### Nuprl Lemma : sum-unroll-1

`∀[n:ℕ]. ∀[f:ℕn ⟶ ℤ].  (Σ(f[x] | x < n) ~ if (n =z 0) then 0 else Σ(f[x] | x < n - 1) + f[n - 1] fi )`

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]` subtract: `n - m` add: `n + m` natural_number: `\$n` int: `ℤ` sqequal: `s ~ t`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` nat: `ℕ` all: `∀x:A. B[x]` implies: `P `` Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` ifthenelse: `if b then t else f fi ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` bfalse: `ff` exists: `∃x:A. B[x]` prop: `ℙ` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` false: `False` le: `A ≤ B` less_than': `less_than'(a;b)` not: `¬A` ge: `i ≥ j ` int_upper: `{i...}` subtype_rel: `A ⊆r B` decidable: `Dec(P)` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` int_seg: `{i..j-}` lelt: `i ≤ j < k` less_than: `a < b`
Lemmas referenced :  eq_int_wf bool_wf eqtt_to_assert assert_of_eq_int sum-as-primrec int_seg_wf eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int int_upper_subtype_nat false_wf le_wf nat_properties nequal-le-implies zero-add subtract_wf int_upper_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermSubtract_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_subtract_lemma int_term_value_var_lemma int_formula_prop_wf decidable__lt intformless_wf int_formula_prop_less_lemma lelt_wf nat_wf int_subtype_base primrec0_lemma primrec-unroll intformeq_wf int_formula_prop_eq_lemma
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin setElimination rename because_Cache hypothesis natural_numberEquality lambdaFormation unionElimination equalityElimination equalityTransitivity equalitySymmetry productElimination independent_isectElimination sqequalRule hypothesisEquality lambdaEquality applyEquality functionExtensionality dependent_pairFormation promote_hyp dependent_functionElimination instantiate cumulativity independent_functionElimination voidElimination hypothesis_subsumption dependent_set_memberEquality independent_pairFormation int_eqEquality intEquality isect_memberEquality voidEquality computeAll sqequalAxiom functionEquality

Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[f:\mBbbN{}n  {}\mrightarrow{}  \mBbbZ{}].
(\mSigma{}(f[x]  |  x  <  n)  \msim{}  if  (n  =\msubz{}  0)  then  0  else  \mSigma{}(f[x]  |  x  <  n  -  1)  +  f[n  -  1]  fi  )

Date html generated: 2017_04_14-AM-09_19_44
Last ObjectModification: 2017_02_27-PM-03_56_00

Theory : int_2

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