Nuprl Lemma : rat-cube-third-exists
∀k:ℕ. ∀c:ℚCube(k). ((↑Inhabited(c)) ⇒ (∃p:ℝ^k. (in-rat-cube(k;p;c) ∧ rat-cube-third(k;p;c))))
Proof
Definitions occuring in Statement :
rat-cube-third: rat-cube-third(k;p;c),
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],
implies: P ⇒ Q,
and: P ∧ Q,
inhabited-rat-cube: Inhabited(c),
rational-cube: ℚCube(k)
Definitions unfolded in proof :
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
sq_type: SQType(T),
decidable: Dec(P),
rev_uimplies: rev_uimplies(P;Q),
satisfiable_int_formula: satisfiable_int_formula(fmla),
ge: i ≥ j ,
top: Top,
rat-interval-third: rat-interval-third(p;I),
rat-cube-third: rat-cube-third(k;p;c),
not: ¬A,
false: False,
inhabited-rat-interval: Inhabited(I),
rational-cube: ℚCube(k),
in-rat-cube: in-rat-cube(k;p;c),
cand: A c∧ B,
nat: ℕ,
prop: ℙ,
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
pi2: snd(t),
pi1: fst(t),
rational-interval: ℚInterval,
real-vec: ℝ^n,
exists: ∃x:A. B[x],
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 :
real_term_value_add_lemma,
int-rinv-cancel,
rmul-rinv3,
radd_functionality,
req_transitivity,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_mul_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
rleq_functionality,
req-iff-rsub-is-0,
rsub_wf,
rleq-implies-rleq,
nequal_wf,
int_term_value_mul_lemma,
int_formula_prop_not_lemma,
intformnot_wf,
decidable__equal_int,
int_subtype_base,
subtype_base_sq,
itermAdd_wf,
rinv_wf2,
rmul_preserves_rleq,
itermVar_wf,
itermMultiply_wf,
itermSubtract_wf,
istype-nat,
rational-cube_wf,
inhabited-rat-cube_wf,
rat-cube-third_wf,
in-rat-cube_wf,
rleq_wf,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_formula_prop_eq_lemma,
istype-int,
itermConstant_wf,
intformeq_wf,
full-omega-unsat,
nat_properties,
int_seg_properties,
rneq-int,
req_wf,
req_weakening,
istype-void,
member_rccint_lemma,
q_le_wf,
istype-assert,
iff_weakening_equal,
assert-q_le-eq,
qle_wf,
istype-false,
rleq-int,
rmul_preserves_rleq2,
rleq-rat2real,
int_seg_wf,
rless_wf,
rless-int,
rat2real_wf,
int-to-real_wf,
rmul_wf,
radd_wf,
rdiv_wf,
assert-inhabited-rat-cube
Rules used in proof :
int_eqEquality,
dependent_set_memberEquality_alt,
unionElimination,
intEquality,
cumulativity,
instantiate,
productIsType,
sqequalBase,
approximateComputation,
imageElimination,
inlFormation_alt,
voidElimination,
isect_memberEquality_alt,
applyEquality,
rename,
setElimination,
universeIsType,
baseClosed,
imageMemberEquality,
independent_pairFormation,
inrFormation_alt,
independent_functionElimination,
dependent_functionElimination,
equalitySymmetry,
equalityTransitivity,
equalityIstype,
inhabitedIsType,
because_Cache,
natural_numberEquality,
closedConclusion,
lambdaEquality_alt,
sqequalRule,
dependent_pairFormation_alt,
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:\mBbbQ{}Cube(k). ((\muparrow{}Inhabited(c)) {}\mRightarrow{} (\mexists{}p:\mBbbR{}\^{}k. (in-rat-cube(k;p;c) \mwedge{} rat-cube-third(k;p;c))))
Date html generated:
2019_10_31-AM-06_03_51
Last ObjectModification:
2019_10_30-PM-03_36_41
Theory : real!vectors
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