Nuprl Lemma : m-k-regular-mcauchy
∀[X:Type]. ∀[d:metric(X)]. ∀[s:ℕ ⟶ X]. ∀b:ℕ+. (m-k-regular(d;b;s) ⇒ mcauchy(d;n.s n))
Proof
Definitions occuring in Statement :
m-k-regular: m-k-regular(d;k;s),
mcauchy: mcauchy(d;n.x[n]),
metric: metric(X),
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
m-k-regular: m-k-regular(d;k;s),
mcauchy: mcauchy(d;n.x[n]),
sq_exists: ∃x:A [B[x]],
member: t ∈ T,
nat: ℕ,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
rneq: x ≠ y,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
uiff: uiff(P;Q)
Lemmas referenced :
multiply_nat_wf,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
istype-le,
rleq_wf,
mdist_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_properties,
decidable__lt,
rless_wf,
m-k-regular_wf,
nat_plus_wf,
istype-nat,
metric_wf,
istype-universe,
radd_wf,
itermAdd_wf,
int_term_value_add_lemma,
implies_weakening_uimplies,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rleq-int-fractions,
istype-less_than,
mul_bounds_1b,
mul_nat_plus,
radd_functionality_wrt_rleq,
rleq_functionality,
radd-int-fractions,
req_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalHypSubstitution,
dependent_set_memberFormation_alt,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
dependent_set_memberEquality_alt,
multiplyEquality,
natural_numberEquality,
setElimination,
rename,
because_Cache,
hypothesis,
hypothesisEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
functionIsType,
applyEquality,
closedConclusion,
inrFormation_alt,
productElimination,
instantiate,
universeEquality,
addEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)]. \mforall{}[s:\mBbbN{} {}\mrightarrow{} X]. \mforall{}b:\mBbbN{}\msupplus{}. (m-k-regular(d;b;s) {}\mRightarrow{} mcauchy(d;n.s n))
Date html generated:
2019_10_30-AM-06_59_08
Last ObjectModification:
2019_10_09-AM-08_53_30
Theory : reals
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