Nuprl Lemma : filter_is_nil2
∀[T:Type]. ∀[L:T List]. ∀[P:{x:T| (x ∈ L)} ⟶ 𝔹]. filter(P;L) ~ [] supposing (∀x∈L.¬↑P[x])
Proof
Definitions occuring in Statement :
l_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
filter: filter(P;l),
nil: [],
list: T List,
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
not: ¬A,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
so_apply: x[s],
so_lambda: λ2x.t[x]
Lemmas referenced :
filter_is_nil,
l_member_wf,
list-subtype,
l_all_wf,
not_wf,
assert_wf,
bool_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setEquality,
hypothesisEquality,
hypothesis,
cumulativity,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
sqequalRule,
sqequalAxiom,
lambdaEquality,
applyEquality,
isect_memberEquality,
because_Cache,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[P:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbB{}]. filter(P;L) \msim{} [] supposing (\mforall{}x\mmember{}L.\mneg{}\muparrow{}P[x])
Date html generated:
2016_05_14-AM-06_51_28
Last ObjectModification:
2015_12_26-PM-00_22_06
Theory : list_0
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