Nuprl Lemma : finite-type-list
∀[T:Type]. ((∀x,y:T. Dec(x = y ∈ T)) ⇒ (∀L:T List. finite-type({x:T| (x ∈ L)} )))
Proof
Definitions occuring in Statement :
finite-type: finite-type(T),
l_member: (x ∈ l),
list: T List,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
cardinality-le-finite,
l_member_wf,
length_wf_nat,
list_wf,
all_wf,
decidable_wf,
equal_wf,
cardinality-le-list-set
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setEquality,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
independent_functionElimination,
sqequalRule,
lambdaEquality,
universeEquality
Latex:
\mforall{}[T:Type]. ((\mforall{}x,y:T. Dec(x = y)) {}\mRightarrow{} (\mforall{}L:T List. finite-type(\{x:T| (x \mmember{} L)\} )))
Date html generated:
2016_05_14-PM-03_31_44
Last ObjectModification:
2015_12_26-PM-06_01_41
Theory : decidable!equality
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