Nuprl Lemma : center-point-in-cube-interior
∀[k:ℕ]. ∀[a:ℚCube(k)]. rat-point-in-cube-interior(k;λj.qavg(fst((a j));snd((a j)));a) supposing ↑Inhabited(a)
Proof
Definitions occuring in Statement :
rat-point-in-cube-interior: rat-point-in-cube-interior(k;x;a),
inhabited-rat-cube: Inhabited(c),
rational-cube: ℚCube(k),
qavg: qavg(a;b),
nat: ℕ,
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
apply: f a,
lambda: λx.A[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
rat-point-in-cube-interior: rat-point-in-cube-interior(k;x;a),
all: ∀x:A. B[x],
cand: A c∧ B,
rational-cube: ℚCube(k),
implies: P ⇒ Q,
rational-interval: ℚInterval,
pi1: fst(t),
pi2: snd(t),
rev_uimplies: rev_uimplies(P;Q),
inhabited-rat-interval: Inhabited(I),
prop: ℙ,
guard: {T},
iff: P ⇐⇒ Q,
nat: ℕ
Lemmas referenced :
assert-inhabited-rat-cube,
qle-qavg-iff-1,
qle_wf,
assert-q_le-eq,
iff_weakening_equal,
istype-assert,
q_le_wf,
qavg-qle-iff-1,
qless-qavg-iff-1,
qavg-qless-iff-1,
qless_wf,
int_seg_wf,
qle_witness,
qavg_wf,
qless_witness,
inhabited-rat-cube_wf,
rational-cube_wf,
istype-nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_isectElimination,
lambdaFormation_alt,
sqequalRule,
applyEquality,
inhabitedIsType,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
because_Cache,
universeIsType,
independent_pairFormation,
natural_numberEquality,
setElimination,
rename,
lambdaEquality_alt,
independent_pairEquality,
functionIsTypeImplies,
isect_memberEquality_alt,
isectIsTypeImplies
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[a:\mBbbQ{}Cube(k)].
rat-point-in-cube-interior(k;\mlambda{}j.qavg(fst((a j));snd((a j)));a) supposing \muparrow{}Inhabited(a)
Date html generated:
2020_05_20-AM-09_18_55
Last ObjectModification:
2020_01_04-PM-10_30_18
Theory : rationals
Home
Index