Nuprl Lemma : comparison-test-ext
∀y:ℕ ⟶ ℝ. (Σn.y[n]↓ ⇒ (∀x:ℕ ⟶ ℝ. Σn.x[n]↓ supposing ∀n:ℕ. (|x[n]| ≤ y[n])))
Proof
Definitions occuring in Statement :
series-converges: Σn.x[n]↓,
rleq: x ≤ y,
rabs: |x|,
real: ℝ,
nat: ℕ,
uimplies: b supposing a,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
member: t ∈ T,
so_apply: x[s],
so_lambda: λ2x.t[x],
accelerate: accelerate(k;f),
comparison-test,
converges-iff-cauchy-ext,
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],
top: Top,
uimplies: b supposing a,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
strict4: strict4(F),
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
has-value: (a)↓,
prop: ℙ,
or: P ∨ Q,
squash: ↓T
Lemmas referenced :
comparison-test,
lifting-strict-spread,
istype-void,
strict4-apply,
value-type-has-value,
int-value-type,
has-value_wf_base,
istype-base,
is-exception_wf,
istype-universe,
converges-iff-cauchy-ext
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,
independent_pairFormation,
lambdaFormation_alt,
callbyvalueAdd,
baseApply,
closedConclusion,
hypothesisEquality,
productElimination,
intEquality,
universeIsType,
addExceptionCases,
exceptionSqequal,
inrFormation_alt,
imageMemberEquality,
imageElimination,
inlFormation_alt
Latex:
\mforall{}y:\mBbbN{} {}\mrightarrow{} \mBbbR{}. (\mSigma{}n.y[n]\mdownarrow{} {}\mRightarrow{} (\mforall{}x:\mBbbN{} {}\mrightarrow{} \mBbbR{}. \mSigma{}n.x[n]\mdownarrow{} supposing \mforall{}n:\mBbbN{}. (|x[n]| \mleq{} y[n])))
Date html generated:
2019_10_29-AM-10_25_45
Last ObjectModification:
2019_04_02-AM-00_16_31
Theory : reals
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