Nuprl Lemma : compatible-rat-intervals-iff
∀I,J:ℚInterval.
((↑Inhabited(I))
⇒ (↑Inhabited(J))
⇒ (↑Inhabited(I ⋂ J))
⇒ (I ⋂ J ≤ I ∧ I ⋂ J ≤ J ⇐⇒ (I = J ∈ ℚInterval) ∨ ((snd(I)) = (fst(J)) ∈ ℚ) ∨ ((snd(J)) = (fst(I)) ∈ ℚ)))
Proof
Definitions occuring in Statement :
rat-interval-intersection: I ⋂ J,
inhabited-rat-interval: Inhabited(I),
rat-interval-face: I ≤ J,
rational-interval: ℚInterval,
rationals: ℚ,
assert: ↑b,
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
assert: ↑b,
bnot: ¬bb,
exists: ∃x:A. B[x],
it: ⋅,
unit: Unit,
bool: 𝔹,
squash: ↓T,
true: True,
qmax: qmax(x;y),
bfalse: ff,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
sq_type: SQType(T),
false: False,
not: ¬A,
qmin: qmin(x;y),
top: Top,
so_apply: x[s],
so_lambda: λ2x.t[x],
rat-point-interval: [a],
guard: {T},
uimplies: b supposing a,
prop: ℙ,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
member: t ∈ T,
or: P ∨ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
rat-interval-face: I ≤ J,
pi1: fst(t),
pi2: snd(t),
inhabited-rat-interval: Inhabited(I),
rat-interval-intersection: I ⋂ J,
rational-interval: ℚInterval,
all: ∀x:A. B[x]
Lemmas referenced :
qle_antisymmetry,
assert-bnot,
bool_cases_sqequal,
qmin-idempotent,
qmax-idempotent,
istype-universe,
true_wf,
squash_wf,
equal_wf,
qle_weakening_eq_qorder,
qle_transitivity_qorder,
subtype_rel_self,
assert_of_bnot,
iff_weakening_uiff,
iff_transitivity,
eqff_to_assert,
eqtt_to_assert,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases,
not_wf,
bnot_wf,
assert_wf,
istype-void,
pi1_wf_top,
rationals_wf,
pi2_wf,
rational-interval_wf,
q_le_wf,
istype-assert,
iff_weakening_equal,
assert-q_le-eq,
qle_wf,
rat-point-interval_wf,
qmin_wf,
qmax_wf
Rules used in proof :
unionEquality,
dependent_pairFormation_alt,
equalityElimination,
hyp_replacement,
promote_hyp,
baseClosed,
imageMemberEquality,
universeEquality,
imageElimination,
natural_numberEquality,
cumulativity,
instantiate,
dependent_functionElimination,
functionIsType,
inlFormation_alt,
inrFormation_alt,
voidElimination,
isect_memberEquality_alt,
lambdaEquality_alt,
applyLambdaEquality,
unionElimination,
independent_isectElimination,
equalitySymmetry,
equalityTransitivity,
independent_functionElimination,
universeIsType,
inhabitedIsType,
applyEquality,
hypothesis,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
independent_pairEquality,
because_Cache,
equalityIstype,
unionIsType,
productIsType,
independent_pairFormation,
cut,
sqequalRule,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}I,J:\mBbbQ{}Interval.
((\muparrow{}Inhabited(I))
{}\mRightarrow{} (\muparrow{}Inhabited(J))
{}\mRightarrow{} (\muparrow{}Inhabited(I \mcap{} J))
{}\mRightarrow{} (I \mcap{} J \mleq{} I \mwedge{} I \mcap{} J \mleq{} J \mLeftarrow{}{}\mRightarrow{} (I = J) \mvee{} ((snd(I)) = (fst(J))) \mvee{} ((snd(J)) = (fst(I)))))
Date html generated:
2019_10_29-AM-07_53_56
Last ObjectModification:
2019_10_19-AM-01_36_30
Theory : rationals
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