### Nuprl Lemma : sum_difference

`∀[n:ℕ]. ∀[f,g:ℕn ⟶ ℤ]. ∀[d:ℤ].  Σ(f[x] | x < n) = (Σ(g[x] | x < n) + d) ∈ ℤ supposing Σ(f[x] - g[x] | x < n) = d ∈ 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]` function: `x:A ⟶ B[x]` subtract: `n - m` add: `n + m` 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` sq_type: `SQType(T)` all: `∀x:A. B[x]` implies: `P `` Q` guard: `{T}` so_lambda: `λ2x.t[x]` so_apply: `x[s]` nat: `ℕ` subtype_rel: `A ⊆r B` prop: `ℙ` squash: `↓T` true: `True` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` int_seg: `{i..j-}` ge: `i ≥ j ` lelt: `i ≤ j < k` 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 :  subtype_base_sq int_subtype_base equal-wf-T-base sum_wf subtract_wf int_seg_wf nat_wf equal_wf squash_wf true_wf sum_linear subtype_rel_self iff_weakening_equal sum_functionality int_seg_properties nat_properties decidable__equal_int lelt_wf full-omega-unsat intformnot_wf intformeq_wf itermVar_wf itermAdd_wf itermSubtract_wf int_formula_prop_not_lemma int_formula_prop_eq_lemma int_term_value_var_lemma int_term_value_add_lemma int_term_value_subtract_lemma int_formula_prop_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut thin instantiate extract_by_obid sqequalHypSubstitution isectElimination cumulativity intEquality independent_isectElimination hypothesis dependent_functionElimination equalityTransitivity equalitySymmetry hypothesisEquality independent_functionElimination Error :universeIsType,  sqequalRule lambdaEquality applyEquality natural_numberEquality setElimination rename isect_memberEquality axiomEquality because_Cache Error :inhabitedIsType,  functionEquality Error :functionIsType,  imageElimination universeEquality imageMemberEquality baseClosed productElimination addEquality lambdaFormation unionElimination dependent_set_memberEquality independent_pairFormation approximateComputation dependent_pairFormation int_eqEquality voidElimination voidEquality

Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[f,g:\mBbbN{}n  {}\mrightarrow{}  \mBbbZ{}].  \mforall{}[d:\mBbbZ{}].
\mSigma{}(f[x]  |  x  <  n)  =  (\mSigma{}(g[x]  |  x  <  n)  +  d)  supposing  \mSigma{}(f[x]  -  g[x]  |  x  <  n)  =  d

Date html generated: 2019_06_20-PM-01_18_08
Last ObjectModification: 2018_09_26-PM-02_37_38

Theory : int_2

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