Nuprl Lemma : member-free-dl-meet
∀[X:Type]
∀as,bs:X List List. ∀x:X List.
((x ∈ free-dl-meet(as;bs)) ⇐⇒ ∃u,v:X List. ((u ∈ as) ∧ (v ∈ bs) ∧ (x = (u @ v) ∈ (X List))))
Proof
Definitions occuring in Statement :
free-dl-meet: free-dl-meet(as;bs),
l_member: (x ∈ l),
append: as @ bs,
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
free-dl-meet: free-dl-meet(as;bs),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
or: P ∨ Q,
member: t ∈ T,
uimplies: b supposing a,
not: ¬A,
false: False,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
guard: {T},
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
exists: ∃x:A. B[x],
top: Top,
cand: A c∧ B
Lemmas referenced :
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
list_wf,
nil_wf,
btrue_neq_bfalse,
or_wf,
l_member_wf,
exists_wf,
equal_wf,
append_wf,
list_accum_wf,
map_wf,
all_wf,
iff_wf,
list_induction,
list_accum_nil_lemma,
list_accum_cons_lemma,
cons_wf,
member_wf,
member_append,
member_map,
cons_member,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
sqequalRule,
cut,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
unionElimination,
thin,
introduction,
extract_by_obid,
hypothesis,
isectElimination,
cumulativity,
hypothesisEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
voidElimination,
lambdaEquality,
productEquality,
inrFormation,
addLevel,
allFunctionality,
productElimination,
impliesFunctionality,
dependent_functionElimination,
because_Cache,
universeEquality,
isect_memberEquality,
voidEquality,
rename,
inlFormation,
applyEquality,
dependent_pairFormation,
orFunctionality,
levelHypothesis,
promote_hyp,
existsFunctionality,
andLevelFunctionality,
existsLevelFunctionality,
dependent_set_memberEquality,
applyLambdaEquality,
setElimination
Latex:
\mforall{}[X:Type]
\mforall{}as,bs:X List List. \mforall{}x:X List.
((x \mmember{} free-dl-meet(as;bs)) \mLeftarrow{}{}\mRightarrow{} \mexists{}u,v:X List. ((u \mmember{} as) \mwedge{} (v \mmember{} bs) \mwedge{} (x = (u @ v))))
Date html generated:
2020_05_20-AM-08_26_59
Last ObjectModification:
2017_07_28-AM-09_13_21
Theory : lattices
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