Nuprl Lemma : three-intersecting-wait-set
∀t:ℕ. ∀A:Id List.
({a:Id| (a ∈ A)} ~ ℕ(3 * t) + 1
⇒ (∀W:{a:Id| (a ∈ A)} List List
((∀ws:{a:Id| (a ∈ A)} List. ((ws ∈ W) ⇐⇒ (||ws|| = ((2 * t) + 1) ∈ ℤ) ∧ no_repeats({a:Id| (a ∈ A)} ;ws)))
⇒ three-intersection(A;W))))
Proof
Definitions occuring in Statement :
three-intersection: three-intersection(A;W),
equipollent: A ~ B,
Id: Id,
no_repeats: no_repeats(T;l),
l_member: (x ∈ l),
length: ||as||,
list: T List,
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
multiply: n * m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
three-intersection: three-intersection(A;W),
n-intersecting: n-intersecting(A;T;n),
subtract: n - m,
cand: A c∧ B,
combination: Combination(n;T),
length: ||as||,
list_ind: list_ind,
cons: [a / b],
nil: [],
it: ⋅,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
guard: {T}
Latex:
\mforall{}t:\mBbbN{}. \mforall{}A:Id List.
(\{a:Id| (a \mmember{} A)\} \msim{} \mBbbN{}(3 * t) + 1
{}\mRightarrow{} (\mforall{}W:\{a:Id| (a \mmember{} A)\} List List
((\mforall{}ws:\{a:Id| (a \mmember{} A)\} List
((ws \mmember{} W) \mLeftarrow{}{}\mRightarrow{} (||ws|| = ((2 * t) + 1)) \mwedge{} no\_repeats(\{a:Id| (a \mmember{} A)\} ;ws)))
{}\mRightarrow{} three-intersection(A;W))))
Date html generated:
2016_05_16-PM-00_01_36
Last ObjectModification:
2015_12_29-PM-01_23_32
Theory : event-ordering
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