Nuprl Lemma : select-map2
∀[T:Type]
∀[A,B:Type]. ∀[f:A ⟶ B ⟶ T]. ∀[as:A List]. ∀[bs:B List].
∀[i:ℕ||as||]. (map2(f;as;bs)[i] = (f as[i] bs[i]) ∈ T) supposing ||as|| = ||bs|| ∈ ℤ
supposing value-type(T)
Proof
Definitions occuring in Statement :
map2: map2(f;as;bs),
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
value-type: value-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
prop: ℙ,
int_seg: {i..j-},
sq_stable: SqStable(P),
implies: P ⇒ Q,
lelt: i ≤ j < k,
and: P ∧ Q,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
all: ∀x:A. B[x],
so_apply: x[s],
map2: map2(f;as;bs),
nil: [],
it: ⋅,
exists: ∃x:A. B[x],
false: False,
select: L[n],
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
cons: [a / b],
subtract: n - m,
le: A ≤ B,
uiff: uiff(P;Q),
sq_type: SQType(T),
ge: i ≥ j ,
nat: ℕ,
has-value: (a)↓,
bool: 𝔹,
unit: Unit,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
or: P ∨ Q,
bnot: ¬bb,
assert: ↑b,
not: ¬A,
decidable: Dec(P),
less_than': less_than'(a;b)
Lemmas referenced :
list_induction,
uall_wf,
list_wf,
isect_wf,
equal_wf,
length_wf,
int_seg_wf,
select_wf,
map2_wf,
sq_stable__le,
less_than_wf,
squash_wf,
true_wf,
length-map2,
iff_weakening_equal,
less_than_transitivity1,
le_weakening,
equal-wf-base-T,
nil_wf,
length-nil,
length_of_nil_lemma,
non_neg_length,
length_wf_nat,
nat_wf,
subtype_rel-equal,
base_wf,
less_than_irreflexivity,
stuck-spread,
equal-wf-base,
length_of_cons_lemma,
cons_wf,
spread_cons_lemma,
equal-wf-T-base,
value-type_wf,
subtract_wf,
minus-one-mul,
zero-add,
add-mul-special,
zero-mul,
trivial-cancel,
subtype_base_sq,
int_subtype_base,
int_seg_properties,
nat_properties,
value-type-has-value,
list-value-type,
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
le_wf,
decidable__equal_int,
false_wf,
not-equal-2,
le_antisymmetry_iff,
condition-implies-le,
add-associates,
minus-add,
add-swap,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
le-add-cancel2,
decidable__le,
not-le-2,
minus-zero,
add-zero,
minus-minus,
le-add-cancel,
decidable__lt,
not-lt-2,
less-iff-le,
lelt_wf,
select-cons
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
cumulativity,
hypothesis,
intEquality,
because_Cache,
natural_numberEquality,
independent_isectElimination,
functionExtensionality,
applyEquality,
setElimination,
rename,
independent_functionElimination,
productElimination,
imageMemberEquality,
baseClosed,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
dependent_functionElimination,
lambdaFormation,
dependent_pairFormation,
sqequalIntensionalEquality,
voidElimination,
promote_hyp,
voidEquality,
isect_memberEquality,
axiomEquality,
addEquality,
functionEquality,
multiplyEquality,
minusEquality,
instantiate,
callbyvalueReduce,
unionElimination,
equalityElimination,
independent_pairFormation,
dependent_set_memberEquality
Latex:
\mforall{}[T:Type]
\mforall{}[A,B:Type]. \mforall{}[f:A {}\mrightarrow{} B {}\mrightarrow{} T]. \mforall{}[as:A List]. \mforall{}[bs:B List].
\mforall{}[i:\mBbbN{}||as||]. (map2(f;as;bs)[i] = (f as[i] bs[i])) supposing ||as|| = ||bs||
supposing value-type(T)
Date html generated:
2017_04_14-AM-08_49_00
Last ObjectModification:
2017_02_27-PM-03_36_13
Theory : list_0
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