Nuprl Lemma : mtb-point-cantor-seq-regular
∀[X:Type]. ∀[d:metric(X)]. ∀mtb:m-TB(X;d). ∀p:X. m-k-regular(d;1;mtb-seq(mtb;mtb-point-cantor(mtb;p)))
Proof
Definitions occuring in Statement :
m-k-regular: m-k-regular(d;k;s),
mtb-point-cantor: mtb-point-cantor(mtb;p),
mtb-seq: mtb-seq(mtb;s),
m-TB: m-TB(X;d),
metric: metric(X),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
m-TB: m-TB(X;d),
spreadn: spread3,
mtb-point-cantor: mtb-point-cantor(mtb;p),
mtb-seq: mtb-seq(mtb;s),
m-k-regular: m-k-regular(d;k;s),
member: t ∈ T,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
and: P ∧ Q,
uimplies: b supposing a,
nat: ℕ,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
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,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
uiff: uiff(P;Q)
Lemmas referenced :
le_witness_for_triv,
istype-nat,
m-TB_wf,
metric_wf,
istype-universe,
mdist_wf,
radd_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
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,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
radd_functionality_wrt_rleq,
mdist-triangle-inequality,
rleq_functionality,
req_weakening,
radd_functionality,
mdist-symm
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
sqequalRule,
introduction,
cut,
dependent_functionElimination,
hypothesisEquality,
lambdaEquality_alt,
extract_by_obid,
isectElimination,
equalityTransitivity,
hypothesis,
equalitySymmetry,
independent_isectElimination,
functionIsTypeImplies,
inhabitedIsType,
universeIsType,
instantiate,
universeEquality,
applyEquality,
closedConclusion,
natural_numberEquality,
addEquality,
because_Cache,
inrFormation_alt,
independent_functionElimination,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
equalityIstype
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)].
\mforall{}mtb:m-TB(X;d). \mforall{}p:X. m-k-regular(d;1;mtb-seq(mtb;mtb-point-cantor(mtb;p)))
Date html generated:
2019_10_30-AM-07_04_29
Last ObjectModification:
2019_10_09-AM-09_20_05
Theory : reals
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