Nuprl Lemma : metric-symmetry
∀[X:Type]. ∀[d:metric(X)]. ∀x,y:X. ((d x y) = (d y x))
Proof
Definitions occuring in Statement :
metric: metric(X),
req: x = y,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
metric: metric(X),
sq_stable: SqStable(P),
implies: P ⇒ Q,
and: P ∧ Q,
squash: ↓T,
guard: {T},
uimplies: b supposing a,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top
Lemmas referenced :
sq_stable__req,
req_witness,
metric_wf,
istype-universe,
radd_wf,
int-to-real_wf,
rleq_antisymmetry,
rleq_functionality,
req_weakening,
radd_functionality,
rleq_weakening,
itermSubtract_wf,
itermAdd_wf,
itermConstant_wf,
itermVar_wf,
req-iff-rsub-is-0,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
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,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
sqequalHypSubstitution,
setElimination,
thin,
rename,
extract_by_obid,
isectElimination,
applyEquality,
hypothesisEquality,
hypothesis,
independent_functionElimination,
productElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
inhabitedIsType,
universeIsType,
lambdaEquality_alt,
dependent_functionElimination,
because_Cache,
functionIsTypeImplies,
isect_memberEquality_alt,
isectIsTypeImplies,
instantiate,
universeEquality,
natural_numberEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
approximateComputation,
int_eqEquality,
voidElimination
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)]. \mforall{}x,y:X. ((d x y) = (d y x))
Date html generated:
2019_10_29-AM-10_52_35
Last ObjectModification:
2019_10_02-AM-09_34_20
Theory : reals
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