Nuprl Lemma : filter_filter2
∀[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[L:T List]. (filter(P;L) = filter2(λi.(P L[i]);L) ∈ (T List))
Proof
Definitions occuring in Statement :
filter2: filter2(P;L),
select: L[n],
filter: filter(P;l),
list: T List,
bool: 𝔹,
uall: ∀[x:A]. B[x],
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
prop: ℙ,
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,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
ge: i ≥ j ,
le: A ≤ B,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
less_than: a < b,
squash: ↓T,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
bfalse: ff,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
select: L[n],
cons: [a / b]
Lemmas referenced :
bool_wf,
equal_wf,
cons_wf,
filter2_wf,
select_wf,
int_seg_properties,
length_wf,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_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_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
length_of_cons_lemma,
non_neg_length,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
equal-wf-T-base,
assert_wf,
bnot_wf,
not_wf,
list_induction,
list_wf,
filter_wf5,
subtype_rel_dep_function,
l_member_wf,
set_wf,
int_seg_wf,
filter_nil_lemma,
filter2_nil_lemma,
nil_wf,
filter_cons_lemma,
squash_wf,
true_wf,
eqtt_to_assert,
cons_filter2,
iff_weakening_equal,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
select-cons-tl,
add-subtract-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
applyEquality,
functionExtensionality,
hypothesisEquality,
cumulativity,
cut,
introduction,
extract_by_obid,
hypothesis,
because_Cache,
thin,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
sqequalHypSubstitution,
isectElimination,
lambdaEquality,
addEquality,
setElimination,
rename,
independent_isectElimination,
natural_numberEquality,
productElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
baseClosed,
isect_memberFormation,
setEquality,
lambdaFormation,
imageElimination,
independent_functionElimination,
equalityTransitivity,
equalityElimination,
imageMemberEquality,
universeEquality,
axiomEquality,
functionEquality
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[L:T List]. (filter(P;L) = filter2(\mlambda{}i.(P L[i]);L))
Date html generated:
2017_10_01-AM-08_35_11
Last ObjectModification:
2017_07_26-PM-04_25_44
Theory : list!
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