Nuprl Lemma : tuple-type-concat
∀[T:Type]. ∀f:T ⟶ (Type List). ∀L:T List. tuple-type(map(λi.tuple-type(f i);L)) ~ tuple-type(concat(map(f;L)))
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
tuple-type: tuple-type(L),
concat: concat(ll),
map: map(f;as),
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
concat: concat(ll),
top: Top,
implies: P ⇒ Q,
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x.t[x],
member: t ∈ T,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
not: ¬A,
false: False,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
cons: [a / b],
subtype_rel: A ⊆r B,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
sq_type: SQType(T),
guard: {T},
true: True
Lemmas referenced :
tupletype_cons_lemma,
map_cons_lemma,
unit_wf2,
equipollent_same,
reduce_nil_lemma,
tupletype_nil_lemma,
istype-void,
map_nil_lemma,
list_wf,
concat_wf,
istype-universe,
map_wf,
tuple-type_wf,
equipollent_wf,
list_induction,
reduce_cons_lemma,
null_wf,
equal-wf-T-base,
bool_wf,
assert_wf,
bnot_wf,
not_wf,
istype-assert,
length_wf,
length_of_nil_lemma,
length-map,
uiff_transitivity,
eqtt_to_assert,
assert_of_null,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
list-cases,
product_subtype_list,
subtype_rel_list,
top_wf,
append-nil,
equipollent_weakening_ext-eq,
ext-eq_weakening,
list_ind_nil_lemma,
list_ind_cons_lemma,
subtype_base_sq,
int_subtype_base,
append_wf,
tuple-type-append-equipollent,
equipollent_functionality_wrt_equipollent,
product_functionality_wrt_equipollent_right
Rules used in proof :
Error :functionIsType,
Error :inhabitedIsType,
rename,
voidElimination,
Error :isect_memberEquality_alt,
dependent_functionElimination,
independent_functionElimination,
Error :universeIsType,
because_Cache,
hypothesis,
applyEquality,
universeEquality,
cumulativity,
instantiate,
Error :lambdaEquality_alt,
sqequalRule,
hypothesisEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
thin,
cut,
Error :lambdaFormation_alt,
Error :isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
equalityTransitivity,
equalitySymmetry,
baseClosed,
Error :equalityIstype,
sqequalBase,
applyLambdaEquality,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
independent_pairFormation,
promote_hyp,
hypothesis_subsumption,
natural_numberEquality,
intEquality,
productEquality
Latex:
\mforall{}[T:Type]
\mforall{}f:T {}\mrightarrow{} (Type List). \mforall{}L:T List.
tuple-type(map(\mlambda{}i.tuple-type(f i);L)) \msim{} tuple-type(concat(map(f;L)))
Date html generated:
2019_06_20-PM-02_19_27
Last ObjectModification:
2019_01_12-AM-11_44_15
Theory : equipollence!!cardinality!
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