Nuprl Lemma : select_concat_sum
∀[T:Type]. ∀[ll:T List List]. ∀[i:ℕ||ll||]. ∀[j:ℕ||ll[i]||].  (ll[i][j] = concat(ll)[Σ(||ll[k]|| | k < i) + j] ∈ T)
Proof
Definitions occuring in Statement : 
sum: Σ(f[x] | x < k)
, 
select: L[n]
, 
length: ||as||
, 
concat: concat(ll)
, 
list: T List
, 
int_seg: {i..j-}
, 
uall: ∀[x:A]. B[x]
, 
add: n + m
, 
natural_number: $n
, 
universe: Type
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
member: t ∈ T
, 
uall: ∀[x:A]. B[x]
, 
int_seg: {i..j-}
, 
uimplies: b supposing a
, 
guard: {T}
, 
lelt: i ≤ j < k
, 
and: P ∧ Q
, 
all: ∀x:A. B[x]
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
exists: ∃x:A. B[x]
, 
false: False
, 
implies: P 
⇒ Q
, 
not: ¬A
, 
top: Top
, 
prop: ℙ
, 
less_than: a < b
, 
squash: ↓T
, 
subtype_rel: A ⊆r B
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
uiff: uiff(P;Q)
, 
le: A ≤ B
, 
less_than': less_than'(a;b)
, 
nat: ℕ
, 
ge: i ≥ j 
, 
true: True
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
sq_type: SQType(T)
, 
cand: A c∧ B
, 
int_iseg: {i...j}
, 
gt: i > j
Lemmas referenced : 
int_seg_wf, 
length_wf, 
select_wf, 
list_wf, 
int_seg_properties, 
decidable__le, 
satisfiable-full-omega-tt, 
intformand_wf, 
intformnot_wf, 
intformle_wf, 
itermConstant_wf, 
itermVar_wf, 
int_formula_prop_and_lemma, 
int_formula_prop_not_lemma, 
int_formula_prop_le_lemma, 
int_term_value_constant_lemma, 
int_term_value_var_lemma, 
int_formula_prop_wf, 
decidable__lt, 
intformless_wf, 
int_formula_prop_less_lemma, 
select_concat, 
add-member-int_seg1, 
sum_wf, 
int_seg_subtype_nat, 
concat_wf, 
lelt_wf, 
subtract_wf, 
false_wf, 
sum_lower_bound, 
le_wf, 
non_neg_length, 
itermMultiply_wf, 
int_term_value_mul_lemma, 
itermSubtract_wf, 
int_term_value_subtract_lemma, 
sum_split, 
length_wf_nat, 
itermAdd_wf, 
int_term_value_add_lemma, 
less_than_wf, 
squash_wf, 
true_wf, 
length_concat, 
iff_weakening_equal, 
subtype_base_sq, 
int_subtype_base, 
sum1, 
equal_wf, 
zero-add, 
add-is-int-iff, 
firstn_wf, 
length_firstn, 
subtype_rel_sets, 
sum_functionality, 
length_firstn_eq, 
select_firstn, 
decidable__or, 
equal-wf-base, 
or_wf, 
decidable__equal_int, 
intformor_wf, 
intformeq_wf, 
int_formula_prop_or_lemma, 
int_formula_prop_eq_lemma, 
sum_split+, 
subtract-is-int-iff, 
zero-le-nat
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
cut, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
natural_numberEquality, 
cumulativity, 
hypothesisEquality, 
because_Cache, 
hypothesis, 
setElimination, 
rename, 
independent_isectElimination, 
productElimination, 
dependent_functionElimination, 
unionElimination, 
dependent_pairFormation, 
lambdaEquality, 
int_eqEquality, 
intEquality, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
sqequalRule, 
independent_pairFormation, 
computeAll, 
imageElimination, 
universeEquality, 
isect_memberFormation, 
axiomEquality, 
applyEquality, 
dependent_set_memberEquality, 
lambdaFormation, 
addEquality, 
equalityTransitivity, 
equalitySymmetry, 
imageMemberEquality, 
baseClosed, 
independent_functionElimination, 
instantiate, 
pointwiseFunctionality, 
promote_hyp, 
baseApply, 
closedConclusion, 
hyp_replacement, 
applyLambdaEquality, 
productEquality, 
setEquality
Latex:
\mforall{}[T:Type].  \mforall{}[ll:T  List  List].  \mforall{}[i:\mBbbN{}||ll||].  \mforall{}[j:\mBbbN{}||ll[i]||].
    (ll[i][j]  =  concat(ll)[\mSigma{}(||ll[k]||  |  k  <  i)  +  j])
Date html generated:
2017_04_17-AM-08_50_50
Last ObjectModification:
2017_02_27-PM-05_12_23
Theory : list_1
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