### Nuprl Lemma : isolate_summand

`∀[n:ℕ]. ∀[f:ℕn ⟶ ℤ]. ∀[m:ℕn].  (Σ(f[x] | x < n) = (f[m] + Σ(if (x =z m) then 0 else f[x] fi  | x < n)) ∈ ℤ)`

Proof

Definitions occuring in Statement :  sum: `Σ(f[x] | x < k)` int_seg: `{i..j-}` nat: `ℕ` ifthenelse: `if b then t else f fi ` eq_int: `(i =z j)` uall: `∀[x:A]. B[x]` so_apply: `x[s]` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` all: `∀x:A. B[x]` top: `Top` and: `P ∧ Q` prop: `ℙ` guard: `{T}` int_seg: `{i..j-}` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` subtype_rel: `A ⊆r B` le: `A ≤ B` less_than': `less_than'(a;b)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` uiff: `uiff(P;Q)` subtract: `n - m` true: `True` sq_type: `SQType(T)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` less_than: `a < b` squash: `↓T` bfalse: `ff` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` nequal: `a ≠ b ∈ T ` so_lambda: `λ2x.t[x]` so_apply: `x[s]`
Lemmas referenced :  nat_properties full-omega-unsat intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf ge_wf less_than_wf int_seg_wf int_seg_properties decidable__le subtract_wf intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma subtype_rel_function int_seg_subtype false_wf not-le-2 condition-implies-le add-associates minus-add minus-one-mul add-swap minus-one-mul-top add-mul-special zero-mul add-zero add-commutes le-add-cancel2 subtype_rel_self nat_wf decidable__equal_int subtype_base_sq int_subtype_base lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int top_wf eq_int_wf assert_of_eq_int eqff_to_assert equal_wf bool_cases_sqequal bool_subtype_base assert-bnot neg_assert_of_eq_int intformeq_wf int_formula_prop_eq_lemma not_functionality_wrt_uiff assert_wf sum-unroll decidable__lt lelt_wf itermAdd_wf int_term_value_add_lemma sum_functionality le_wf add-is-int-iff
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis setElimination rename intWeakElimination lambdaFormation natural_numberEquality independent_isectElimination approximateComputation independent_functionElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation axiomEquality functionEquality because_Cache productElimination Error :universeIsType,  Error :functionIsType,  Error :inhabitedIsType,  unionElimination applyEquality addEquality minusEquality multiplyEquality instantiate cumulativity equalityTransitivity equalitySymmetry equalityElimination lessCases axiomSqEquality imageMemberEquality baseClosed imageElimination promote_hyp dependent_set_memberEquality functionExtensionality pointwiseFunctionality baseApply closedConclusion

Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[f:\mBbbN{}n  {}\mrightarrow{}  \mBbbZ{}].  \mforall{}[m:\mBbbN{}n].
(\mSigma{}(f[x]  |  x  <  n)  =  (f[m]  +  \mSigma{}(if  (x  =\msubz{}  m)  then  0  else  f[x]  fi    |  x  <  n)))

Date html generated: 2019_06_20-PM-01_18_23
Last ObjectModification: 2018_09_26-PM-02_40_38

Theory : int_2

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