Nuprl Lemma : filter_trivial
∀[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[L:T List].  filter(P;L) ~ L supposing (∀x∈L.↑P[x])
Proof
Definitions occuring in Statement : 
l_all: (∀x∈L.P[x])
, 
filter: filter(P;l)
, 
list: T List
, 
assert: ↑b
, 
bool: 𝔹
, 
uimplies: b supposing a
, 
uall: ∀[x:A]. B[x]
, 
so_apply: x[s]
, 
function: x:A ⟶ B[x]
, 
universe: Type
, 
sqequal: s ~ t
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
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
subtype_rel: A ⊆r B
, 
or: P ∨ Q
, 
top: Top
, 
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)
, 
and: 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: ⋅
, 
sq_type: SQType(T)
, 
less_than: a < b
, 
bool: 𝔹
, 
unit: Unit
, 
btrue: tt
, 
ifthenelse: if b then t else f fi 
, 
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, 
l_all_wf, 
assert_wf, 
l_member_wf, 
equal-wf-T-base, 
nat_wf, 
colength_wf_list, 
int_subtype_base, 
list-cases, 
filter_nil_lemma, 
l_all_wf_nil, 
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, 
filter_cons_lemma, 
bool_wf, 
eqtt_to_assert, 
eqff_to_assert, 
bool_cases_sqequal, 
bool_subtype_base, 
assert-bnot, 
cons_wf, 
list_wf, 
l_all_cons
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
Error :isect_memberFormation_alt, 
introduction, 
cut, 
thin, 
lambdaFormation, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
hypothesisEquality, 
hypothesis, 
setElimination, 
rename, 
intWeakElimination, 
natural_numberEquality, 
independent_isectElimination, 
independent_functionElimination, 
voidElimination, 
sqequalRule, 
lambdaEquality, 
dependent_functionElimination, 
isect_memberEquality, 
axiomSqEquality, 
applyEquality, 
setEquality, 
equalityTransitivity, 
equalitySymmetry, 
because_Cache, 
unionElimination, 
voidEquality, 
Error :universeIsType, 
promote_hyp, 
hypothesis_subsumption, 
productElimination, 
applyLambdaEquality, 
imageMemberEquality, 
baseClosed, 
imageElimination, 
addEquality, 
dependent_set_memberEquality, 
independent_pairFormation, 
minusEquality, 
intEquality, 
instantiate, 
cumulativity, 
equalityElimination, 
dependent_pairFormation, 
Error :functionIsType, 
functionEquality, 
universeEquality
Latex:
\mforall{}[T:Type].  \mforall{}[P:T  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[L:T  List].    filter(P;L)  \msim{}  L  supposing  (\mforall{}x\mmember{}L.\muparrow{}P[x])
Date html generated:
2019_06_20-PM-00_43_39
Last ObjectModification:
2018_09_26-PM-02_16_52
Theory : list_0
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