Nuprl Lemma : implies-filter-equal
∀[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[L1,L2:T List].
(filter(P;L1) = L2 ∈ (T List)) supposing
(((∀x:T. ((x ∈ L2) ⇐⇒ (x ∈ L1) ∧ (↑(P x)))) ∧ (∀x,y:T. (x before y ∈ L2 ⇒ x before y ∈ L1))) and
no_repeats(T;L1))
Proof
Definitions occuring in Statement :
l_before: x before y ∈ l,
no_repeats: no_repeats(T;l),
l_member: (x ∈ l),
filter: filter(P;l),
list: T List,
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
not: ¬A,
false: False,
guard: {T}
Lemmas referenced :
filter-equals,
all_wf,
iff_wf,
l_member_wf,
assert_wf,
l_before_wf,
no_repeats_wf,
list_wf,
bool_wf,
no_repeats_iff,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
extract_by_obid,
isectElimination,
hypothesisEquality,
dependent_functionElimination,
independent_isectElimination,
hypothesis,
independent_functionElimination,
independent_pairFormation,
productEquality,
cumulativity,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
functionEquality,
isect_memberEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
universeEquality,
lambdaFormation,
voidElimination
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[L1,L2:T List].
(filter(P;L1) = L2) supposing
(((\mforall{}x:T. ((x \mmember{} L2) \mLeftarrow{}{}\mRightarrow{} (x \mmember{} L1) \mwedge{} (\muparrow{}(P x))))
\mwedge{} (\mforall{}x,y:T. (x before y \mmember{} L2 {}\mRightarrow{} x before y \mmember{} L1))) and
no\_repeats(T;L1))
Date html generated:
2017_04_17-AM-08_35_35
Last ObjectModification:
2017_02_27-PM-04_55_03
Theory : list_1
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