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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