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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