Nuprl Lemma : in-rat-half-cube
∀k:ℕ. ∀c,h:ℚCube(k). ((↑is-half-cube(k;h;c)) ⇒ (∀x:ℝ^k. (in-rat-cube(k;x;h) ⇒ in-rat-cube(k;x;c))))
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],
implies: P ⇒ Q,
is-half-cube: is-half-cube(k;h;c),
rational-cube: ℚCube(k)
Definitions unfolded in proof :
rge: x ≥ y,
rev_uimplies: rev_uimplies(P;Q),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
ifthenelse: if b then t else f fi ,
band: p ∧b q,
bfalse: ff,
guard: {T},
sq_type: SQType(T),
or: P ∨ Q,
cand: A c∧ B,
prop: ℙ,
nat: ℕ,
pi2: snd(t),
pi1: fst(t),
is-half-interval: is-half-interval(I;J),
rational-interval: ℚInterval,
rational-cube: ℚCube(k),
real-vec: ℝ^n,
in-rat-cube: in-rat-cube(k;p;c),
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
rleq_weakening_equal,
rleq_functionality_wrt_implies,
qle-qavg-iff-1,
qavg-qle-iff-1,
rleq-rat2real,
rleq_transitivity,
assert_of_band,
assert_of_bor,
iff_weakening_uiff,
iff_transitivity,
rationals_wf,
equal_wf,
bfalse_wf,
assert-qeq,
btrue_wf,
band_wf,
eqtt_to_assert,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases,
qeq_wf2,
bor_wf,
assert_wf,
qavg_wf,
rleq_wf,
rat2real_wf,
istype-nat,
rational-cube_wf,
is-half-cube_wf,
istype-assert,
real-vec_wf,
in-rat-cube_wf,
int_seg_wf,
assert-is-half-cube
Rules used in proof :
inrFormation_alt,
inlFormation_alt,
isect_memberEquality_alt,
productEquality,
because_Cache,
unionEquality,
cumulativity,
instantiate,
unionIsType,
promote_hyp,
applyLambdaEquality,
productIsType,
independent_pairFormation,
dependent_set_memberEquality_alt,
hyp_replacement,
unionElimination,
rename,
setElimination,
natural_numberEquality,
universeIsType,
independent_functionElimination,
equalitySymmetry,
equalityTransitivity,
equalityIstype,
sqequalRule,
inhabitedIsType,
applyEquality,
dependent_functionElimination,
independent_isectElimination,
productElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}k:\mBbbN{}. \mforall{}c,h:\mBbbQ{}Cube(k).
((\muparrow{}is-half-cube(k;h;c)) {}\mRightarrow{} (\mforall{}x:\mBbbR{}\^{}k. (in-rat-cube(k;x;h) {}\mRightarrow{} in-rat-cube(k;x;c))))
Date html generated:
2019_11_04-PM-04_43_03
Last ObjectModification:
2019_10_31-AM-10_26_12
Theory : real!vectors
Home
Index