Nuprl Lemma : member-list-diff
∀[T:Type]. ∀eq:EqDecider(T). ∀as,bs:T List. ∀x:T. ((x ∈ as-bs) ⇐⇒ (x ∈ as) ∧ (¬(x ∈ bs)))
Proof
Definitions occuring in Statement :
list-diff: as-bs,
l_member: (x ∈ l),
list: T List,
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
and: P ∧ Q
Lemmas referenced :
list-diff-property,
list_wf,
deq_wf
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
dependent_functionElimination,
productElimination,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}eq:EqDecider(T). \mforall{}as,bs:T List. \mforall{}x:T. ((x \mmember{} as-bs) \mLeftarrow{}{}\mRightarrow{} (x \mmember{} as) \mwedge{} (\mneg{}(x \mmember{} bs)))
Date html generated:
2016_05_14-PM-03_29_52
Last ObjectModification:
2015_12_26-PM-06_02_51
Theory : decidable!equality
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