Nuprl Lemma : member-union-list2
∀[T:Type]. ∀eq:EqDecider(T). ∀ll:T List List. ∀x:T. ((x ∈ union-list2(eq;ll)) ⇐⇒ ∃l:T List. ((l ∈ ll) ∧ (x ∈ l)))
Proof
Definitions occuring in Statement :
union-list2: union-list2(eq;ll),
l_member: (x ∈ l),
list: T List,
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
implies: P ⇒ Q,
union-list2: union-list2(eq;ll),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
iff: P ⇐⇒ Q,
uimplies: b supposing a,
not: ¬A,
false: False,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
cand: A c∧ B,
or: P ∨ Q,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
list_induction,
list_wf,
all_wf,
iff_wf,
l_member_wf,
union-list2_wf,
exists_wf,
list_ind_nil_lemma,
list_ind_cons_lemma,
deq_wf,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
nil_wf,
btrue_neq_bfalse,
null_wf,
bool_wf,
eqtt_to_assert,
assert_of_null,
or_wf,
equal_wf,
and_wf,
cons_member,
cons_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
equal-wf-T-base,
member-union,
l-union_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
productEquality,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
rename,
because_Cache,
universeEquality,
independent_pairFormation,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
unionElimination,
equalityElimination,
dependent_pairFormation,
inlFormation,
addLevel,
hyp_replacement,
dependent_set_memberEquality,
applyLambdaEquality,
setElimination,
levelHypothesis,
impliesFunctionality,
existsFunctionality,
andLevelFunctionality,
existsLevelFunctionality,
promote_hyp,
instantiate,
baseClosed,
inrFormation
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T). \mforall{}ll:T List List. \mforall{}x:T.
((x \mmember{} union-list2(eq;ll)) \mLeftarrow{}{}\mRightarrow{} \mexists{}l:T List. ((l \mmember{} ll) \mwedge{} (x \mmember{} l)))
Date html generated:
2017_04_17-AM-09_10_01
Last ObjectModification:
2017_02_27-PM-05_18_53
Theory : decidable!equality
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