Nuprl Lemma : decidable__list-closed
∀[T:Type]. ∀L:T List. ∀f:T ⟶ (T List).  ((∀x,y:T.  Dec(x = y ∈ T)) 
⇒ Dec(list-closed(T;L;f)))
Proof
Definitions occuring in Statement : 
list-closed: list-closed(T;L;f)
, 
list: T List
, 
decidable: Dec(P)
, 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
function: x:A ⟶ B[x]
, 
universe: Type
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
list-closed: list-closed(T;L;f)
, 
member: t ∈ T
, 
so_lambda: λ2x.t[x]
, 
prop: ℙ
, 
so_apply: x[s]
Lemmas referenced : 
decidable__l_all, 
l_all_wf, 
l_member_wf, 
decidable__l_member, 
decidable_wf, 
equal_wf, 
list_wf, 
istype-universe
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
Error :isect_memberFormation_alt, 
Error :lambdaFormation_alt, 
cut, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
dependent_functionElimination, 
sqequalRule, 
Error :lambdaEquality_alt, 
applyEquality, 
setElimination, 
rename, 
hypothesis, 
Error :setIsType, 
Error :universeIsType, 
independent_functionElimination, 
because_Cache, 
Error :functionIsType, 
Error :inhabitedIsType, 
instantiate, 
universeEquality
Latex:
\mforall{}[T:Type].  \mforall{}L:T  List.  \mforall{}f:T  {}\mrightarrow{}  (T  List).    ((\mforall{}x,y:T.    Dec(x  =  y))  {}\mRightarrow{}  Dec(list-closed(T;L;f)))
Date html generated:
2019_06_20-PM-01_50_58
Last ObjectModification:
2019_05_13-PM-03_36_46
Theory : list_1
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