Nuprl Lemma : approx-zero
∀I:{I:Interval| icompact(I)} . ∀n:ℕ. ∀f:{f:I^n ⟶ ℝ| ∀a,b:I^n. (req-vec(n;a;b) ⇒ ((f a) = (f b)))} .
((¬(∀x:I^n. f x ≠ r0)) ⇒ (∀e:{e:ℝ| r0 < e} . ∃x:I^n. (|f x| < e)))
Proof
Definitions occuring in Statement :
interval-vec: I^n,
req-vec: req-vec(n;x;y),
icompact: icompact(I),
interval: Interval,
rneq: x ≠ y,
rless: x < y,
rabs: |x|,
req: x = y,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
exists: ∃x:A. B[x],
not: ¬A,
false: False,
interval-vec: I^n,
sq_stable: SqStable(P),
rdiv: (x/y),
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
top: Top,
subtype_rel: A ⊆r B,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
rdiv_wf,
int-to-real_wf,
rless-int,
rless_wf,
rmul_preserves_rless,
rless_transitivity2,
rabs_wf,
real_wf,
rneq_wf,
istype-void,
interval-vec_wf,
req-vec_wf,
req_wf,
istype-nat,
interval_wf,
icompact_wf,
rmul_wf,
itermSubtract_wf,
itermMultiply_wf,
itermConstant_wf,
rinv_wf2,
itermVar_wf,
sq_stable__rless,
radd-preserves-rless,
rminus_wf,
radd_wf,
itermAdd_wf,
itermMinus_wf,
rless_functionality,
req_transitivity,
rmul-rinv3,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_minus_lemma,
rabs_functionality,
rleq_antisymmetry,
infn_wf,
rleq-iff-all-rless,
rleq_functionality,
req_weakening,
not-rless,
rneq-iff-rabs,
rsub_wf,
infn-rleq,
rleq_weakening_rless,
rless_transitivity1,
infn-nonneg,
zero-rleq-rabs,
infn-property,
rleq_wf,
rleq-implies-rleq,
radd_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
dependent_set_memberEquality_alt,
introduction,
extract_by_obid,
isectElimination,
setElimination,
rename,
because_Cache,
closedConclusion,
natural_numberEquality,
independent_isectElimination,
sqequalRule,
inrFormation_alt,
productElimination,
independent_pairFormation,
imageMemberEquality,
baseClosed,
universeIsType,
dependent_pairFormation_alt,
applyEquality,
setIsType,
functionIsType,
imageElimination,
approximateComputation,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
int_eqEquality,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType
Latex:
\mforall{}I:\{I:Interval| icompact(I)\} . \mforall{}n:\mBbbN{}. \mforall{}f:\{f:I\^{}n {}\mrightarrow{} \mBbbR{}| \mforall{}a,b:I\^{}n. (req-vec(n;a;b) {}\mRightarrow{} ((f a) = (f b)))\}\000C .
((\mneg{}(\mforall{}x:I\^{}n. f x \mneq{} r0)) {}\mRightarrow{} (\mforall{}e:\{e:\mBbbR{}| r0 < e\} . \mexists{}x:I\^{}n. (|f x| < e)))
Date html generated:
2019_10_30-AM-08_26_48
Last ObjectModification:
2019_05_29-AM-09_07_48
Theory : reals
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