Nuprl Lemma : filter_of_filter2
∀[T:Type]. ∀[L:T List]. ∀[P:ℕ||L|| ⟶ 𝔹]. ∀[Q:T ⟶ 𝔹].
(filter(Q;filter2(P;L)) = filter2(λi.((P i) ∧b (Q L[i]));L) ∈ (T List))
Proof
Definitions occuring in Statement :
filter2: filter2(P;L),
select: L[n],
length: ||as||,
filter: filter(P;l),
list: T List,
band: p ∧b q,
int_seg: {i..j-},
bool: 𝔹,
uall: ∀[x:A]. B[x],
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
prop: ℙ,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
less_than: a < b,
squash: ↓T,
bfalse: ff,
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
subtract: n - m,
true: True,
nat_plus: ℕ+,
less_than': less_than'(a;b),
cons: [a / b],
select: L[n],
le: A ≤ B,
ge: i ≥ j ,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q
Lemmas referenced :
list_induction,
uall_wf,
int_seg_wf,
length_wf,
bool_wf,
equal_wf,
list_wf,
filter_wf5,
filter2_wf,
subtype_rel_dep_function,
l_member_wf,
subtype_rel_self,
set_wf,
eqtt_to_assert,
select_wf,
int_seg_properties,
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,
length_of_nil_lemma,
filter2_nil_lemma,
filter_nil_lemma,
nil_wf,
length_of_cons_lemma,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
add-member-int_seg2,
lelt_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
add-is-int-iff,
nat_plus_properties,
nat_plus_wf,
less_than_wf,
length_wf_nat,
add_nat_plus,
false_wf,
int_term_value_add_lemma,
itermAdd_wf,
non_neg_length,
cons_wf,
cons_filter2,
iff_weakening_equal,
true_wf,
squash_wf,
add-subtract-cancel,
select-cons-tl,
filter2_functionality,
filter_cons_lemma,
equal-wf-T-base,
assert_wf,
bnot_wf,
not_wf,
uiff_transitivity,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
natural_numberEquality,
cumulativity,
hypothesis,
because_Cache,
functionExtensionality,
applyEquality,
setEquality,
independent_isectElimination,
setElimination,
rename,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
dependent_functionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
imageElimination,
equalityTransitivity,
equalitySymmetry,
functionIsType,
universeIsType,
axiomEquality,
inhabitedIsType,
addEquality,
universeEquality,
instantiate,
closedConclusion,
baseApply,
promote_hyp,
pointwiseFunctionality,
baseClosed,
imageMemberEquality,
dependent_set_memberEquality,
applyLambdaEquality,
hyp_replacement
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[P:\mBbbN{}||L|| {}\mrightarrow{} \mBbbB{}]. \mforall{}[Q:T {}\mrightarrow{} \mBbbB{}].
(filter(Q;filter2(P;L)) = filter2(\mlambda{}i.((P i) \mwedge{}\msubb{} (Q L[i]));L))
Date html generated:
2019_10_15-AM-10_55_12
Last ObjectModification:
2018_09_27-AM-10_44_55
Theory : list!
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