Nuprl Lemma : metric-nonneg
∀[X:Type]. ∀[d:metric(X)]. ∀[x,y:X]. (r0 ≤ (d x y))
Proof
Definitions occuring in Statement :
metric: metric(X),
rleq: x ≤ y,
int-to-real: r(n),
uall: ∀[x:A]. B[x],
apply: f a,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
metric: metric(X),
sq_stable: SqStable(P),
implies: P ⇒ Q,
and: P ∧ Q,
squash: ↓T,
rleq: x ≤ y,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
le: A ≤ B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
top: Top,
false: False,
not: ¬A
Lemmas referenced :
sq_stable__rleq,
int-to-real_wf,
le_witness_for_triv,
metric_wf,
istype-universe,
radd_wf,
rmul_preserves_rleq,
rless-int,
rmul_wf,
itermSubtract_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
rleq-implies-rleq,
rsub_wf,
itermAdd_wf,
req-iff-rsub-is-0,
rleq_functionality,
req_weakening,
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,
real_term_value_add_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
extract_by_obid,
isectElimination,
natural_numberEquality,
hypothesis,
applyEquality,
hypothesisEquality,
independent_functionElimination,
productElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
lambdaEquality_alt,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
functionIsTypeImplies,
inhabitedIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
universeIsType,
instantiate,
universeEquality,
because_Cache,
independent_pairFormation,
approximateComputation,
voidElimination,
int_eqEquality
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)]. \mforall{}[x,y:X]. (r0 \mleq{} (d x y))
Date html generated:
2019_10_29-AM-10_52_55
Last ObjectModification:
2019_10_02-AM-09_34_38
Theory : reals
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