Nuprl Lemma : inhabited-rat-cube-face
∀[k:ℕ]. ∀[c:ℚCube(k)]. ((↑Inhabited(c)) ⇒ (∀f:ℚCube(k). (f ≤ c ⇒ (↑Inhabited(f)))))
Proof
Definitions occuring in Statement :
inhabited-rat-cube: Inhabited(c),
rat-cube-face: c ≤ d,
rational-cube: ℚCube(k),
nat: ℕ,
assert: ↑b,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
nat: ℕ,
subtype_rel: A ⊆r B,
guard: {T},
true: True,
prop: ℙ,
squash: ↓T,
or: P ∨ Q,
rat-interval-face: I ≤ J,
rational-interval: ℚInterval,
rational-cube: ℚCube(k),
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
rat-cube-face: c ≤ d,
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
istype-nat,
rational-cube_wf,
assert_witness,
inhabited-rat-cube_wf,
rat-cube-face_wf,
int_seg_wf,
istype-assert,
subtype_rel_self,
rational-interval_wf,
assert_of_tt,
inhabited-rat-point-interval,
true_wf,
squash_wf,
rat-point-interval_wf,
inhabited-rat-interval_wf,
assert_functionality_wrt_uiff,
assert-inhabited-rat-cube
Rules used in proof :
isectIsTypeImplies,
isect_memberEquality_alt,
functionIsTypeImplies,
rename,
setElimination,
independent_functionElimination,
equalityIstype,
unionIsType,
independent_pairEquality,
baseClosed,
imageMemberEquality,
natural_numberEquality,
because_Cache,
universeIsType,
equalitySymmetry,
equalityTransitivity,
imageElimination,
lambdaEquality_alt,
unionElimination,
sqequalRule,
inhabitedIsType,
applyEquality,
dependent_functionElimination,
independent_isectElimination,
productElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
extract_by_obid,
sqequalHypSubstitution,
lambdaFormation_alt,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[c:\mBbbQ{}Cube(k)]. ((\muparrow{}Inhabited(c)) {}\mRightarrow{} (\mforall{}f:\mBbbQ{}Cube(k). (f \mleq{} c {}\mRightarrow{} (\muparrow{}Inhabited(f)))))
Date html generated:
2019_10_29-AM-07_51_49
Last ObjectModification:
2019_10_17-PM-05_49_25
Theory : rationals
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