Nuprl Lemma : consensus-safety1
∀V:Type
((∀v1,v2:V. Dec(v1 = v2 ∈ V))
⇒ {∃v,v':V. (¬(v = v' ∈ V))}
⇒ (∀L:V List. Dec(∃v:V. (¬(v ∈ L))))
⇒ (∀A:Id List. ∀W:{a:Id| (a ∈ A)} List List.
((||W|| ≥ 1 )
⇒ two-intersection(A;W)
⇒ (∃f:ConsensusState ⟶ consensus-state1(V)
((∀v:V. ∀s:ts-reachable(consensus-ts4(V;A;W)).
((f s) = Decided[v] ∈ consensus-state1(V) ⇐⇒ ∃i:ℕ. in state s, inning i has committed v))
∧ ts-refinement(consensus-ts1(V);consensus-ts4(V;A;W);f))))))
Proof
Definitions occuring in Statement :
two-intersection: two-intersection(A;W),
cs-inning-committed: in state s, inning i has committed v,
consensus-ts4: consensus-ts4(V;A;W),
consensus-state4: ConsensusState,
consensus-ts1: consensus-ts1(T),
cs-decided: Decided[v],
consensus-state1: consensus-state1(V),
Id: Id,
l_member: (x ∈ l),
length: ||as||,
list: T List,
nat: ℕ,
decidable: Dec(P),
guard: {T},
ge: i ≥ j ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T,
ts-refinement: ts-refinement(ts1;ts2;f),
ts-reachable: ts-reachable(ts)
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
uimplies: b supposing a,
exists: ∃x:A. B[x],
guard: {T},
prop: ℙ,
ge: i ≥ j ,
so_lambda: λ2x.t[x],
so_apply: x[s],
consensus-ts2: consensus-ts2(T),
ts-type: ts-type(ts),
consensus-ts3: consensus-ts3(T),
pi1: fst(t),
consensus-ts1: consensus-ts1(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
subtype_rel: A ⊆r B,
consensus-state2: consensus-state2(T),
consensus-state1: consensus-state1(V),
top: Top,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
consensus-ts4: consensus-ts4(V;A;W),
ts-reachable: ts-reachable(ts),
infix_ap: x f y,
consensus-state4: ConsensusState,
nat: ℕ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
compose: f o g,
list: T List,
cs-ambivalent: AMBIVALENT,
cs-is-decided: cs-is-decided(x),
isl: isl(x),
squash: ↓T,
true: True,
cs-predecided: PREDECIDED[v],
cs-decided: Decided[v],
not: ¬A,
l_member: (x ∈ l),
cand: A c∧ B,
cs-ref-map-constraints: cs-ref-map-constraints(V;A;W;f),
cs-inning-committed: in state s, inning i has committed v,
one-intersection: one-intersection(A;W),
cs-archived: by state s, a archived v in inning i,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
cs-undecided: UNDECIDED
Latex:
\mforall{}V:Type
((\mforall{}v1,v2:V. Dec(v1 = v2))
{}\mRightarrow{} \{\mexists{}v,v':V. (\mneg{}(v = v'))\}
{}\mRightarrow{} (\mforall{}L:V List. Dec(\mexists{}v:V. (\mneg{}(v \mmember{} L))))
{}\mRightarrow{} (\mforall{}A:Id List. \mforall{}W:\{a:Id| (a \mmember{} A)\} List List.
((||W|| \mgeq{} 1 )
{}\mRightarrow{} two-intersection(A;W)
{}\mRightarrow{} (\mexists{}f:ConsensusState {}\mrightarrow{} consensus-state1(V)
((\mforall{}v:V. \mforall{}s:ts-reachable(consensus-ts4(V;A;W)).
((f s) = Decided[v] \mLeftarrow{}{}\mRightarrow{} \mexists{}i:\mBbbN{}. in state s, inning i has committed v))
\mwedge{} ts-refinement(consensus-ts1(V);consensus-ts4(V;A;W);f))))))
Date html generated:
2016_05_16-PM-00_15_39
Last ObjectModification:
2016_01_17-PM-03_56_29
Theory : event-ordering
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