Nuprl Lemma : consensus-accum-num-property1
∀[V:Type]
∀A:Id List. ∀t:ℕ+. ∀f:(V List) ⟶ V. ∀v0:V. ∀L:consensus-rcv(V;A) List.
let b,i,as,vs,v = consensus-accum-num-state(t;f;v0;L) in (filter(λr.i - 1 <z inning(r);L)
= []
∈ (consensus-rcv(V;A) List))
∧ (||as|| = ||vs|| ∈ ℤ)
∧ (zip(as;vs) = votes-from-inning(i - 1;L) ∈ (({b:Id| (b ∈ A)} × V) List))
∧ (0 ≤ i)
∧ 1 ≤ i supposing ¬↑null(L)
∧ (||values-for-distinct(IdDeq;votes-from-inning(i - 1;L))|| ≤ (2 * t))
Proof
Definitions occuring in Statement :
consensus-accum-num-state: consensus-accum-num-state(t;f;v0;L),
votes-from-inning: votes-from-inning(i;L),
rcvd-inning-gt: i <z inning(r),
consensus-rcv: consensus-rcv(V;A),
id-deq: IdDeq,
Id: Id,
values-for-distinct: values-for-distinct(eq;L),
zip: zip(as;bs),
l_member: (x ∈ l),
length: ||as||,
null: null(as),
filter: filter(P;l),
nil: [],
list: T List,
nat_plus: ℕ+,
assert: ↑b,
spreadn: let a,b,c,d,e = u in v[a; b; c; d; e],
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
not: ¬A,
and: P ∧ Q,
set: {x:A| B[x]} ,
lambda: λx.A[x],
function: x:A ⟶ B[x],
product: x:A × B[x],
multiply: n * m,
subtract: n - m,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
prop: ℙ,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
spreadn: let a,b,c,d,e = u in v[a; b; c; d; e],
and: P ∧ Q,
top: Top,
nat_plus: ℕ+,
consensus-accum-num-state: consensus-accum-num-state(t;f;v0;L),
list_accum: list_accum,
nil: [],
it: ⋅,
subtract: n - m,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
votes-from-inning: votes-from-inning(i;L),
values-for-distinct: values-for-distinct(eq;L),
cand: A c∧ B,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
true: True,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
consensus-rcv: consensus-rcv(V;A),
consensus-accum-num: consensus-accum-num(num;f;s;r),
cs-initial-rcv: Init[v],
spreadn: spread3,
let: let,
cs-rcv-vote: Vote[a;i;v],
guard: {T},
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
rcvd-inning-eq: inning(r) =z i,
rcvd-vote: rcvd-vote(x),
mapfilter: mapfilter(f;P;L),
rcv-vote?: rcv-vote?(x),
outr: outr(x),
bfalse: ff,
band: p ∧b q,
rcvd-inning-gt: i <z inning(r),
nat: ℕ,
ge: i ≥ j ,
uiff: uiff(P;Q),
pi1: fst(t),
pi2: snd(t),
bool: 𝔹,
unit: Unit,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
squash: ↓T,
unzip: unzip(as),
Id: Id,
sq_type: SQType(T)
Latex:
\mforall{}[V:Type]
\mforall{}A:Id List. \mforall{}t:\mBbbN{}\msupplus{}. \mforall{}f:(V List) {}\mrightarrow{} V. \mforall{}v0:V. \mforall{}L:consensus-rcv(V;A) List.
let b,i,as,vs,v = consensus-accum-num-state(t;f;v0;L) in (filter(\mlambda{}r.i - 1 <z inning(r);L) = [])
\mwedge{} (||as|| = ||vs||)
\mwedge{} (zip(as;vs) = votes-from-inning(i - 1;L))
\mwedge{} (0 \mleq{} i)
\mwedge{} 1 \mleq{} i supposing \mneg{}\muparrow{}null(L)
\mwedge{} (||values-for-distinct(IdDeq;votes-from-inning(i - 1;L))|| \mleq{} (2 * t))
Date html generated:
2016_05_16-PM-00_40_48
Last ObjectModification:
2016_01_17-PM-08_04_31
Theory : event-ordering
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