Nuprl Lemma : absval_sum
∀[n:ℕ]. ∀[f:ℕn ⟶ ℤ]. (|Σ(f[x] | x < n)| ≤ Σ(|f[x]| | x < n))
Proof
Definitions occuring in Statement :
sum: Σ(f[x] | x < k),
absval: |i|,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
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,
le: A ≤ B,
absval: |i|,
less_than': less_than'(a;b),
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
le_weakening,
int-triangle-inequality,
le_functionality,
int_term_value_add_lemma,
itermAdd_wf,
lelt_wf,
decidable__lt,
sum_wf,
assert_of_bnot,
eqff_to_assert,
iff_weakening_uiff,
not_wf,
bnot_wf,
bfalse_wf,
iff_transitivity,
assert_of_eq_int,
eqtt_to_assert,
assert_wf,
btrue_wf,
equal_wf,
uiff_transitivity,
bool_wf,
eq_int_wf,
primrec-unroll,
subtype_rel_self,
int_seg_subtype,
subtype_rel_dep_function,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformnot_wf,
subtract_wf,
decidable__le,
int_seg_properties,
le_wf,
false_wf,
primrec0_lemma,
nat_wf,
primrec_wf,
less_than'_wf,
less_than_wf,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
satisfiable-full-omega-tt,
nat_properties,
absval_wf,
int_seg_wf,
sum-as-primrec
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality,
applyEquality,
natural_numberEquality,
setElimination,
rename,
hypothesis,
because_Cache,
lambdaFormation,
intWeakElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
productElimination,
independent_pairEquality,
addEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
minusEquality,
dependent_set_memberEquality,
setEquality,
unionElimination,
equalityElimination,
impliesFunctionality,
equalityEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[f:\mBbbN{}n {}\mrightarrow{} \mBbbZ{}]. (|\mSigma{}(f[x] | x < n)| \mleq{} \mSigma{}(|f[x]| | x < n))
Date html generated:
2016_05_14-PM-04_27_53
Last ObjectModification:
2016_01_14-PM-11_34_45
Theory : num_thy_1
Home
Index