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 :
rev_uimplies: rev_uimplies(P;Q),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
bfalse: ff,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
or: P ∨ Q,
decidable: Dec(P),
lelt: i ≤ j < k,
int_seg: {i..j-},
guard: {T},
less_than': less_than'(a;b),
absval: |i|,
le: A ≤ B,
prop: ℙ,
and: P ∧ Q,
top: Top,
not: ¬A,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
uimplies: b supposing a,
ge: i ≥ j ,
false: False,
implies: P ⇒ Q,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
nat: ℕ,
so_apply: x[s],
so_lambda: λ2x.t[x],
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
le_weakening,
int-triangle-inequality,
le_functionality,
int_term_value_add_lemma,
itermAdd_wf,
lelt_wf,
decidable__lt,
sum_wf,
equal_wf,
assert_of_bnot,
eqff_to_assert,
iff_weakening_uiff,
not_wf,
bnot_wf,
iff_transitivity,
assert_of_eq_int,
eqtt_to_assert,
assert_wf,
int_subtype_base,
equal-wf-base,
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 :
impliesFunctionality,
baseClosed,
closedConclusion,
baseApply,
equalityElimination,
unionElimination,
applyLambdaEquality,
dependent_set_memberEquality,
minusEquality,
functionEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
addEquality,
independent_pairEquality,
productElimination,
independent_functionElimination,
computeAll,
independent_pairFormation,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
intEquality,
int_eqEquality,
dependent_pairFormation,
independent_isectElimination,
intWeakElimination,
lambdaFormation,
hypothesis,
because_Cache,
rename,
setElimination,
natural_numberEquality,
functionExtensionality,
applyEquality,
lambdaEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[f:\mBbbN{}n {}\mrightarrow{} \mBbbZ{}]. (|\mSigma{}(f[x] | x < n)| \mleq{} \mSigma{}(|f[x]| | x < n))
Date html generated:
2017_04_14-AM-09_21_48
Last ObjectModification:
2017_04_13-AM-00_46_41
Theory : int_2
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