Nuprl Lemma : not-not-in-0-dim-cube
∀[k:ℕ]. ∀[c:ℚCube(k)].
∀[p:ℝ^k]. uiff(¬¬in-rat-cube(k;p;c);req-vec(k;p;λj.rat2real(fst((c j))))) supposing dim(c) = 0 ∈ ℤ
Proof
Definitions occuring in Statement :
in-rat-cube: in-rat-cube(k;p;c),
req-vec: req-vec(n;x;y),
real-vec: ℝ^n,
rat2real: rat2real(q),
nat: ℕ,
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
pi1: fst(t),
not: ¬A,
apply: f a,
lambda: λx.A[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T,
rat-cube-dimension: dim(c),
rational-cube: ℚCube(k)
Definitions unfolded in proof :
stable: Stable{P},
so_apply: x[s],
so_lambda: λ2x.t[x],
int_seg: {i..j-},
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
false: False,
prop: ℙ,
nat: ℕ,
not: ¬A,
pi1: fst(t),
rational-interval: ℚInterval,
implies: P ⇒ Q,
rational-cube: ℚCube(k),
real-vec: ℝ^n,
all: ∀x:A. B[x],
req-vec: req-vec(n;x;y),
member: t ∈ T,
and: P ∧ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Lemmas referenced :
stable_req,
req_wf,
stable__all,
istype-nat,
rational-cube_wf,
int_subtype_base,
lelt_wf,
set_subtype_base,
rat-cube-dimension_wf,
istype-int,
real-vec_wf,
in-0-dim-cube,
in-rat-cube_wf,
iff_weakening_uiff,
istype-void,
int_seg_wf,
req-vec_wf,
rat2real_wf,
req_witness
Rules used in proof :
sqequalBase,
baseClosed,
addEquality,
minusEquality,
intEquality,
promote_hyp,
independent_isectElimination,
because_Cache,
voidElimination,
rename,
setElimination,
natural_numberEquality,
universeIsType,
functionIsType,
functionIsTypeImplies,
independent_functionElimination,
equalitySymmetry,
equalityTransitivity,
equalityIstype,
productElimination,
lambdaFormation_alt,
hypothesis,
inhabitedIsType,
applyEquality,
isectElimination,
extract_by_obid,
hypothesisEquality,
thin,
dependent_functionElimination,
lambdaEquality_alt,
sqequalHypSubstitution,
sqequalRule,
introduction,
independent_pairFormation,
cut,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[c:\mBbbQ{}Cube(k)].
\mforall{}[p:\mBbbR{}\^{}k]. uiff(\mneg{}\mneg{}in-rat-cube(k;p;c);req-vec(k;p;\mlambda{}j.rat2real(fst((c j))))) supposing dim(c) = 0
Date html generated:
2019_10_30-AM-10_12_58
Last ObjectModification:
2019_10_28-PM-04_28_41
Theory : real!vectors
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