Nuprl Lemma : mdist-difference
∀[X:Type]. ∀[d:metric(X)]. ∀[x,a,b:X]. (|mdist(d;x;a) - mdist(d;x;b)| ≤ mdist(d;a;b))
Proof
Definitions occuring in Statement :
mdist: mdist(d;x;y),
metric: metric(X),
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
uall: ∀[x:A]. B[x],
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,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
cand: A c∧ B,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
uimplies: b supposing a,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top
Lemmas referenced :
rabs-difference-bound-rleq,
mdist_wf,
mdist-triangle-inequality1,
le_witness_for_triv,
metric_wf,
istype-universe,
mdist-triangle-inequality,
rleq-implies-rleq,
rsub_wf,
radd_wf,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
req-iff-rsub-is-0,
real_polynomial_null,
int-to-real_wf,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
hypothesis,
productElimination,
independent_functionElimination,
independent_pairFormation,
sqequalRule,
lambdaEquality_alt,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
functionIsTypeImplies,
inhabitedIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
universeIsType,
instantiate,
universeEquality,
natural_numberEquality,
because_Cache,
approximateComputation,
int_eqEquality,
voidElimination
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)]. \mforall{}[x,a,b:X]. (|mdist(d;x;a) - mdist(d;x;b)| \mleq{} mdist(d;a;b))
Date html generated:
2019_10_29-AM-11_14_36
Last ObjectModification:
2019_10_02-AM-09_55_04
Theory : reals
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