Nuprl Lemma : m-reg-test_wf
∀[X:Type]
∀d:metric(X). ∀b:ℕ. ∀s:ℕb ⟶ X. ∀x:X.
(m-reg-test(d;b;s;x) ∈ (∃n:ℕb. (((r(2)/r(n + 1)) + (r(2)/r(b + 1))) < mdist(d;s n;x)))
∨ (∀n:ℕb. (mdist(d;s n;x) < ((r(3)/r(n + 1)) + (r(3)/r(b + 1))))))
Proof
Definitions occuring in Statement :
m-reg-test: m-reg-test(d;b;s;x),
mdist: mdist(d;x;y),
metric: metric(X),
rdiv: (x/y),
rless: x < y,
radd: a + b,
int-to-real: r(n),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
or: P ∨ Q,
member: t ∈ T,
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],
member: t ∈ T,
all: ∀x:A. B[x],
m-reg-test: m-reg-test(d;b;s;x),
nat: ℕ,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
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),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
so_apply: x[s],
nat_plus: ℕ+,
uiff: uiff(P;Q)
Lemmas referenced :
int-seg-case_wf,
rless_wf,
radd_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
int_seg_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,
mdist_wf,
rless-case_wf,
rlessw_wf,
int_seg_wf,
istype-nat,
metric_wf,
istype-universe,
radd_functionality_wrt_rless1,
rleq-int-fractions,
istype-less_than,
decidable__le,
itermMultiply_wf,
int_term_value_mul_lemma,
rless-int-fractions
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
closedConclusion,
natural_numberEquality,
setElimination,
rename,
because_Cache,
hypothesis,
lambdaEquality_alt,
addEquality,
independent_isectElimination,
inrFormation_alt,
dependent_functionElimination,
productElimination,
independent_functionElimination,
imageElimination,
hypothesisEquality,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
applyEquality,
functionIsType,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
instantiate,
universeEquality,
dependent_set_memberEquality_alt,
multiplyEquality
Latex:
\mforall{}[X:Type]
\mforall{}d:metric(X). \mforall{}b:\mBbbN{}. \mforall{}s:\mBbbN{}b {}\mrightarrow{} X. \mforall{}x:X.
(m-reg-test(d;b;s;x) \mmember{} (\mexists{}n:\mBbbN{}b. (((r(2)/r(n + 1)) + (r(2)/r(b + 1))) < mdist(d;s n;x)))
\mvee{} (\mforall{}n:\mBbbN{}b. (mdist(d;s n;x) < ((r(3)/r(n + 1)) + (r(3)/r(b + 1))))))
Date html generated:
2019_10_30-AM-06_59_49
Last ObjectModification:
2019_10_09-AM-08_58_55
Theory : reals
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