Nuprl Lemma : decidable__exists-unit-ball-approx-1-ext
∀k,n:ℕ.  ∀[P:unit-ball-approx(n;k) ⟶ ℙ]. ((∀p:unit-ball-approx(n;k). Dec(P[p])) 
⇒ Dec(∃p:unit-ball-approx(n;k). P[p]))
Proof
Definitions occuring in Statement : 
unit-ball-approx: unit-ball-approx(n;k)
, 
nat: ℕ
, 
decidable: Dec(P)
, 
uall: ∀[x:A]. B[x]
, 
prop: ℙ
, 
so_apply: x[s]
, 
all: ∀x:A. B[x]
, 
exists: ∃x:A. B[x]
, 
implies: P 
⇒ Q
, 
function: x:A ⟶ B[x]
Definitions unfolded in proof : 
member: t ∈ T
, 
it: ⋅
, 
subtract: n - m
, 
extend-approx-ball: extend-approx-ball(n;p;z)
, 
ifthenelse: if b then t else f fi 
, 
lt_int: i <z j
, 
btrue: tt
, 
bfalse: ff
, 
int_seg_decide: int_seg_decide(d;i;j)
, 
genrec-ap: genrec-ap, 
spreadn: spread4, 
decidable__exists-unit-ball-approx-1, 
unit-ball-approx0, 
decidable__exists_int_seg, 
decidable__cand, 
decidable__le, 
decidable__and2, 
decidable__and, 
decidable__not, 
decidable__less_than', 
decidable__implies, 
decidable__false, 
any: any x
, 
uall: ∀[x:A]. B[x]
, 
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w])
, 
so_apply: x[s1;s2;s3;s4]
, 
so_lambda: λ2x.t[x]
, 
top: Top
, 
so_apply: x[s]
, 
uimplies: b supposing a
Lemmas referenced : 
decidable__exists-unit-ball-approx-1, 
lifting-strict-decide, 
istype-void, 
strict4-decide, 
lifting-strict-less, 
unit-ball-approx0, 
decidable__exists_int_seg, 
decidable__cand, 
decidable__le, 
decidable__and2, 
decidable__and, 
decidable__not, 
decidable__less_than', 
decidable__implies, 
decidable__false
Rules used in proof : 
introduction, 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
cut, 
instantiate, 
extract_by_obid, 
hypothesis, 
sqequalRule, 
thin, 
sqequalHypSubstitution, 
equalityTransitivity, 
equalitySymmetry, 
isectElimination, 
baseClosed, 
isect_memberEquality_alt, 
voidElimination, 
independent_isectElimination
Latex:
\mforall{}k,n:\mBbbN{}.
    \mforall{}[P:unit-ball-approx(n;k)  {}\mrightarrow{}  \mBbbP{}]
        ((\mforall{}p:unit-ball-approx(n;k).  Dec(P[p]))  {}\mRightarrow{}  Dec(\mexists{}p:unit-ball-approx(n;k).  P[p]))
Date html generated:
2019_10_30-AM-11_28_24
Last ObjectModification:
2019_07_30-AM-11_34_07
Theory : real!vectors
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