Nuprl Lemma : assert-inhabited-rat-cube
∀[k:ℕ]. ∀[c:ℚCube(k)]. uiff(↑Inhabited(c);∀i:ℕk. (↑Inhabited(c i)))
Proof
Definitions occuring in Statement :
inhabited-rat-cube: Inhabited(c),
rational-cube: ℚCube(k),
inhabited-rat-interval: Inhabited(I),
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
natural_number: $n
Definitions unfolded in proof :
guard: {T},
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
rational-cube: ℚCube(k),
nat: ℕ,
all: ∀x:A. B[x],
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
inhabited-rat-cube: Inhabited(c),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
istype-nat,
rational-cube_wf,
inhabited-rat-cube_wf,
bdd-all_wf,
assert-bdd-all,
istype-assert,
inhabited-rat-interval_wf,
assert_witness,
int_seg_wf
Rules used in proof :
isectIsTypeImplies,
isect_memberEquality_alt,
independent_pairEquality,
promote_hyp,
independent_isectElimination,
productElimination,
because_Cache,
functionIsType,
inhabitedIsType,
functionIsTypeImplies,
independent_functionElimination,
applyEquality,
dependent_functionElimination,
lambdaEquality_alt,
sqequalRule,
hypothesis,
hypothesisEquality,
rename,
setElimination,
natural_numberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
universeIsType,
lambdaFormation_alt,
independent_pairFormation,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[c:\mBbbQ{}Cube(k)]. uiff(\muparrow{}Inhabited(c);\mforall{}i:\mBbbN{}k. (\muparrow{}Inhabited(c i)))
Date html generated:
2019_10_29-AM-07_51_43
Last ObjectModification:
2019_10_17-PM-04_38_45
Theory : rationals
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