Nuprl Lemma : mdist_functionality
∀[X:Type]. ∀[d:metric(X)]. ∀[x,y,x',y':X]. (mdist(d;x;y) = mdist(d;x';y')) supposing (x ≡ x' and y ≡ y')
Proof
Definitions occuring in Statement :
mdist: mdist(d;x;y),
meq: x ≡ y,
metric: metric(X),
req: x = y,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
universe: Type
Definitions unfolded in proof :
meq: x ≡ y,
mdist: mdist(d;x;y),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
implies: P ⇒ Q,
prop: ℙ,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
guard: {T},
uiff: uiff(P;Q),
and: P ∧ Q,
all: ∀x:A. B[x],
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top
Lemmas referenced :
rleq_antisymmetry,
mdist_wf,
req_witness,
req_wf,
int-to-real_wf,
metric_wf,
istype-universe,
uimplies_transitivity,
rleq_wf,
radd_wf,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
mdist-triangle-inequality,
radd_functionality_wrt_rleq,
rleq_functionality,
radd_functionality,
req_weakening,
req_functionality,
mdist-symm,
rleq_weakening,
itermSubtract_wf,
itermAdd_wf,
itermConstant_wf,
itermVar_wf,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_add_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_isectElimination,
independent_functionElimination,
universeIsType,
natural_numberEquality,
isect_memberEquality_alt,
because_Cache,
isectIsTypeImplies,
inhabitedIsType,
instantiate,
universeEquality,
equalityTransitivity,
equalitySymmetry,
productElimination,
dependent_functionElimination,
approximateComputation,
lambdaEquality_alt,
int_eqEquality,
voidElimination
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)]. \mforall{}[x,y,x',y':X].
(mdist(d;x;y) = mdist(d;x';y')) supposing (x \mequiv{} x' and y \mequiv{} y')
Date html generated:
2019_10_29-AM-10_59_19
Last ObjectModification:
2019_10_02-AM-09_40_52
Theory : reals
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