Nuprl Lemma : rat-cube-third-half
∀k:ℕ. ∀p:ℝ^k. ∀c:ℚCube(k). (rat-cube-third(k;p;c) ⇒ (∀h:ℚCube(k). ((↑is-half-cube(k;h;c)) ⇒ rat-cube-third(k;p;h))))
Proof
Definitions occuring in Statement :
rat-cube-third: rat-cube-third(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 :
subtype_rel: A ⊆r B,
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
rdiv: (x/y),
ifthenelse: if b then t else f fi ,
band: p ∧b q,
bfalse: ff,
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
sq_type: SQType(T),
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
not: ¬A,
false: False,
le: A ≤ B,
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
rneq: x ≠ y,
guard: {T},
prop: ℙ,
nat: ℕ,
pi2: snd(t),
pi1: fst(t),
is-half-interval: is-half-interval(I;J),
rat-interval-third: rat-interval-third(p;I),
rational-interval: ℚInterval,
real-vec: ℝ^n,
rational-cube: ℚCube(k),
in-rat-cube: in-rat-cube(k;p;c),
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
member: t ∈ T,
rat-cube-third: rat-cube-third(k;p;c),
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
req_inversion,
true_wf,
real_wf,
rneq_wf,
squash_wf,
iff_weakening_equal,
subtype_rel_self,
qavg-same,
rleq_weakening,
rmul-rinv3,
rdiv_functionality,
req_functionality,
rmul_preserves_req,
or_wf,
qle_wf,
real_term_value_minus_lemma,
int-rinv-cancel,
rat2real-qavg-2,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_mul_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
req-iff-rsub-is-0,
rinv-mul-as-rdiv,
req_weakening,
rmul_functionality,
radd_functionality,
req_transitivity,
rleq_functionality,
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,
bool_cases,
qeq_wf2,
bor_wf,
assert_wf,
qle_antisymmetry,
itermMinus_wf,
rminus_wf,
radd-preserves-rleq,
nequal_wf,
int_formula_prop_wf,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_formula_prop_eq_lemma,
istype-void,
int_formula_prop_not_lemma,
istype-int,
intformeq_wf,
intformnot_wf,
full-omega-unsat,
decidable__equal_int,
int_subtype_base,
subtype_base_sq,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
itermMultiply_wf,
itermSubtract_wf,
rinv_wf2,
req_wf,
rmul_wf,
radd_wf,
rleq_wf,
istype-false,
rleq-int,
qavg_wf,
int-to-real_wf,
rless_wf,
rless-int,
rdiv_wf,
rmul_preserves_rleq2,
rleq-rat2real,
rat2real_wf,
rleq_transitivity,
istype-nat,
real-vec_wf,
rational-cube_wf,
rat-cube-third_wf,
is-half-cube_wf,
istype-assert,
in-rat-cube_wf,
int_seg_wf,
assert-is-half-cube,
in-rat-half-cube
Rules used in proof :
universeEquality,
imageElimination,
applyLambdaEquality,
hyp_replacement,
int_eqEquality,
promote_hyp,
inlFormation_alt,
productEquality,
unionEquality,
sqequalBase,
dependent_set_memberEquality_alt,
voidElimination,
isect_memberEquality_alt,
lambdaEquality_alt,
dependent_pairFormation_alt,
approximateComputation,
unionElimination,
intEquality,
cumulativity,
instantiate,
productIsType,
unionIsType,
closedConclusion,
baseClosed,
imageMemberEquality,
independent_pairFormation,
inrFormation_alt,
because_Cache,
setElimination,
natural_numberEquality,
universeIsType,
equalitySymmetry,
equalityTransitivity,
equalityIstype,
sqequalRule,
rename,
inhabitedIsType,
applyEquality,
independent_isectElimination,
productElimination,
isectElimination,
hypothesis,
independent_functionElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
extract_by_obid,
introduction,
cut,
sqequalHypSubstitution,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}k:\mBbbN{}. \mforall{}p:\mBbbR{}\^{}k. \mforall{}c:\mBbbQ{}Cube(k).
(rat-cube-third(k;p;c) {}\mRightarrow{} (\mforall{}h:\mBbbQ{}Cube(k). ((\muparrow{}is-half-cube(k;h;c)) {}\mRightarrow{} rat-cube-third(k;p;h))))
Date html generated:
2019_11_04-PM-04_43_25
Last ObjectModification:
2019_11_04-PM-03_13_09
Theory : real!vectors
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