Nuprl Lemma : inhabited-half-cube
∀k:ℕ. ∀a,b:ℚCube(k). ((↑is-half-cube(k;a;b)) ⇒ (↑Inhabited(a) ⇐⇒ ↑Inhabited(b)))
Proof
Definitions occuring in Statement :
inhabited-rat-cube: Inhabited(c),
is-half-cube: is-half-cube(k;h;c),
rational-cube: ℚCube(k),
nat: ℕ,
assert: ↑b,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
uimplies: b supposing a,
rev_uimplies: rev_uimplies(P;Q),
rational-cube: ℚCube(k),
rev_implies: P ⇐ Q,
nat: ℕ,
prop: ℙ,
guard: {T},
rational-interval: ℚInterval,
inhabited-rat-interval: Inhabited(I),
is-half-interval: is-half-interval(I;J),
or: P ∨ Q,
sq_type: SQType(T),
bfalse: ff,
band: p ∧b q,
ifthenelse: if b then t else f fi
Lemmas referenced :
assert-inhabited-rat-cube,
istype-assert,
inhabited-rat-cube_wf,
int_seg_wf,
is-half-interval_wf,
iff_weakening_uiff,
assert_wf,
is-half-cube_wf,
assert-is-half-cube,
rational-cube_wf,
istype-nat,
qle_wf,
qle-qavg-iff-1,
qavg_wf,
qavg-qle-iff-1,
bor_wf,
qeq_wf2,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
band_wf,
btrue_wf,
assert-qeq,
bfalse_wf,
equal_wf,
rationals_wf,
q_le_wf,
assert-q_le-eq,
iff_weakening_equal,
iff_transitivity,
assert_of_bor,
assert_of_band
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
independent_pairFormation,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesis,
productElimination,
independent_isectElimination,
dependent_functionElimination,
hypothesisEquality,
applyEquality,
inhabitedIsType,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
sqequalRule,
functionIsType,
universeIsType,
natural_numberEquality,
setElimination,
rename,
functionEquality,
unionElimination,
promote_hyp,
hyp_replacement,
applyLambdaEquality,
unionIsType,
productIsType,
instantiate,
cumulativity,
unionEquality,
productEquality,
isect_memberEquality_alt,
inlFormation_alt,
inrFormation_alt
Latex:
\mforall{}k:\mBbbN{}. \mforall{}a,b:\mBbbQ{}Cube(k). ((\muparrow{}is-half-cube(k;a;b)) {}\mRightarrow{} (\muparrow{}Inhabited(a) \mLeftarrow{}{}\mRightarrow{} \muparrow{}Inhabited(b)))
Date html generated:
2020_05_20-AM-09_20_13
Last ObjectModification:
2019_11_01-PM-10_54_15
Theory : rationals
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