Nuprl Lemma : select_firstn
∀[T:Type]. ∀[as:T List]. ∀[n:{0...||as||}]. ∀[i:ℕn]. (firstn(n;as)[i] = as[i] ∈ T)
Proof
Definitions occuring in Statement :
firstn: firstn(n;as),
select: L[n],
length: ||as||,
list: T List,
int_iseg: {i...j},
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
true: True,
less_than': less_than'(a;b),
le: A ≤ B,
top: Top,
subtract: n - m,
squash: ↓T,
lelt: i ≤ j < k,
sq_stable: SqStable(P),
uiff: uiff(P;Q),
false: False,
rev_implies: P ⇐ Q,
not: ¬A,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
int_seg: {i..j-},
cand: A c∧ B,
implies: P ⇒ Q,
all: ∀x:A. B[x],
uimplies: b supposing a,
so_apply: x[s],
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
nat: ℕ,
int_iseg: {i...j},
subtype_rel: A ⊆r B,
member: t ∈ T,
guard: {T},
less_than: a < b,
firstn: firstn(n;as),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
select: L[n],
nil: [],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
cons: [a / b]
Lemmas referenced :
list_wf,
int_iseg_wf,
int_seg_wf,
le-add-cancel,
add-associates,
add_functionality_wrt_le,
add-commutes,
minus-one-mul-top,
add-swap,
minus-one-mul,
minus-add,
condition-implies-le,
sq_stable__le,
not-le-2,
false_wf,
decidable__le,
length_wf,
le_wf,
subtype_rel_sets,
nat_wf,
subtract_wf,
istype-universe,
select_wf,
firstn_wf,
less_than_wf,
squash_wf,
true_wf,
istype-int,
length_firstn_eq,
istype-false,
less-iff-le,
zero-add,
istype-void,
minus-minus,
add-zero,
subtype_rel_self,
iff_weakening_equal,
less_than_transitivity1,
primrec-wf2,
all_wf,
equal_wf,
lt_int_wf,
equal-wf-base,
bool_wf,
assert_wf,
le_int_wf,
bnot_wf,
less_than_irreflexivity,
int_subtype_base,
uiff_transitivity,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
list-cases,
length_of_nil_lemma,
list_ind_nil_lemma,
stuck-spread,
istype-base,
product_subtype_list,
length_of_cons_lemma,
list_ind_cons_lemma,
decidable__equal_int,
le_weakening,
select_cons_hd,
decidable__lt,
not-lt-2,
not-equal-2,
minus-zero,
le-add-cancel2,
select_cons_tl,
le-add-cancel-alt
Rules used in proof :
axiomEquality,
isect_memberFormation,
universeEquality,
minusEquality,
voidEquality,
isect_memberEquality,
addEquality,
imageElimination,
baseClosed,
imageMemberEquality,
independent_functionElimination,
voidElimination,
independent_pairFormation,
unionElimination,
dependent_functionElimination,
productElimination,
lambdaFormation,
setEquality,
rename,
setElimination,
independent_isectElimination,
cumulativity,
hypothesis,
natural_numberEquality,
productEquality,
lambdaEquality,
because_Cache,
intEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
sqequalRule,
applyEquality,
hypothesisEquality,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
cut,
Error :lambdaFormation_alt,
Error :universeIsType,
Error :functionIsType,
Error :equalityIsType1,
Error :lambdaEquality_alt,
equalityTransitivity,
equalitySymmetry,
Error :inhabitedIsType,
Error :dependent_set_memberEquality_alt,
Error :isect_memberEquality_alt,
Error :productIsType,
instantiate,
Error :setIsType,
functionEquality,
baseApply,
closedConclusion,
equalityElimination,
promote_hyp,
hypothesis_subsumption
Latex:
\mforall{}[T:Type]. \mforall{}[as:T List]. \mforall{}[n:\{0...||as||\}]. \mforall{}[i:\mBbbN{}n]. (firstn(n;as)[i] = as[i])
Date html generated:
2019_06_20-PM-00_43_28
Last ObjectModification:
2018_10_08-PM-00_19_06
Theory : list_0
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