Nuprl Lemma : two-intersecting-wait-set-exists'
∀t:ℕ. ∀A:Id List.
(∃W:Id List List
((∀ws:Id List. ((ws ∈ W) ⇐⇒ (||ws|| = (t + 1) ∈ ℤ) ∧ no_repeats(Id;ws) ∧ (∀x∈ws.(x ∈ A))))
∧ (∀ws1∈W.(∀ws2∈W.∃a:Id. ((a ∈ ws1) ∧ (a ∈ ws2)))))) supposing
(no_repeats(Id;A) and
(||A|| = ((2 * t) + 1) ∈ ℤ))
Proof
Definitions occuring in Statement :
Id: Id,
l_all: (∀x∈L.P[x]),
no_repeats: no_repeats(T;l),
l_member: (x ∈ l),
length: ||as||,
list: T List,
nat: ℕ,
uimplies: b supposing a,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
multiply: n * m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
and: P ∧ Q,
subtype_rel: A ⊆r B,
prop: ℙ,
cand: A c∧ B,
so_lambda: λ2x.t[x],
nat: ℕ,
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
squash: ↓T,
sq_stable: SqStable(P),
l_all: (∀x∈L.P[x]),
top: Top,
not: ¬A,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
guard: {T},
sq_type: SQType(T),
Id: Id,
l_member: (x ∈ l),
uiff: uiff(P;Q),
less_than: a < b,
lelt: i ≤ j < k,
int_seg: {i..j-},
two-intersection: two-intersection(A;W)
Latex:
\mforall{}t:\mBbbN{}. \mforall{}A:Id List.
(\mexists{}W:Id List List
((\mforall{}ws:Id List. ((ws \mmember{} W) \mLeftarrow{}{}\mRightarrow{} (||ws|| = (t + 1)) \mwedge{} no\_repeats(Id;ws) \mwedge{} (\mforall{}x\mmember{}ws.(x \mmember{} A))))
\mwedge{} (\mforall{}ws1\mmember{}W.(\mforall{}ws2\mmember{}W.\mexists{}a:Id. ((a \mmember{} ws1) \mwedge{} (a \mmember{} ws2)))))) supposing
(no\_repeats(Id;A) and
(||A|| = ((2 * t) + 1)))
Date html generated:
2016_05_16-PM-00_02_18
Last ObjectModification:
2016_04_03-PM-05_04_43
Theory : event-ordering
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