Nuprl Lemma : mdist-difference2
∀[X:Type]. ∀[d:metric(X)]. ∀[x,a,b,y:X]. (|mdist(d;x;y) - mdist(d;a;b)| ≤ (mdist(d;x;a) + mdist(d;y;b)))
Proof
Definitions occuring in Statement :
mdist: mdist(d;x;y),
metric: metric(X),
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
radd: a + b,
uall: ∀[x:A]. B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
rge: x ≥ y,
guard: {T},
rleq: x ≤ y,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
le: A ≤ B,
and: P ∧ Q,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top
Lemmas referenced :
rleq_functionality_wrt_implies,
rabs_wf,
rsub_wf,
mdist_wf,
radd_wf,
rleq_weakening_equal,
r-triangle-inequality2,
le_witness_for_triv,
metric_wf,
istype-universe,
radd-preserves-rleq,
rminus_wf,
itermSubtract_wf,
itermAdd_wf,
itermMinus_wf,
itermVar_wf,
rleq_weakening,
mdist-symm,
radd_functionality_wrt_rleq,
mdist-difference,
rleq_functionality,
radd_functionality,
rabs_functionality,
rsub_functionality,
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_minus_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,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
because_Cache,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
lambdaEquality_alt,
dependent_functionElimination,
productElimination,
functionIsTypeImplies,
inhabitedIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
universeIsType,
instantiate,
universeEquality,
natural_numberEquality,
approximateComputation,
int_eqEquality,
voidElimination
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)]. \mforall{}[x,a,b,y:X].
(|mdist(d;x;y) - mdist(d;a;b)| \mleq{} (mdist(d;x;a) + mdist(d;y;b)))
Date html generated:
2019_10_29-AM-11_14_59
Last ObjectModification:
2019_10_02-AM-09_55_21
Theory : reals
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