Nuprl Lemma : is-half-interval-sub-interval
∀I,J:ℚInterval. (rat-sub-interval(I;J)) supposing ((↑is-half-interval(I;J)) and (↑Inhabited(J)))
Proof
Definitions occuring in Statement :
is-half-interval: is-half-interval(I;J),
inhabited-rat-interval: Inhabited(I),
rat-sub-interval: rat-sub-interval(I;J),
rational-interval: ℚInterval,
assert: ↑b,
uimplies: b supposing a,
all: ∀x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
rational-interval: ℚInterval,
is-half-interval: is-half-interval(I;J),
inhabited-rat-interval: Inhabited(I),
rat-sub-interval: rat-sub-interval(I;J),
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
uiff: uiff(P;Q),
and: P ∧ Q,
bfalse: ff,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
prop: ℙ,
cand: A c∧ B,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B
Lemmas referenced :
assert_witness,
inhabited-rat-interval_wf,
is-half-interval_wf,
istype-assert,
rational-interval_wf,
assert_wf,
bor_wf,
qeq_wf2,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
band_wf,
btrue_wf,
assert-qeq,
qavg_wf,
bfalse_wf,
equal_wf,
rationals_wf,
qle_reflexivity,
qless_wf,
iff_weakening_uiff,
qle_wf,
qavg-qle-iff-1,
qless-qavg-iff-1,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
qle-qavg-iff-1,
qavg-qless-iff-1,
iff_transitivity,
assert_of_bor,
assert_of_band,
assert-q_le-eq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_functionElimination,
rename,
productElimination,
sqequalRule,
inhabitedIsType,
universeIsType,
dependent_functionElimination,
unionElimination,
instantiate,
cumulativity,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
unionEquality,
because_Cache,
productEquality,
isect_memberEquality_alt,
productIsType,
equalityIstype,
unionIsType,
independent_pairFormation,
promote_hyp,
applyEquality,
lambdaEquality_alt,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
inlFormation_alt,
inrFormation_alt
Latex:
\mforall{}I,J:\mBbbQ{}Interval. (rat-sub-interval(I;J)) supposing ((\muparrow{}is-half-interval(I;J)) and (\muparrow{}Inhabited(J)))
Date html generated:
2020_05_20-AM-09_17_55
Last ObjectModification:
2019_11_14-PM-10_58_51
Theory : rationals
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