Nuprl Lemma : sum-nat
∀[n:ℕ]. ∀[f:ℕn ⟶ ℕ]. (Σ(f[x] | x < n) ∈ ℕ)
Proof
Definitions occuring in Statement :
sum: Σ(f[x] | x < k),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
all: ∀x:A. B[x],
guard: {T},
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
prop: ℙ,
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
Lemmas referenced :
int_formula_prop_wf,
int_formula_prop_not_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
intformnot_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
satisfiable-full-omega-tt,
le_wf,
lelt_wf,
decidable__le,
nat_properties,
int_seg_properties,
non_neg_sum,
sum_wf,
nat_wf,
int_seg_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
dependent_set_memberEquality,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
lemma_by_obid,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
isect_memberEquality,
because_Cache,
lambdaEquality,
applyEquality,
independent_isectElimination,
lambdaFormation,
productElimination,
dependent_functionElimination,
independent_pairFormation,
unionElimination,
setEquality,
intEquality,
dependent_pairFormation,
int_eqEquality,
voidElimination,
voidEquality,
computeAll
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[f:\mBbbN{}n {}\mrightarrow{} \mBbbN{}]. (\mSigma{}(f[x] | x < n) \mmember{} \mBbbN{})
Date html generated:
2016_05_14-AM-07_31_57
Last ObjectModification:
2016_01_14-PM-09_56_36
Theory : int_2
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