Nuprl Lemma : in-bar-converges
∀[T:Type]. ∀[b,a:T]. (in-bar(a)↓b
⇐⇒ a = b ∈ T)
Proof
Definitions occuring in Statement :
bar-converges: x↓a
,
in-bar: in-bar(b)
,
uall: ∀[x:A]. B[x]
,
iff: P
⇐⇒ Q
,
universe: Type
,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x]
,
bar-converges: x↓a
,
exists: ∃x:A. B[x]
,
member: t ∈ T
,
nat: ℕ
,
le: A ≤ B
,
and: P ∧ Q
,
less_than': less_than'(a;b)
,
false: False
,
not: ¬A
,
implies: P
⇒ Q
,
prop: ℙ
,
bar-val: bar-val(n;x)
,
in-bar: in-bar(b)
,
iff: P
⇐⇒ Q
,
rev_implies: P
⇐ Q
,
guard: {T}
Lemmas referenced :
false_wf,
le_wf,
unit_wf2,
equal_wf,
bar-val_wf,
in-bar_wf,
bar-converges-unique,
bar-converges_wf,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
dependent_pairFormation,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
lambdaFormation,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
inlEquality,
unionEquality,
cumulativity,
because_Cache,
independent_functionElimination,
equalitySymmetry,
universeEquality,
hyp_replacement,
applyEquality,
lambdaEquality,
setElimination,
rename,
productElimination,
setEquality
Latex:
\mforall{}[T:Type]. \mforall{}[b,a:T]. (in-bar(a)\mdownarrow{}b \mLeftarrow{}{}\mRightarrow{} a = b)
Date html generated:
2016_10_21-AM-09_47_34
Last ObjectModification:
2016_07_12-AM-05_07_42
Theory : co-recursion
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