Nuprl Lemma : firstn-filter
∀[T:Type]. ∀P:T ⟶ 𝔹. ∀L:T List. ∀n:ℕ. ∃m:ℕ||L|| + 1. (firstn(n;filter(P;L)) = filter(P;firstn(m;L)) ∈ (T List))
Proof
Definitions occuring in Statement :
firstn: firstn(n;as),
length: ||as||,
filter: filter(P;l),
list: T List,
int_seg: {i..j-},
nat: ℕ,
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀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,
nat: ℕ,
int_seg: {i..j-},
implies: P ⇒ Q,
top: Top,
firstn: firstn(n;as),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
exists: ∃x:A. B[x],
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
less_than: a < b,
squash: ↓T,
true: True,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
subtract: n - m,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
lt_int: i <z j
Lemmas referenced :
list_induction,
all_wf,
nat_wf,
exists_wf,
int_seg_wf,
length_wf,
equal_wf,
list_wf,
firstn_wf,
filter_wf5,
subtype_rel_dep_function,
bool_wf,
l_member_wf,
subtype_rel_self,
set_wf,
length_of_nil_lemma,
filter_nil_lemma,
list_ind_nil_lemma,
false_wf,
lelt_wf,
nil_wf,
equal-wf-base,
length_of_cons_lemma,
filter_cons_lemma,
eqtt_to_assert,
list_ind_cons_lemma,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
lt_int_wf,
assert_of_lt_int,
less_than_wf,
subtract_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
add-member-int_seg2,
decidable__lt,
add-is-int-iff,
itermAdd_wf,
int_term_value_add_lemma,
squash_wf,
true_wf,
cons_wf,
iff_weakening_equal,
add-subtract-cancel,
int_seg_properties,
ifthenelse_wf,
non_neg_length,
equal-wf-base-T
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
hypothesis,
natural_numberEquality,
addEquality,
cumulativity,
because_Cache,
applyEquality,
setEquality,
independent_isectElimination,
setElimination,
rename,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_pairFormation,
dependent_set_memberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
functionExtensionality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
promote_hyp,
instantiate,
functionEquality,
universeEquality,
int_eqEquality,
intEquality,
computeAll,
pointwiseFunctionality,
imageElimination,
baseApply,
closedConclusion,
equalityUniverse,
levelHypothesis
Latex:
\mforall{}[T:Type]
\mforall{}P:T {}\mrightarrow{} \mBbbB{}. \mforall{}L:T List. \mforall{}n:\mBbbN{}. \mexists{}m:\mBbbN{}||L|| + 1. (firstn(n;filter(P;L)) = filter(P;firstn(m;L)))
Date html generated:
2017_04_14-AM-09_24_34
Last ObjectModification:
2017_02_27-PM-03_59_21
Theory : list_1
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