Nuprl Lemma : in-some-half-cube
∀k:ℕ. ∀x:ℝ^k. ∀c:ℚCube(k). (in-rat-cube(k;x;c) ⇒ (¬¬(∃h:ℚCube(k). ((↑is-half-cube(k;h;c)) ∧ in-rat-cube(k;x;h)))))
Proof
Definitions occuring in Statement :
in-rat-cube: in-rat-cube(k;p;c),
real-vec: ℝ^n,
nat: ℕ,
assert: ↑b,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
is-half-cube: is-half-cube(k;h;c),
rational-cube: ℚCube(k)
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
band: p ∧b q,
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
bfalse: ff,
is-half-interval: is-half-interval(I;J),
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
isl: isl(x),
squash: ↓T,
less_than: a < b,
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
stable: Stable{P},
uimplies: b supposing a,
guard: {T},
cand: A c∧ B,
pi2: snd(t),
real-vec: ℝ^n,
pi1: fst(t),
rational-interval: ℚInterval,
rational-cube: ℚCube(k),
or: P ∨ Q,
nat: ℕ,
prop: ℙ,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
member: t ∈ T,
exists: ∃x:A. B[x],
in-rat-cube: in-rat-cube(k;p;c),
false: False,
not: ¬A,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
assert_of_band,
assert_of_bor,
iff_weakening_uiff,
iff_transitivity,
equal_wf,
rationals_wf,
member_wf,
assert-qeq,
band_wf,
bool_cases,
qeq_wf2,
bor_wf,
assert_wf,
assert-bnot,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
eqtt_to_assert,
assert-is-half-cube,
rational-interval_wf,
bfalse_wf,
btrue_wf,
ifthenelse_wf,
Error :finite-double-negation-shift2,
minimal-not-not-excluded-middle,
minimal-double-negation-hyp-elim,
not-rless,
rleq_weakening_rless,
rless_wf,
false_wf,
qavg_wf,
rat2real_wf,
rleq_wf,
not_wf,
stable__not,
int_seg_wf,
istype-nat,
real-vec_wf,
rational-cube_wf,
istype-void,
in-rat-cube_wf,
is-half-cube_wf,
istype-assert
Rules used in proof :
isect_memberEquality_alt,
cumulativity,
instantiate,
promote_hyp,
equalityElimination,
independent_pairEquality,
imageElimination,
functionExtensionality,
dependent_pairFormation_alt,
lambdaEquality_alt,
unionElimination,
inrFormation_alt,
unionIsType,
independent_isectElimination,
independent_pairFormation,
inlFormation_alt,
functionEquality,
because_Cache,
equalitySymmetry,
equalityTransitivity,
equalityIstype,
applyEquality,
productEquality,
unionEquality,
productElimination,
rename,
setElimination,
natural_numberEquality,
dependent_functionElimination,
universeIsType,
isectElimination,
extract_by_obid,
introduction,
hypothesisEquality,
inhabitedIsType,
productIsType,
functionIsType,
sqequalRule,
voidElimination,
independent_functionElimination,
hypothesis,
sqequalHypSubstitution,
thin,
cut,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}k:\mBbbN{}. \mforall{}x:\mBbbR{}\^{}k. \mforall{}c:\mBbbQ{}Cube(k).
(in-rat-cube(k;x;c) {}\mRightarrow{} (\mneg{}\mneg{}(\mexists{}h:\mBbbQ{}Cube(k). ((\muparrow{}is-half-cube(k;h;c)) \mwedge{} in-rat-cube(k;x;h)))))
Date html generated:
2019_11_04-PM-04_43_09
Last ObjectModification:
2019_10_31-AM-11_04_42
Theory : real!vectors
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