDeq,
eq_atom: x =a y,
new_23_sig_headers_type: new_23_sig_headers_type{i:l}(Cmd;notify;propose),
so_apply: x[s],
sq_stable: SqStable(P),
new_23_sig_headers: new_23_sig_headers(notify;propose),
new_23_sig_headers_fun: new_23_sig_headers_fun(Cmd;notify;propose),
encodes-msg-type: hdr encodes T,
nil: [],
null: null(as),
new_23_sig_vote2retry: new_23_sig_vote2retry(Cmd),
less_than: a < b
Latex:
\mforall{}Cmd:ValueAllType. \mforall{}notify,propose:Atom List. \mforall{}f:new\_23\_sig\_headers\_type\{i:l\}(Cmd;notify;propose).
\mforall{}es:EO+(Message(f)). \mforall{}e1,e2:E. \mforall{}n,round1,round2:\mBbbZ{}.
((\mexists{}z:\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd
\mexists{}round:\mBbbZ{}
\mexists{}e:E
(e1 \mleq{}loc e
\mwedge{} (e <loc e2)
\mwedge{} z \mmember{} new\_23\_sig\_RoundInfo(Cmd;notify;propose;f)(e)
\mwedge{} round \mmember{} new\_23\_sig\_NewRoundsState(Cmd;notify;propose;f) n(e)
\mwedge{} let z,cmd = z
in let n',round' = z
in (n' = n) \mwedge{} round < round'))
{}\mRightarrow{} (e1 <loc e2)
{}\mRightarrow{} round1 \mmember{} new\_23\_sig\_NewRoundsState(Cmd;notify;propose;f) n(e1)
{}\mRightarrow{} round2 \mmember{} new\_23\_sig\_NewRoundsState(Cmd;notify;propose;f) n(e2)
{}\mRightarrow{} round1 < round2)
Date html generated:
2016_05_17-PM-02_05_07
Last ObjectModification:
2016_01_18-AM-09_34_28
Theory : 2!3!consensus!with!signatures
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