### Nuprl Lemma : sum-as-primrec

`∀[k:ℕ]. ∀[f:ℕk ⟶ ℤ].  (Σ(f[x] | x < k) ~ primrec(k;0;λj,x. (x + f[j])))`

Proof

Definitions occuring in Statement :  sum: `Σ(f[x] | x < k)` primrec: `primrec(n;b;c)` int_seg: `{i..j-}` nat: `ℕ` uall: `∀[x:A]. B[x]` so_apply: `x[s]` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n` int: `ℤ` sqequal: `s ~ t`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` sum: `Σ(f[x] | x < k)` sq_type: `SQType(T)` all: `∀x:A. B[x]` implies: `P `` Q` guard: `{T}` nat: `ℕ` so_apply: `x[s]` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` and: `P ∧ Q` prop: `ℙ` squash: `↓T` int_seg: `{i..j-}` subtype_rel: `A ⊆r B` le: `A ≤ B` less_than': `less_than'(a;b)` lelt: `i ≤ j < k` true: `True` so_lambda: `λ2x.t[x]`
Lemmas referenced :  sum_aux-as-primrec lelt_wf int_seg_properties false_wf int_seg_subtype zero-add le_wf int_term_value_subtract_lemma int_formula_prop_eq_lemma itermSubtract_wf intformeq_wf subtract_wf decidable__equal_int true_wf squash_wf primrec_wf int_formula_prop_wf int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermVar_wf itermConstant_wf intformle_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__le nat_properties nat_wf int_seg_wf int_subtype_base subtype_base_sq
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut thin instantiate lemma_by_obid sqequalHypSubstitution isectElimination because_Cache independent_isectElimination hypothesis dependent_functionElimination equalityTransitivity equalitySymmetry independent_functionElimination sqequalAxiom functionEquality natural_numberEquality setElimination rename hypothesisEquality intEquality sqequalRule isect_memberEquality lambdaEquality applyEquality unionElimination dependent_pairFormation int_eqEquality voidElimination voidEquality independent_pairFormation computeAll imageElimination universeEquality dependent_set_memberEquality functionExtensionality addEquality lambdaFormation setEquality productElimination imageMemberEquality baseClosed

Latex:
\mforall{}[k:\mBbbN{}].  \mforall{}[f:\mBbbN{}k  {}\mrightarrow{}  \mBbbZ{}].    (\mSigma{}(f[x]  |  x  <  k)  \msim{}  primrec(k;0;\mlambda{}j,x.  (x  +  f[j])))

Date html generated: 2016_05_14-AM-07_31_20
Last ObjectModification: 2016_01_14-PM-09_56_40

Theory : int_2

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