### 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: `ℕ` uimplies: `b 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: `ℤ`
Definitions unfolded in proof :  guard: `{T}` uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` all: `∀x:A. B[x]` nat: `ℕ` le: `A ≤ B` and: `P ∧ Q` not: `¬A` implies: `P `` Q` false: `False` so_apply: `x[s]` subtype_rel: `A ⊆r B` prop: `ℙ` so_lambda: `λ2x.t[x]` int_seg: `{i..j-}` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f 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 ≥ j ` 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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