Nuprl Lemma : append-tuple-split-tuple
∀[L:Type List]. ∀[x:tuple-type(L)]. ∀[n:ℕ||L||].
  (append-tuple(n;||L|| - n;fst(split-tuple(x;n));snd(split-tuple(x;n))) ~ x)
Proof
Definitions occuring in Statement : 
append-tuple: append-tuple(n;m;x;y)
, 
split-tuple: split-tuple(x;n)
, 
tuple-type: tuple-type(L)
, 
length: ||as||
, 
list: T List
, 
int_seg: {i..j-}
, 
uall: ∀[x:A]. B[x]
, 
pi1: fst(t)
, 
pi2: snd(t)
, 
subtract: n - m
, 
natural_number: $n
, 
universe: Type
, 
sqequal: s ~ t
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
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
exists: ∃x:A. B[x]
, 
not: ¬A
, 
top: Top
, 
and: P ∧ Q
, 
prop: ℙ
, 
subtype_rel: A ⊆r B
, 
guard: {T}
, 
or: P ∨ Q
, 
int_seg: {i..j-}
, 
lelt: i ≤ j < k
, 
cons: [a / b]
, 
colength: colength(L)
, 
so_lambda: λ2x y.t[x; y]
, 
so_apply: x[s1;s2]
, 
decidable: Dec(P)
, 
nil: []
, 
it: ⋅
, 
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)
, 
append-tuple: append-tuple(n;m;x;y)
, 
bool: 𝔹
, 
unit: Unit
, 
btrue: tt
, 
uiff: uiff(P;Q)
, 
ifthenelse: if b then t else f fi 
, 
bfalse: ff
, 
bnot: ¬bb
, 
assert: ↑b
, 
split-tuple: split-tuple(x;n)
, 
pi2: snd(t)
, 
nequal: a ≠ b ∈ T 
, 
pi1: fst(t)
Lemmas referenced : 
nat_properties, 
satisfiable-full-omega-tt, 
intformand_wf, 
intformle_wf, 
itermConstant_wf, 
itermVar_wf, 
intformless_wf, 
int_formula_prop_and_lemma, 
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, 
less_than_wf, 
int_seg_wf, 
length_wf, 
tuple-type_wf, 
equal-wf-T-base, 
nat_wf, 
colength_wf_list, 
less_than_transitivity1, 
less_than_irreflexivity, 
list_wf, 
list-cases, 
tupletype_nil_lemma, 
length_of_nil_lemma, 
int_seg_properties, 
unit_wf2, 
product_subtype_list, 
spread_cons_lemma, 
intformeq_wf, 
itermAdd_wf, 
int_formula_prop_eq_lemma, 
int_term_value_add_lemma, 
decidable__le, 
intformnot_wf, 
int_formula_prop_not_lemma, 
le_wf, 
equal_wf, 
subtract_wf, 
itermSubtract_wf, 
int_term_value_subtract_lemma, 
subtype_base_sq, 
set_subtype_base, 
int_subtype_base, 
decidable__equal_int, 
tupletype_cons_lemma, 
length_of_cons_lemma, 
ifthenelse_wf, 
null_wf, 
le_int_wf, 
bool_wf, 
eqtt_to_assert, 
assert_of_le_int, 
eqff_to_assert, 
bool_cases_sqequal, 
bool_subtype_base, 
assert-bnot, 
eq_int_wf, 
assert_of_eq_int, 
neg_assert_of_eq_int, 
nequal-le-implies, 
equal-wf-base-T, 
null_nil_lemma, 
null_cons_lemma, 
add-is-int-iff, 
false_wf, 
decidable__lt, 
lelt_wf, 
cons_wf, 
split-tuple_wf, 
firstn_wf, 
nth_tl_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
thin, 
lambdaFormation, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
hypothesisEquality, 
hypothesis, 
setElimination, 
rename, 
intWeakElimination, 
natural_numberEquality, 
independent_isectElimination, 
dependent_pairFormation, 
lambdaEquality, 
int_eqEquality, 
intEquality, 
dependent_functionElimination, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
sqequalRule, 
independent_pairFormation, 
computeAll, 
independent_functionElimination, 
sqequalAxiom, 
instantiate, 
universeEquality, 
applyEquality, 
because_Cache, 
unionElimination, 
productElimination, 
promote_hyp, 
hypothesis_subsumption, 
equalityTransitivity, 
equalitySymmetry, 
applyLambdaEquality, 
dependent_set_memberEquality, 
addEquality, 
baseClosed, 
cumulativity, 
imageElimination, 
productEquality, 
equalityElimination, 
pointwiseFunctionality, 
baseApply, 
closedConclusion
Latex:
\mforall{}[L:Type  List].  \mforall{}[x:tuple-type(L)].  \mforall{}[n:\mBbbN{}||L||].
    (append-tuple(n;||L||  -  n;fst(split-tuple(x;n));snd(split-tuple(x;n)))  \msim{}  x)
Date html generated:
2017_04_17-AM-09_30_11
Last ObjectModification:
2017_02_27-PM-05_31_16
Theory : tuples
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