Nuprl Lemma : sum-nat

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

Proof

Definitions occuring in Statement :  sum: `Σ(f[x] | x < k)` 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` so_lambda: `λ2x.t[x]` so_apply: `x[s]` subtype_rel: `A ⊆r B` uimplies: `b supposing a` all: `∀x:A. B[x]` guard: `{T}` int_seg: `{i..j-}` ge: `i ≥ j ` lelt: `i ≤ j < k` and: `P ∧ Q` le: `A ≤ B` prop: `ℙ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` implies: `P `` Q` not: `¬A` top: `Top`
Lemmas referenced :  int_formula_prop_wf int_formula_prop_not_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_and_lemma intformnot_wf itermVar_wf itermConstant_wf intformle_wf intformand_wf satisfiable-full-omega-tt le_wf lelt_wf decidable__le nat_properties int_seg_properties non_neg_sum sum_wf nat_wf int_seg_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut dependent_set_memberEquality sqequalHypSubstitution hypothesis sqequalRule axiomEquality equalityTransitivity equalitySymmetry functionEquality lemma_by_obid isectElimination thin natural_numberEquality setElimination rename hypothesisEquality isect_memberEquality because_Cache lambdaEquality applyEquality independent_isectElimination lambdaFormation productElimination dependent_functionElimination independent_pairFormation unionElimination setEquality intEquality dependent_pairFormation int_eqEquality voidElimination voidEquality computeAll

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

Date html generated: 2016_05_14-AM-07_31_57
Last ObjectModification: 2016_01_14-PM-09_56_36

Theory : int_2

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