Nuprl Lemma : product-equipollent-tuple2
∀[A:Type]. ∀L:Type List. A × tuple-type(L) ~ tuple-type([A / L])
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
tuple-type: tuple-type(L),
cons: [a / b],
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
top: Top,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
tupletype_cons_lemma,
null_wf3,
subtype_rel_list,
top_wf,
bool_wf,
eqtt_to_assert,
assert_of_null,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
equal-wf-T-base,
list_wf,
tuple-type_wf,
equipollent_weakening_ext-eq,
ext-eq_weakening,
length_wf_nat,
nat_wf,
tupletype_nil_lemma,
equipollent_wf,
unit_wf2,
equipollent-identity,
equipollent_functionality_wrt_equipollent,
equipollent-product-com
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalTransitivity,
computationStep,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
isect_memberFormation,
lambdaFormation,
isectElimination,
because_Cache,
applyEquality,
instantiate,
universeEquality,
cumulativity,
independent_isectElimination,
lambdaEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
hypothesisEquality,
dependent_pairFormation,
promote_hyp,
independent_functionElimination,
baseClosed,
productEquality,
dependent_set_memberEquality,
hyp_replacement,
applyLambdaEquality,
setElimination,
rename
Latex:
\mforall{}[A:Type]. \mforall{}L:Type List. A \mtimes{} tuple-type(L) \msim{} tuple-type([A / L])
Date html generated:
2018_05_21-PM-08_04_07
Last ObjectModification:
2017_07_26-PM-05_40_13
Theory : general
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