Nuprl Lemma : map_select
∀[A,B:Type]. ∀[f:A ⟶ B]. ∀[as:A List]. ∀[n:ℕ||as||]. (map(f;as)[n] = (f as[n]) ∈ B)
Proof
Definitions occuring in Statement :
select: L[n],
length: ||as||,
map: map(f;as),
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
int_seg: {i..j-},
uimplies: b supposing a,
sq_stable: SqStable(P),
implies: P ⇒ Q,
lelt: i ≤ j < k,
and: P ∧ Q,
squash: ↓T,
top: Top,
so_apply: x[s],
all: ∀x:A. B[x],
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
prop: ℙ,
false: False,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
not: ¬A,
uiff: uiff(P;Q),
le: A ≤ B,
less_than': less_than'(a;b),
subtract: n - m,
nat: ℕ
Lemmas referenced :
int_seg_wf,
length_wf,
list_wf,
list_induction,
all_wf,
equal_wf,
select_wf,
map_wf,
sq_stable__le,
map-length,
length_of_nil_lemma,
map_nil_lemma,
stuck-spread,
base_wf,
length_of_cons_lemma,
map_cons_lemma,
less_than_transitivity1,
less_than_irreflexivity,
decidable__int_equal,
le_weakening,
squash_wf,
true_wf,
select_cons_hd,
iff_weakening_equal,
decidable__lt,
false_wf,
not-lt-2,
not-equal-2,
add_functionality_wrt_le,
zero-add,
add-zero,
le-add-cancel,
condition-implies-le,
add-commutes,
minus-add,
minus-zero,
le_wf,
map_length_nat,
nat_wf,
decidable__le,
not-le-2,
less-iff-le,
add-associates,
minus-one-mul,
add-swap,
minus-one-mul-top,
le-add-cancel2,
subtract_wf,
minus-minus,
lelt_wf,
select_cons_tl
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
cumulativity,
hypothesisEquality,
hypothesis,
because_Cache,
functionEquality,
universeEquality,
isect_memberFormation,
sqequalRule,
isect_memberEquality,
axiomEquality,
lambdaEquality,
functionExtensionality,
applyEquality,
setElimination,
rename,
independent_isectElimination,
independent_functionElimination,
productElimination,
imageMemberEquality,
baseClosed,
imageElimination,
voidElimination,
voidEquality,
dependent_functionElimination,
lambdaFormation,
addEquality,
unionElimination,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
minusEquality,
intEquality,
dependent_set_memberEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[f:A {}\mrightarrow{} B]. \mforall{}[as:A List]. \mforall{}[n:\mBbbN{}||as||]. (map(f;as)[n] = (f as[n]))
Date html generated:
2017_04_14-AM-08_38_49
Last ObjectModification:
2017_02_27-PM-03_29_52
Theory : list_0
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