Nuprl Lemma : listify_select_id
∀[T:Type]. ∀[as:T List]. ((λi:ℕ||as||. as[i])[ℕ||as||] = as ∈ (T List))
Proof
Definitions occuring in Statement :
select: L[n],
length: ||as||,
listify: listify(f;m;n),
list: T List,
tlambda: λx:T. b[x],
int_seg: {i..j-},
uall: ∀[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,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
nat: ℕ,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
uimplies: b supposing a,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
lelt: i ≤ j < k,
and: P ∧ Q,
subtract: n - m,
sq_type: SQType(T),
guard: {T},
le: A ≤ B,
uiff: uiff(P;Q),
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
not: ¬A,
false: False,
decidable: Dec(P),
or: P ∨ Q,
listify: listify(f;m;n),
bool: 𝔹,
unit: Unit,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
ge: i ≥ j ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_stable: SqStable(P),
cand: A c∧ B,
tlambda: λx:T. b[x]
Lemmas referenced :
list_wf,
equal_wf,
length_wf,
nat_wf,
list_induction,
all_wf,
listify_wf,
select_wf,
subtract_wf,
non_neg_length,
length_wf_nat,
set_subtype_base,
le_wf,
int_subtype_base,
int_seg_wf,
length_of_nil_lemma,
stuck-spread,
base_wf,
length_of_cons_lemma,
minus-one-mul,
add-associates,
add-mul-special,
add-swap,
two-mul,
add-commutes,
mul-distributes-right,
zero-mul,
zero-add,
add-zero,
one-mul,
subtype_base_sq,
mul-distributes,
mul-associates,
add-is-int-iff,
add_functionality_wrt_le,
le_reflexive,
minus-one-mul-top,
not-le-2,
minus-zero,
omega-shadow,
less_than_wf,
mul-swap,
mul-commutes,
le-add-cancel-alt,
less-iff-le,
not-lt-2,
minus-add,
le-add-cancel,
int_seg_properties,
nat_properties,
decidable__le,
decidable__lt,
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
nil_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
condition-implies-le,
le_antisymmetry_iff,
assert_wf,
lt_int_wf,
bnot_wf,
uiff_transitivity,
assert_functionality_wrt_uiff,
bnot_of_le_int,
assert_of_lt_int,
subtype_rel-equal,
minus-minus,
cons_wf,
squash_wf,
true_wf,
select-cons-hd,
false_wf,
select_cons_tl,
iff_weakening_equal,
sq_stable__le
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
universeEquality,
lambdaFormation,
intEquality,
addEquality,
setElimination,
rename,
lambdaEquality,
functionEquality,
independent_isectElimination,
dependent_pairFormation,
sqequalIntensionalEquality,
applyEquality,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
productElimination,
promote_hyp,
baseClosed,
voidElimination,
voidEquality,
multiplyEquality,
minusEquality,
instantiate,
baseApply,
closedConclusion,
dependent_set_memberEquality,
independent_pairFormation,
imageMemberEquality,
unionElimination,
equalityElimination,
applyLambdaEquality,
imageElimination,
hyp_replacement,
functionExtensionality,
productEquality
Latex:
\mforall{}[T:Type]. \mforall{}[as:T List]. ((\mlambda{}i:\mBbbN{}||as||. as[i])[\mBbbN{}||as||] = as)
Date html generated:
2017_04_14-AM-08_39_02
Last ObjectModification:
2017_02_27-PM-03_30_40
Theory : list_0
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