Nuprl Lemma : equipollent-list-as-product
∀[T:Type]. T List ~ k:ℕ × (T^k)
Proof
Definitions occuring in Statement : 
power-type: (T^k)
, 
equipollent: A ~ B
, 
list: T List
, 
nat: ℕ
, 
uall: ∀[x:A]. B[x]
, 
product: x:A × B[x]
, 
universe: Type
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
equipollent: A ~ B
, 
exists: ∃x:A. B[x]
, 
member: t ∈ T
, 
all: ∀x:A. B[x]
, 
prop: ℙ
, 
biject: Bij(A;B;f)
, 
and: P ∧ Q
, 
inject: Inj(A;B;f)
, 
surject: Surj(A;B;f)
, 
implies: P 
⇒ Q
, 
pi2: snd(t)
, 
pi1: fst(t)
, 
top: Top
, 
uimplies: b supposing a
, 
guard: {T}
, 
subtype_rel: A ⊆r B
, 
sq_type: SQType(T)
Lemmas referenced : 
length_wf_nat, 
list-subtype-power-type, 
power-type_wf, 
list_wf, 
biject_wf, 
nat_wf, 
istype-universe, 
istype-nat, 
pi1_wf_top, 
istype-void, 
power-type-subtype-list, 
equal_functionality_wrt_subtype_rel2, 
power-type-length, 
subtype_base_sq, 
int_subtype_base
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation_alt, 
dependent_pairFormation_alt, 
lambdaEquality_alt, 
dependent_pairEquality_alt, 
cut, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
hypothesis, 
dependent_functionElimination, 
universeIsType, 
productEquality, 
instantiate, 
universeEquality, 
independent_pairFormation, 
lambdaFormation_alt, 
sqequalRule, 
productIsType, 
inhabitedIsType, 
applyLambdaEquality, 
productElimination, 
independent_pairEquality, 
isect_memberEquality_alt, 
voidElimination, 
equalityTransitivity, 
equalitySymmetry, 
independent_isectElimination, 
independent_functionElimination, 
equalityIstype, 
because_Cache, 
applyEquality, 
cumulativity, 
intEquality
Latex:
\mforall{}[T:Type].  T  List  \msim{}  k:\mBbbN{}  \mtimes{}  (T\^{}k)
Date html generated:
2019_10_15-AM-11_18_59
Last ObjectModification:
2018_11_30-PM-00_11_24
Theory : general
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