Nuprl Lemma : l_member_decidable
∀[T:Type]. ∀x:T. ∀l:T List. ((∀y:T. Dec(x = y ∈ T)) ⇒ Dec((x ∈ l)))
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
list: T List,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
list_induction,
all_wf,
decidable_wf,
equal_wf,
l_member_wf,
decidable_functionality,
nil_wf,
false_wf,
nil_member,
decidable__false,
cons_wf,
or_wf,
cons_member,
decidable__or,
list_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
Error :lambdaEquality_alt,
functionEquality,
because_Cache,
hypothesis,
independent_functionElimination,
dependent_functionElimination,
productElimination,
Error :functionIsType,
Error :universeIsType,
rename,
instantiate,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}x:T. \mforall{}l:T List. ((\mforall{}y:T. Dec(x = y)) {}\mRightarrow{} Dec((x \mmember{} l)))
Date html generated:
2019_06_20-PM-01_19_50
Last ObjectModification:
2018_10_29-PM-06_24_45
Theory : list_1
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