Nuprl Lemma : meq-equiv
∀[X:Type]. ∀[d:metric(X)]. EquivRel(X;x,y.x ≡ y)
Proof
Definitions occuring in Statement :
meq: x ≡ y,
metric: metric(X),
equiv_rel: EquivRel(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
meq: x ≡ y,
equiv_rel: EquivRel(T;x,y.E[x; y]),
and: P ∧ Q,
refl: Refl(T;x,y.E[x; y]),
all: ∀x:A. B[x],
metric: metric(X),
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
cand: A c∧ B,
sym: Sym(T;x,y.E[x; y]),
prop: ℙ,
trans: Trans(T;x,y.E[x; y]),
uimplies: b supposing a,
guard: {T},
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
req_int_terms: t1 ≡ t2,
top: Top
Lemmas referenced :
metric-symmetry,
sq_stable__req,
int-to-real_wf,
req_witness,
req_wf,
rleq_antisymmetry,
metric-nonneg,
rleq_wf,
radd_wf,
metric_wf,
istype-universe,
req_functionality,
req_weakening,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rleq_functionality,
radd_functionality,
rleq_weakening,
itermSubtract_wf,
itermAdd_wf,
itermConstant_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
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
independent_pairFormation,
lambdaFormation_alt,
applyEquality,
setElimination,
rename,
natural_numberEquality,
independent_functionElimination,
productElimination,
imageMemberEquality,
baseClosed,
imageElimination,
universeIsType,
because_Cache,
independent_isectElimination,
dependent_set_memberEquality_alt,
productIsType,
functionIsType,
inhabitedIsType,
instantiate,
universeEquality,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
approximateComputation,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)]. EquivRel(X;x,y.x \mequiv{} y)
Date html generated:
2019_10_29-AM-10_54_54
Last ObjectModification:
2019_10_02-AM-09_36_20
Theory : reals
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