Nuprl Lemma : remove-singularity-max-seq_wf
∀[X:Type]. ∀[k:ℕ]. ∀[p:ℝ^k]. ∀[f:{p:ℝ^k| r0 < mdist(max-metric(k);p;λi.r0)} ⟶ X]. ∀[z:X].
(remove-singularity-max-seq(k;p;f;z) ∈ ℕ ⟶ X)
Proof
Definitions occuring in Statement :
remove-singularity-max-seq: remove-singularity-max-seq(k;p;f;z),
max-metric: max-metric(n),
real-vec: ℝ^n,
mdist: mdist(d;x;y),
rless: x < y,
int-to-real: r(n),
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
real-vec: ℝ^n,
member: t ∈ T,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
nat: ℕ,
remove-singularity-max-seq: remove-singularity-max-seq(k;p;f;z),
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
bfalse: ff
Lemmas referenced :
int-to-real_wf,
int_seg_wf,
realvec-max-ibs_wf,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
realvec-max-ibs-property,
set_subtype_base,
lelt_wf,
int_subtype_base,
rless_wf,
mdist_wf,
real-vec_wf,
max-metric_wf,
istype-nat,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
sqequalRule,
lambdaEquality_alt,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
productElimination,
hypothesis,
universeIsType,
natural_numberEquality,
hypothesisEquality,
applyEquality,
because_Cache,
closedConclusion,
inhabitedIsType,
lambdaFormation_alt,
unionElimination,
equalityElimination,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
dependent_pairFormation_alt,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
intEquality,
baseClosed,
sqequalBase,
dependent_set_memberEquality_alt,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
functionIsType,
setIsType,
instantiate,
universeEquality
Latex:
\mforall{}[X:Type]. \mforall{}[k:\mBbbN{}]. \mforall{}[p:\mBbbR{}\^{}k]. \mforall{}[f:\{p:\mBbbR{}\^{}k| r0 < mdist(max-metric(k);p;\mlambda{}i.r0)\} {}\mrightarrow{} X]. \mforall{}[z:X].
(remove-singularity-max-seq(k;p;f;z) \mmember{} \mBbbN{} {}\mrightarrow{} X)
Date html generated:
2019_10_30-AM-10_16_28
Last ObjectModification:
2019_07_03-PM-04_27_14
Theory : real!vectors
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