Nuprl Lemma : scale-metric-cauchy
∀[X:Type]. ∀[d:metric(X)]. ∀c:{c:ℝ| r0 < c} . ∀x:ℕ ⟶ X. (mcauchy(d;n.x n) ⇐⇒ mcauchy(c*d;n.x n))
Proof
Definitions occuring in Statement :
mcauchy: mcauchy(d;n.x[n]),
scale-metric: c*d,
metric: metric(X),
rless: x < y,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
metric-leq: d1 ≤ d2,
uimplies: b supposing a,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
metric: metric(X),
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
rev_implies: P ⇐ Q,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
sq_stable: SqStable(P),
squash: ↓T,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
rdiv: (x/y),
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
scale-metric: c*d,
mdist: mdist(d;x;y),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
metric-leq-cauchy,
scale-metric_wf,
rleq_weakening_equal,
mdist_wf,
le_witness_for_triv,
mcauchy_wf,
istype-nat,
rdiv_wf,
int-to-real_wf,
sq_stable__rless,
rless_wf,
subtype_rel_sets_simple,
real_wf,
rleq_wf,
rleq_weakening_rless,
metric_wf,
istype-universe,
rmul_preserves_rless,
rmul_wf,
itermSubtract_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
rinv_wf2,
rless-int,
rless_functionality,
req_transitivity,
rmul-rinv,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
rleq_functionality,
req_weakening,
rmul_functionality
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation_alt,
independent_pairFormation,
applyEquality,
because_Cache,
sqequalRule,
dependent_functionElimination,
independent_functionElimination,
independent_isectElimination,
lambdaEquality_alt,
productElimination,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
universeIsType,
setElimination,
rename,
dependent_set_memberEquality_alt,
closedConclusion,
natural_numberEquality,
inrFormation_alt,
imageMemberEquality,
baseClosed,
imageElimination,
functionIsType,
setIsType,
instantiate,
universeEquality,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
voidElimination
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)]. \mforall{}c:\{c:\mBbbR{}| r0 < c\} . \mforall{}x:\mBbbN{} {}\mrightarrow{} X. (mcauchy(d;n.x n) \mLeftarrow{}{}\mRightarrow{} mcauchy(c*d;n.x n))
Date html generated:
2019_10_30-AM-06_44_55
Last ObjectModification:
2019_10_02-AM-10_56_50
Theory : reals
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