Nuprl Lemma : assert-exists_sublist
∀[T:Type]. ∀L:T List. ∀P:(T List) ⟶ 𝔹. (↑exists_sublist(L;P) ⇐⇒ ∃LL:T List. (LL ⊆ L ∧ (↑(P LL))))
Proof
Definitions occuring in Statement :
exists_sublist: exists_sublist(L;P),
sublist: L1 ⊆ L2,
list: T List,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
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,
exists_sublist: exists_sublist(L;P),
ifthenelse: if b then t else f fi ,
btrue: tt,
top: Top,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
bfalse: ff,
iff: P ⇐⇒ Q,
exists: ∃x:A. B[x],
cand: A c∧ B,
rev_implies: P ⇐ Q,
uimplies: b supposing a,
guard: {T},
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
or: P ∨ Q,
cons: [a / b],
sq_type: SQType(T),
assert: ↑b,
true: True
Lemmas referenced :
list_induction,
all_wf,
list_wf,
bool_wf,
iff_wf,
assert_wf,
exists_sublist_wf,
exists_wf,
sublist_wf,
null_nil_lemma,
null_cons_lemma,
spread_cons_lemma,
nil_wf,
nil-sublist,
assert_witness,
sublist_nil,
assert_functionality_wrt_uiff,
or_wf,
cons_wf,
assert_of_bor,
bor_wf,
sublist_tl2,
cons_sublist_cons,
list-cases,
product_subtype_list,
and_wf,
equal_wf,
assert_elim,
subtype_base_sq,
bool_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
cumulativity,
hypothesis,
functionExtensionality,
applyEquality,
productEquality,
independent_functionElimination,
rename,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
universeEquality,
independent_pairFormation,
dependent_pairFormation,
productElimination,
independent_isectElimination,
equalitySymmetry,
unionElimination,
addLevel,
allFunctionality,
impliesFunctionality,
orFunctionality,
orLevelFunctionality,
inlFormation,
promote_hyp,
hypothesis_subsumption,
inrFormation,
dependent_set_memberEquality,
applyLambdaEquality,
setElimination,
equalityTransitivity,
levelHypothesis,
instantiate,
natural_numberEquality
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. \mforall{}P:(T List) {}\mrightarrow{} \mBbbB{}. (\muparrow{}exists\_sublist(L;P) \mLeftarrow{}{}\mRightarrow{} \mexists{}LL:T List. (LL \msubseteq{} L \mwedge{} (\muparrow{}(P LL))))
Date html generated:
2018_05_21-PM-00_34_15
Last ObjectModification:
2017_10_12-AM-10_13_20
Theory : list_1
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