### Nuprl Lemma : sum-nat-le

[n:ℕ]. ∀[f:ℕn ⟶ ℕ]. ∀[b:ℤ].  {∀x:ℕn. (f[x] ≤ b)} supposing Σ(f[x] x < n) ≤ b

Proof

Definitions occuring in Statement :  sum: Σ(f[x] x < k) int_seg: {i..j-} nat: uimplies: supposing a uall: [x:A]. B[x] guard: {T} so_apply: x[s] le: A ≤ B all: x:A. B[x] function: x:A ⟶ B[x] natural_number: \$n int:
Definitions unfolded in proof :  guard: {T} uall: [x:A]. B[x] member: t ∈ T uimplies: supposing a all: x:A. B[x] nat: le: A ≤ B and: P ∧ Q not: ¬A implies:  Q false: False so_apply: x[s] subtype_rel: A ⊆B prop: so_lambda: λ2x.t[x] int_seg: {i..j-} bool: 𝔹 unit: Unit it: btrue: tt ifthenelse: if then else fi  uiff: uiff(P;Q) less_than': less_than'(a;b) bfalse: ff exists: x:A. B[x] or: P ∨ Q sq_type: SQType(T) bnot: ¬bb assert: b ge: i ≥  lelt: i ≤ j < k decidable: Dec(P) satisfiable_int_formula: satisfiable_int_formula(fmla) top: Top
Lemmas referenced :  int_seg_wf less_than'_wf le_wf sum_wf nat_wf isolate_summand sum-nat eq_int_wf bool_wf eqtt_to_assert assert_of_eq_int false_wf eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int nat_properties int_seg_properties decidable__le lelt_wf add-is-int-iff satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermVar_wf itermConstant_wf intformeq_wf itermAdd_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_eq_lemma int_term_value_add_lemma int_formula_prop_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation introduction cut lambdaFormation extract_by_obid sqequalHypSubstitution isectElimination thin natural_numberEquality setElimination rename hypothesisEquality hypothesis lambdaEquality dependent_functionElimination productElimination independent_pairEquality because_Cache applyEquality functionExtensionality axiomEquality equalityTransitivity equalitySymmetry isect_memberEquality intEquality functionEquality voidElimination unionElimination equalityElimination independent_isectElimination dependent_set_memberEquality independent_pairFormation dependent_pairFormation promote_hyp instantiate cumulativity independent_functionElimination applyLambdaEquality pointwiseFunctionality baseApply closedConclusion baseClosed int_eqEquality voidEquality computeAll

Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[f:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}].  \mforall{}[b:\mBbbZ{}].    \{\mforall{}x:\mBbbN{}n.  (f[x]  \mleq{}  b)\}  supposing  \mSigma{}(f[x]  |  x  <  n)  \mleq{}  b

Date html generated: 2017_04_14-AM-09_20_58
Last ObjectModification: 2017_02_27-PM-03_56_40

Theory : int_2

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