Nuprl Lemma : split-tuple_wf
∀[L:Type List]. ∀[x:tuple-type(L)]. ∀[n:ℕ||L||]. (split-tuple(x;n) ∈ tuple-type(firstn(n;L)) × tuple-type(nth_tl(n;L)))
Proof
Definitions occuring in Statement :
split-tuple: split-tuple(x;n),
tuple-type: tuple-type(L),
firstn: firstn(n;as),
length: ||as||,
nth_tl: nth_tl(n;as),
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
member: t ∈ T,
product: x:A × B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
or: P ∨ Q,
firstn: firstn(n;as),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
int_seg: {i..j-},
lelt: i ≤ j < k,
cons: [a / b],
decidable: Dec(P),
colength: colength(L),
nil: [],
it: ⋅,
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
split-tuple: split-tuple(x;n),
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
nth_tl: nth_tl(n;as),
le_int: i ≤z j,
lt_int: i <z j,
bnot: ¬bb,
bfalse: ff,
assert: ↑b,
nequal: a ≠ b ∈ T ,
pi1: fst(t),
pi2: snd(t),
null: null(as),
list_ind: list_ind,
tuple-type: tuple-type(L),
subtract: n - m,
tl: tl(l),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
int_iseg: {i...j},
cand: A c∧ B
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
list-cases,
length_of_nil_lemma,
tupletype_nil_lemma,
list_ind_nil_lemma,
nth_tl_nil,
int_seg_properties,
int_seg_wf,
length_wf,
nil_wf,
tuple-type_wf,
product_subtype_list,
colength-cons-not-zero,
istype-nat,
colength_wf_list,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
istype-le,
list_wf,
subtract-1-ge-0,
subtype_base_sq,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
itermSubtract_wf,
itermAdd_wf,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
le_wf,
length_of_cons_lemma,
tupletype_cons_lemma,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
first0,
cons_wf,
subtype_rel_list,
top_wf,
it_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
null_cons_lemma,
subtype_rel_self,
decidable__lt,
add-is-int-iff,
false_wf,
reduce_tl_cons_lemma,
le_int_wf,
assert_of_le_int,
iff_weakening_uiff,
assert_wf,
list_ind_cons_lemma,
lt_int_wf,
assert_of_lt_int,
null_wf,
firstn_wf,
assert_of_null,
length_firstn,
equal-wf-T-base,
less_than_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
thin,
Error :lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
Error :universeIsType,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
Error :functionIsTypeImplies,
universeEquality,
instantiate,
unionElimination,
productElimination,
voidEquality,
promote_hyp,
hypothesis_subsumption,
Error :equalityIstype,
Error :dependent_set_memberEquality_alt,
because_Cache,
applyLambdaEquality,
imageElimination,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
intEquality,
sqequalBase,
equalityElimination,
cumulativity,
independent_pairEquality,
addEquality,
isect_memberEquality,
pointwiseFunctionality,
Error :productIsType
Latex:
\mforall{}[L:Type List]. \mforall{}[x:tuple-type(L)]. \mforall{}[n:\mBbbN{}||L||].
(split-tuple(x;n) \mmember{} tuple-type(firstn(n;L)) \mtimes{} tuple-type(nth\_tl(n;L)))
Date html generated:
2019_06_20-PM-02_03_31
Last ObjectModification:
2018_12_30-PM-10_15_01
Theory : tuples
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