Nuprl Lemma : cardinality-le-list-set
∀[T:Type]. ((∀x,y:T. Dec(x = y ∈ T)) ⇒ (∀L:T List. |{x:T| (x ∈ L)} | ≤ ||L||))
Proof
Definitions occuring in Statement :
cardinality-le: |T| ≤ n,
l_member: (x ∈ l),
length: ||as||,
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,
sq_stable: SqStable(P),
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
equal_wf,
decidable_wf,
all_wf,
list_wf,
set_wf,
sq_stable__l_member,
l_member-set,
list-subtype,
l_member_wf,
list-cardinality-le
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setEquality,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
cumulativity,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
setElimination,
rename,
because_Cache,
introduction,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
lambdaEquality,
universeEquality
Latex:
\mforall{}[T:Type]. ((\mforall{}x,y:T. Dec(x = y)) {}\mRightarrow{} (\mforall{}L:T List. |\{x:T| (x \mmember{} L)\} | \mleq{} ||L||))
Date html generated:
2016_05_14-PM-03_31_42
Last ObjectModification:
2016_01_14-PM-11_19_33
Theory : decidable!equality
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