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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