Nuprl Lemma : non_neg_sum
∀[n:ℕ]. ∀[f:ℕn ⟶ ℤ]. 0 ≤ Σ(f[x] | x < n) supposing ∀x:ℕn. (0 ≤ f[x])
Proof
Definitions occuring in Statement :
sum: Σ(f[x] | x < k),
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
le: A ≤ B,
less_than': less_than'(a;b),
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
guard: {T}
Lemmas referenced :
sum_wf,
add_nat_wf,
nat_wf,
zero-le-nat,
lelt_wf,
decidable__lt,
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,
false_wf,
primrec0_lemma,
le_wf,
all_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,
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,
lambdaFormation,
intWeakElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
productElimination,
independent_pairEquality,
addEquality,
because_Cache,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
dependent_set_memberEquality,
unionElimination
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[f:\mBbbN{}n {}\mrightarrow{} \mBbbZ{}]. 0 \mleq{} \mSigma{}(f[x] | x < n) supposing \mforall{}x:\mBbbN{}n. (0 \mleq{} f[x])
Date html generated:
2016_05_14-AM-07_31_51
Last ObjectModification:
2016_01_14-PM-09_56_55
Theory : int_2
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