Nuprl Lemma : map_equal3
∀[T,T':Type]. ∀[a:T List+]. ∀[f,g:T ⟶ T'].
map(f;a) = map(g;a) ∈ T' List+ supposing ∀x:T. ((x ∈ a) ⇒ ((f x) = (g x) ∈ T'))
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
listp: A List+,
map: map(f;as),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
listp: A List+,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
top: Top,
or: P ∨ Q,
ge: i ≥ j ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
true: True,
not: ¬A,
false: False,
cons: [a / b],
guard: {T},
nat: ℕ,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
subtype_rel: A ⊆r B
Lemmas referenced :
l_member_wf,
equal_wf,
all_wf,
listp_wf,
map-length,
listp_properties,
list-cases,
length_of_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
length_wf_nat,
nat_wf,
decidable__lt,
false_wf,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
less_than_wf,
length_wf,
map_equal2
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
functionIsType,
universeIsType,
hypothesisEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesis,
applyEquality,
lambdaEquality,
functionEquality,
inhabitedIsType,
because_Cache,
universeEquality,
isect_memberFormation_alt,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
voidElimination,
voidEquality,
dependent_functionElimination,
unionElimination,
productElimination,
independent_functionElimination,
natural_numberEquality,
promote_hyp,
hypothesis_subsumption,
lambdaFormation,
addEquality,
independent_pairFormation,
independent_isectElimination,
intEquality,
minusEquality
Latex:
\mforall{}[T,T':Type]. \mforall{}[a:T List\msupplus{}]. \mforall{}[f,g:T {}\mrightarrow{} T'].
map(f;a) = map(g;a) supposing \mforall{}x:T. ((x \mmember{} a) {}\mRightarrow{} ((f x) = (g x)))
Date html generated:
2019_10_15-AM-10_53_23
Last ObjectModification:
2018_09_27-AM-10_02_48
Theory : list!
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