Nuprl Lemma : length-map2
∀[T:Type]
∀[A,B:Type]. ∀[f:A ⟶ B ⟶ T]. ∀[as:A List]. ∀[bs:B List].
||map2(f;as;bs)|| = ||as|| ∈ ℤ supposing ||as|| = ||bs|| ∈ ℤ
supposing value-type(T)
Proof
Definitions occuring in Statement :
map2: map2(f;as;bs),
length: ||as||,
list: T List,
value-type: value-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
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,
so_apply: x[s],
implies: P ⇒ Q,
map2: map2(f;as;bs),
nil: [],
it: ⋅,
all: ∀x:A. B[x],
top: Top,
cons: [a / b],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
has-value: (a)↓,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
guard: {T},
subtract: n - m,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
squash: ↓T
Lemmas referenced :
list_induction,
uall_wf,
list_wf,
isect_wf,
equal_wf,
length_wf,
map2_wf,
equal-wf-base-T,
equal-wf-T-base,
nil_wf,
length_of_nil_lemma,
equal-wf-base,
length_of_cons_lemma,
cons_wf,
spread_cons_lemma,
value-type_wf,
value-type-has-value,
list-value-type,
decidable__equal_int,
false_wf,
not-equal-2,
le_antisymmetry_iff,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
zero-add,
add-commutes,
add_functionality_wrt_le,
le-add-cancel2,
squash_wf,
true_wf,
add_functionality_wrt_eq,
iff_weakening_equal
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,
independent_isectElimination,
functionExtensionality,
applyEquality,
independent_functionElimination,
baseClosed,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
rename,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
addEquality,
natural_numberEquality,
axiomEquality,
functionEquality,
universeEquality,
callbyvalueReduce,
unionElimination,
independent_pairFormation,
productElimination,
minusEquality,
imageElimination,
imageMemberEquality
Latex:
\mforall{}[T:Type]
\mforall{}[A,B:Type]. \mforall{}[f:A {}\mrightarrow{} B {}\mrightarrow{} T]. \mforall{}[as:A List]. \mforall{}[bs:B List].
||map2(f;as;bs)|| = ||as|| supposing ||as|| = ||bs||
supposing value-type(T)
Date html generated:
2017_04_14-AM-08_48_53
Last ObjectModification:
2017_02_27-PM-03_34_52
Theory : list_0
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