### Nuprl Lemma : sum-partial-nat

`∀[n:ℕ]. ∀[f:ℕn ⟶ partial(ℕ)].  (Σ(f[x] | x < n) ∈ partial(ℕ))`

Proof

Definitions occuring in Statement :  sum: `Σ(f[x] | x < k)` partial: `partial(T)` int_seg: `{i..j-}` nat: `ℕ` uall: `∀[x:A]. B[x]` so_apply: `x[s]` member: `t ∈ T` function: `x:A ⟶ B[x]` natural_number: `\$n`
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` so_lambda: `λ2x.t[x]` so_apply: `x[s]` le: `A ≤ B` less_than': `less_than'(a;b)` decidable: `Dec(P)` or: `P ∨ Q` less_than: `a < b` true: `True` squash: `↓T` subtype_rel: `A ⊆r B` guard: `{T}` int_seg: `{i..j-}` lelt: `i ≤ j < k`
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 partial_wf nat_wf sum-unroll nat-partial-nat false_wf le_wf decidable__le subtract_wf intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma top_wf add-wf-partial-nat int_seg_subtype int_seg_properties decidable__lt lelt_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis setElimination rename sqequalRule intWeakElimination lambdaFormation natural_numberEquality independent_isectElimination approximateComputation independent_functionElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality independent_pairFormation axiomEquality equalityTransitivity equalitySymmetry functionEquality dependent_set_memberEquality because_Cache unionElimination lessCases sqequalAxiom imageMemberEquality baseClosed imageElimination productElimination applyEquality applyLambdaEquality

Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[f:\mBbbN{}n  {}\mrightarrow{}  partial(\mBbbN{})].    (\mSigma{}(f[x]  |  x  <  n)  \mmember{}  partial(\mBbbN{}))

Date html generated: 2018_05_21-PM-00_27_55
Last ObjectModification: 2017_11_03-PM-02_26_53

Theory : int_2

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