Nuprl Lemma : product-equipollent-tuple3
∀L:Type List. ∀[A:Type]. tuple-type(L) × A ~ tuple-type(L @ [A])
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
tuple-type: tuple-type(L),
append: as @ bs,
cons: [a / b],
nil: [],
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],
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
ifthenelse: if b then t else f fi ,
btrue: tt,
or: P ∨ Q,
cons: [a / b],
uimplies: b supposing a,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
and: P ∧ Q,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
nat: ℕ
Lemmas referenced :
list_induction,
uall_wf,
equipollent_wf,
tuple-type_wf,
append_wf,
cons_wf,
nil_wf,
tupletype_nil_lemma,
list_ind_nil_lemma,
istype-void,
tupletype_cons_lemma,
null_nil_lemma,
list_ind_cons_lemma,
list_wf,
equipollent-identity,
unit_wf2,
equipollent_same,
list-cases,
null_cons_lemma,
product_subtype_list,
null_wf3,
subtype_rel_list,
top_wf,
eqtt_to_assert,
assert_of_null,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
equal-wf-T-base,
length_wf_nat,
nat_wf,
set_subtype_base,
le_wf,
istype-int,
int_subtype_base,
equipollent_weakening_ext-eq,
ext-eq_weakening,
equipollent_functionality_wrt_equipollent2,
product_functionality_wrt_equipollent_right,
equipollent_inversion,
equipollent-product-assoc
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
universeEquality,
sqequalRule,
lambdaEquality_alt,
productEquality,
hypothesisEquality,
hypothesis,
applyEquality,
cumulativity,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
because_Cache,
universeIsType,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
rename,
isectIsType,
isect_memberFormation_alt,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
independent_isectElimination,
equalityElimination,
dependent_pairFormation_alt,
equalityIsType1,
baseClosed,
equalityIsType3,
dependent_set_memberEquality_alt,
equalityIsType4,
intEquality,
natural_numberEquality,
hyp_replacement,
applyLambdaEquality,
setElimination
Latex:
\mforall{}L:Type List. \mforall{}[A:Type]. tuple-type(L) \mtimes{} A \msim{} tuple-type(L @ [A])
Date html generated:
2019_10_15-AM-11_15_23
Last ObjectModification:
2018_10_09-PM-02_12_44
Theory : general
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