Nuprl Lemma : not-rat-cube-third
∀[k:ℕ]. ∀[p:ℝ^k]. ∀[c:ℚCube(k)].
(¬rat-cube-third(k;p;c) ⇐⇒ in-rat-cube(k;p;c) ∧ (¬¬(∃i:ℕk. (¬rat-interval-third(p i;c i)))))
Proof
Definitions occuring in Statement :
rat-cube-third: rat-cube-third(k;p;c),
rat-interval-third: rat-interval-third(p;I),
in-rat-cube: in-rat-cube(k;p;c),
real-vec: ℝ^n,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
and: P ∧ Q,
apply: f a,
natural_number: $n,
rational-cube: ℚCube(k)
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
stable: Stable{P},
rat-cube-third: rat-cube-third(k;p;c),
or: P ∨ Q,
uimplies: b supposing a,
le: A ≤ B,
rnonneg: rnonneg(x),
rleq: x ≤ y,
all: ∀x:A. B[x],
in-rat-cube: in-rat-cube(k;p;c),
rev_implies: P ⇐ Q,
prop: ℙ,
rational-cube: ℚCube(k),
real-vec: ℝ^n,
nat: ℕ,
exists: ∃x:A. B[x],
false: False,
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
double-negation-hyp-elim,
Error :not-not-all-int_seg-shift,
stable__not,
minimal-not-not-excluded-middle,
minimal-double-negation-hyp-elim,
not_wf,
false_wf,
stable__in-rat-cube,
istype-nat,
real-vec_wf,
rational-cube_wf,
le_witness_for_triv,
in-rat-cube_wf,
rat-cube-third_wf,
istype-void,
rat-interval-third_wf,
int_seg_wf
Rules used in proof :
dependent_pairFormation_alt,
productEquality,
unionElimination,
unionIsType,
functionEquality,
unionEquality,
isectIsTypeImplies,
isect_memberEquality_alt,
inhabitedIsType,
functionIsTypeImplies,
independent_isectElimination,
equalitySymmetry,
equalityTransitivity,
dependent_functionElimination,
lambdaEquality_alt,
independent_pairEquality,
because_Cache,
productElimination,
applyEquality,
hypothesisEquality,
rename,
setElimination,
natural_numberEquality,
isectElimination,
extract_by_obid,
universeIsType,
productIsType,
functionIsType,
sqequalRule,
voidElimination,
independent_functionElimination,
sqequalHypSubstitution,
hypothesis,
thin,
lambdaFormation_alt,
independent_pairFormation,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[p:\mBbbR{}\^{}k]. \mforall{}[c:\mBbbQ{}Cube(k)].
(\mneg{}rat-cube-third(k;p;c) \mLeftarrow{}{}\mRightarrow{} in-rat-cube(k;p;c) \mwedge{} (\mneg{}\mneg{}(\mexists{}i:\mBbbN{}k. (\mneg{}rat-interval-third(p i;c i)))))
Date html generated:
2019_11_04-PM-04_43_29
Last ObjectModification:
2019_11_04-PM-03_17_25
Theory : real!vectors
Home
Index