Nuprl Lemma : mapfilter-pos-length
∀[T,B:Type]. ∀[L:T List]. ∀[P:{x:T| (x ∈ L)} ⟶ 𝔹]. ∀[f:{x:{x:T| (x ∈ L)} | ↑(P x)} ⟶ B].
0 < ||mapfilter(f;P;L)|| supposing (∃x∈L. ↑(P x))
Proof
Definitions occuring in Statement :
mapfilter: mapfilter(f;P;L),
l_exists: (∃x∈L. P[x]),
l_member: (x ∈ l),
length: ||as||,
list: T List,
assert: ↑b,
bool: 𝔹,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
or: P ∨ Q,
not: ¬A,
false: False,
cons: [a / b],
top: Top,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
and: P ∧ Q,
guard: {T},
decidable: Dec(P),
uiff: uiff(P;Q),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x]
Lemmas referenced :
mapfilter-not-nil,
l_exists_wf,
assert_wf,
l_member_wf,
set_wf,
bool_wf,
list_wf,
mapfilter_wf,
list-subtype,
list-cases,
length_of_nil_lemma,
not_wf,
equal-wf-base,
product_subtype_list,
length_of_cons_lemma,
add_nat_plus,
length_wf_nat,
less_than_wf,
nat_plus_wf,
nat_plus_properties,
decidable__lt,
add-is-int-iff,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
false_wf,
equal_wf,
equal-wf-T-base
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
independent_functionElimination,
cumulativity,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
setEquality,
setElimination,
rename,
dependent_set_memberEquality,
functionEquality,
because_Cache,
lambdaFormation,
universeEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
unionElimination,
voidElimination,
baseClosed,
promote_hyp,
hypothesis_subsumption,
productElimination,
isect_memberEquality,
voidEquality,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
applyLambdaEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll
Latex:
\mforall{}[T,B:Type]. \mforall{}[L:T List]. \mforall{}[P:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbB{}]. \mforall{}[f:\{x:\{x:T| (x \mmember{} L)\} | \muparrow{}(P x)\} {}\mrightarrow{} B].
0 < ||mapfilter(f;P;L)|| supposing (\mexists{}x\mmember{}L. \muparrow{}(P x))
Date html generated:
2017_04_17-AM-07_30_06
Last ObjectModification:
2017_02_27-PM-04_07_21
Theory : list_1
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