Nuprl Lemma : m-TB-iff
∀[X:Type]. ∀[d:metric(X)]. (m-TB(X;d) ⇐⇒ ∀k:ℕ. ∃n:ℕ+. ∃xs:ℕn ⟶ X. ∀x:X. ∃i:ℕn. (mdist(d;x;xs i) ≤ (r1/r(k + 1))))
Proof
Definitions occuring in Statement :
m-TB: m-TB(X;d),
mdist: mdist(d;x;y),
metric: metric(X),
rdiv: (x/y),
rleq: x ≤ y,
int-to-real: r(n),
int_seg: {i..j-},
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
nat_plus: ℕ+,
nat: ℕ,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
ge: i ≥ j ,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
prop: ℙ,
m-TB: m-TB(X;d),
spreadn: spread3,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
pi1: fst(t),
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
istype-nat,
m-TB_wf,
nat_plus_wf,
int_seg_wf,
rleq_wf,
mdist_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
int_seg_properties,
nat_plus_properties,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
rless_wf,
metric_wf,
istype-universe,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rleq_weakening,
itermSubtract_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
nat_wf,
subtype_rel_self,
exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
independent_pairFormation,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
hypothesis,
universeIsType,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
functionIsType,
productIsType,
natural_numberEquality,
setElimination,
rename,
because_Cache,
applyEquality,
closedConclusion,
addEquality,
independent_isectElimination,
inrFormation_alt,
dependent_functionElimination,
productElimination,
independent_functionElimination,
imageElimination,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
instantiate,
universeEquality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
dependent_set_memberEquality_alt,
dependent_pairEquality_alt,
independent_pairEquality,
functionExtensionality,
inhabitedIsType,
equalityIstype,
functionEquality
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)].
(m-TB(X;d) \mLeftarrow{}{}\mRightarrow{} \mforall{}k:\mBbbN{}. \mexists{}n:\mBbbN{}\msupplus{}. \mexists{}xs:\mBbbN{}n {}\mrightarrow{} X. \mforall{}x:X. \mexists{}i:\mBbbN{}n. (mdist(d;x;xs i) \mleq{} (r1/r(k + 1))))
Date html generated:
2019_10_30-AM-06_50_54
Last ObjectModification:
2019_10_10-PM-05_26_48
Theory : reals
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