Nuprl Lemma : length_concat
∀[T:Type]. ∀[ll:T List List]. (||concat(ll)|| = Σ(||ll[i]|| | i < ||ll||) ∈ ℤ)
Proof
Definitions occuring in Statement :
sum: Σ(f[x] | x < k),
select: L[n],
length: ||as||,
concat: concat(ll),
list: T List,
uall: ∀[x:A]. B[x],
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[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,
so_apply: x[s],
concat: concat(ll),
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
sum: Σ(f[x] | x < k),
sum_aux: sum_aux(k;v;i;x.f[x]),
subtype_rel: A ⊆r B,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
ge: i ≥ j ,
uiff: uiff(P;Q),
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_type: SQType(T)
Lemmas referenced :
list_induction,
list_wf,
equal_wf,
length_wf,
concat_wf,
sum_wf,
length_wf_nat,
select_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,
int_seg_wf,
reduce_nil_lemma,
length_of_nil_lemma,
stuck-spread,
base_wf,
length_of_cons_lemma,
concat-cons,
squash_wf,
true_wf,
length_append,
subtype_rel_list,
top_wf,
add_nat_wf,
false_wf,
le_wf,
nat_wf,
nat_properties,
add-is-int-iff,
itermAdd_wf,
intformeq_wf,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
cons_wf,
non_neg_length,
iff_weakening_equal,
subtype_base_sq,
int_subtype_base,
sum_split,
lelt_wf,
sum1,
select-cons-hd,
decidable__equal_int,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
select-cons-tl,
add-subtract-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
intEquality,
because_Cache,
setElimination,
rename,
independent_isectElimination,
natural_numberEquality,
productElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
independent_functionElimination,
baseClosed,
lambdaFormation,
applyEquality,
equalityTransitivity,
equalitySymmetry,
equalityUniverse,
levelHypothesis,
dependent_set_memberEquality,
addEquality,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
imageMemberEquality,
universeEquality,
instantiate,
functionEquality,
axiomEquality
Latex:
\mforall{}[T:Type]. \mforall{}[ll:T List List]. (||concat(ll)|| = \mSigma{}(||ll[i]|| | i < ||ll||))
Date html generated:
2017_04_17-AM-08_50_25
Last ObjectModification:
2017_02_27-PM-05_07_32
Theory : list_1
Home
Index