Nuprl Lemma : filter_is_empty
∀[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[L:T List]. uiff(↑null(filter(P;L));∀[i:ℕ||L||]. (¬↑(P L[i])))
Proof
Definitions occuring in Statement :
select: L[n],
length: ||as||,
null: null(as),
filter: filter(P;l),
list: T List,
int_seg: {i..j-},
assert: ↑b,
bool: 𝔹,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
not: ¬A,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
int_seg: {i..j-},
uimplies: b supposing a,
guard: {T},
lelt: i ≤ j < k,
and: P ∧ Q,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
less_than: a < b,
squash: ↓T,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
l_all: (∀x∈L.P[x])
Lemmas referenced :
assert_wf,
select_wf,
int_seg_properties,
length_wf,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
int_seg_wf,
null_wf,
filter_wf5,
subtype_rel_dep_function,
bool_wf,
l_member_wf,
subtype_rel_self,
set_wf,
uall_wf,
not_wf,
list_wf,
assert_witness,
null_cons_lemma,
false_wf,
true_wf,
l_all_wf,
equal-wf-T-base,
bnot_wf,
list_induction,
iff_wf,
filter_nil_lemma,
null_nil_lemma,
l_all_nil,
l_all_wf_nil,
filter_cons_lemma,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
equal_wf,
l_all_cons,
ifthenelse_wf,
cons_wf,
satisfiable-full-omega-tt
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
applyEquality,
functionExtensionality,
hypothesisEquality,
cumulativity,
because_Cache,
setElimination,
rename,
hypothesis,
independent_isectElimination,
natural_numberEquality,
productElimination,
dependent_functionElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
imageElimination,
setEquality,
lambdaFormation,
functionEquality,
universeEquality,
isect_memberFormation,
independent_pairEquality,
equalityTransitivity,
equalitySymmetry,
productEquality,
baseClosed,
equalityElimination,
computeAll
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[L:T List]. uiff(\muparrow{}null(filter(P;L));\mforall{}[i:\mBbbN{}||L||]. (\mneg{}\muparrow{}(P L[i])))
Date html generated:
2019_06_20-PM-01_25_50
Last ObjectModification:
2018_09_17-PM-10_26_04
Theory : list_1
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