Nuprl Lemma : inhabited-iff-in-rat-cube
∀[k:ℕ]. ∀c:ℚCube(k). (↑Inhabited(c) ⇐⇒ ∃p:ℝ^k. in-rat-cube(k;p;c))
Proof
Definitions occuring in Statement :
in-rat-cube: in-rat-cube(k;p;c),
real-vec: ℝ^n,
nat: ℕ,
assert: ↑b,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
inhabited-rat-cube: Inhabited(c),
rational-cube: ℚCube(k)
Definitions unfolded in proof :
guard: {T},
le: A ≤ B,
rnonneg: rnonneg(x),
rleq: x ≤ y,
cand: A c∧ B,
pi2: snd(t),
inhabited-rat-interval: Inhabited(I),
in-rat-cube: in-rat-cube(k;p;c),
nat: ℕ,
pi1: fst(t),
rational-interval: ℚInterval,
rational-cube: ℚCube(k),
real-vec: ℝ^n,
prop: ℙ,
exists: ∃x:A. B[x],
rev_uimplies: rev_uimplies(P;Q),
rev_implies: P ⇐ Q,
uimplies: b supposing a,
uiff: uiff(P;Q),
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x]
Lemmas referenced :
rleq_wf,
rleq_transitivity,
q_le_wf,
iff_weakening_equal,
assert-q_le-eq,
qle_wf,
le_witness_for_triv,
rleq-rat2real,
rleq_weakening_equal,
int_seg_wf,
rat2real_wf,
istype-nat,
rational-cube_wf,
in-rat-cube_wf,
real-vec_wf,
assert_witness,
inhabited-rat-cube_wf,
istype-assert,
assert-inhabited-rat-cube
Rules used in proof :
functionIsTypeImplies,
independent_pairEquality,
rename,
setElimination,
natural_numberEquality,
dependent_functionElimination,
equalitySymmetry,
equalityTransitivity,
equalityIstype,
inhabitedIsType,
applyEquality,
lambdaEquality_alt,
dependent_pairFormation_alt,
universeIsType,
productIsType,
sqequalRule,
independent_functionElimination,
because_Cache,
independent_isectElimination,
productElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
independent_pairFormation,
lambdaFormation_alt,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}c:\mBbbQ{}Cube(k). (\muparrow{}Inhabited(c) \mLeftarrow{}{}\mRightarrow{} \mexists{}p:\mBbbR{}\^{}k. in-rat-cube(k;p;c))
Date html generated:
2019_10_30-AM-10_13_02
Last ObjectModification:
2019_10_29-PM-01_49_09
Theory : real!vectors
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