Nuprl Lemma : decidable__exists-unit-ball-approx
∀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 :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
subtype_rel: A ⊆r B,
so_apply: x[s],
prop: ℙ,
exists: ∃x:A. B[x]
Lemmas referenced :
unit-ball-ex-decider_wf,
decidable_wf,
unit-ball-approx_wf,
istype-nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
rename,
introduction,
applyEquality,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality_alt,
isectElimination,
equalityTransitivity,
equalitySymmetry,
isectIsType,
because_Cache,
functionIsType,
universeIsType,
productEquality,
universeEquality,
inhabitedIsType
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_32
Last ObjectModification:
2019_07_30-AM-11_37_35
Theory : real!vectors
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