Nuprl Lemma : filter_wf4
∀[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[l:T List]. (filter(λx.P[x];l) ∈ {x:T| (x ∈ l) ∧ (↑P[x])} List)
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
filter: filter(P;l),
list: T List,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
or: P ∨ Q,
top: Top,
and: P ∧ Q,
so_apply: x[s],
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
squash: ↓T,
sq_stable: SqStable(P),
uiff: uiff(P;Q),
le: A ≤ B,
not: ¬A,
less_than': less_than'(a;b),
true: True,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
sq_type: SQType(T),
less_than: a < b,
bool: 𝔹,
unit: Unit,
btrue: tt,
ifthenelse: if b then t else f fi ,
cand: A c∧ B,
bfalse: ff,
exists: ∃x:A. B[x],
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
list-cases,
filter_nil_lemma,
nil_wf,
l_member_wf,
assert_wf,
product_subtype_list,
spread_cons_lemma,
sq_stable__le,
le_antisymmetry_iff,
add_functionality_wrt_le,
add-associates,
add-zero,
zero-add,
le-add-cancel,
decidable__le,
false_wf,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
minus-one-mul-top,
add-commutes,
le_wf,
equal_wf,
subtract_wf,
not-ge-2,
less-iff-le,
minus-minus,
add-swap,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
filter_cons_lemma,
bool_wf,
eqtt_to_assert,
cons_wf,
cons_member,
subtype_rel_list_set,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
lambdaEquality,
dependent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
cumulativity,
applyEquality,
because_Cache,
unionElimination,
isect_memberEquality,
voidEquality,
setEquality,
productEquality,
functionExtensionality,
promote_hyp,
hypothesis_subsumption,
productElimination,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
addEquality,
dependent_set_memberEquality,
independent_pairFormation,
minusEquality,
intEquality,
instantiate,
equalityElimination,
inlFormation,
inrFormation,
dependent_pairFormation,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[l:T List]. (filter(\mlambda{}x.P[x];l) \mmember{} \{x:T| (x \mmember{} l) \mwedge{} (\muparrow{}P[x])\} List)
Date html generated:
2017_04_14-AM-08_51_49
Last ObjectModification:
2017_02_27-PM-03_37_34
Theory : list_0
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