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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