Nuprl Lemma : dlattice-order-iff
∀[X:Type]. ∀as,bs:X List List. (as ⇒ bs ⇐⇒ ∀x:X List. ((x ∈ bs) ⇒ (∃y:X List. ((y ∈ as) ∧ l_subset(X;y;x)))))
Proof
Definitions occuring in Statement :
dlattice-order: as ⇒ bs,
l_subset: l_subset(T;as;bs),
l_member: (x ∈ l),
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
dlattice-order: as ⇒ bs,
exists: ∃x:A. B[x],
cand: A c∧ B
Lemmas referenced :
l_member_wf,
list_wf,
dlattice-order_wf,
all_wf,
exists_wf,
l_subset_wf,
l_all_iff,
l_exists_wf,
l_contains_wf,
l_exists_iff,
l_subset-l_contains
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
functionEquality,
productEquality,
universeEquality,
dependent_functionElimination,
setElimination,
rename,
setEquality,
productElimination,
independent_functionElimination,
dependent_pairFormation,
addLevel,
allFunctionality,
impliesFunctionality,
levelHypothesis,
allLevelFunctionality,
impliesLevelFunctionality
Latex:
\mforall{}[X:Type]
\mforall{}as,bs:X List List.
(as {}\mRightarrow{} bs \mLeftarrow{}{}\mRightarrow{} \mforall{}x:X List. ((x \mmember{} bs) {}\mRightarrow{} (\mexists{}y:X List. ((y \mmember{} as) \mwedge{} l\_subset(X;y;x)))))
Date html generated:
2020_05_20-AM-08_26_30
Last ObjectModification:
2017_07_28-AM-09_13_17
Theory : lattices
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