Nuprl Lemma : sum-is-zero
∀[n:ℕ]. ∀[f:ℕn ⟶ ℤ]. Σ(f[x] | x < n) = 0 ∈ ℤ supposing ∀i:ℕn. (f[i] = 0 ∈ ℤ)
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],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
istype-universe,
sum_functionality,
subtype_rel_self,
iff_weakening_equal,
int_seg_wf,
istype-int,
int_subtype_base,
nat_wf,
sum_constant,
nat_properties,
decidable__equal_int,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
applyEquality,
thin,
Error :lambdaEquality_alt,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
Error :universeIsType,
Error :inhabitedIsType,
instantiate,
universeEquality,
intEquality,
sqequalRule,
because_Cache,
closedConclusion,
natural_numberEquality,
independent_isectElimination,
Error :lambdaFormation_alt,
dependent_functionElimination,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
Error :functionIsType,
setElimination,
rename,
Error :equalityIsType4,
Error :isect_memberEquality_alt,
axiomEquality,
Error :isectIsTypeImplies,
unionElimination,
approximateComputation,
Error :dependent_pairFormation_alt,
int_eqEquality,
voidElimination
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[f:\mBbbN{}n {}\mrightarrow{} \mBbbZ{}]. \mSigma{}(f[x] | x < n) = 0 supposing \mforall{}i:\mBbbN{}n. (f[i] = 0)
Date html generated:
2019_06_20-PM-01_18_04
Last ObjectModification:
2018_10_16-PM-04_30_17
Theory : int_2
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