### Nuprl Lemma : bag-sum_wf_nat

`∀[A:Type]. ∀[f:A ⟶ ℕ]. ∀[ba:bag(A)].  (bag-sum(ba;x.f[x]) ∈ ℕ)`

Proof

Definitions occuring in Statement :  bag-sum: `bag-sum(ba;x.f[x])` bag: `bag(T)` nat: `ℕ` uall: `∀[x:A]. B[x]` so_apply: `x[s]` member: `t ∈ T` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` bag: `bag(T)` so_lambda: `λ2x.t[x]` so_apply: `x[s]` subtype_rel: `A ⊆r B` prop: `ℙ` quotient: `x,y:A//B[x; y]` and: `P ∧ Q` all: `∀x:A. B[x]` implies: `P `` Q` squash: `↓T` nat: `ℕ` bag-sum: `bag-sum(ba;x.f[x])` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` top: `Top` guard: `{T}` le: `A ≤ B` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` int_seg: `{i..j-}` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` less_than': `less_than'(a;b)` less_than: `a < b` cons: `[a / b]` assert: `↑b` ifthenelse: `if b then t else f fi ` bfalse: `ff` iff: `P `⇐⇒` Q` uiff: `uiff(P;Q)` rev_implies: `P `` Q` int_iseg: `{i...j}` cand: `A c∧ B`
Lemmas referenced :  squash_wf le_wf bag-sum_wf list_wf permutation_wf equal_wf equal-wf-base bag_wf nat_wf nat_properties satisfiable-full-omega-tt 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 less_than'_wf list_accum_wf length_wf int_seg_wf int_seg_properties decidable__le subtract_wf intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma decidable__equal_int int_seg_subtype false_wf intformeq_wf int_formula_prop_eq_lemma non_neg_length decidable__lt lelt_wf decidable__assert null_wf list-cases list_accum_nil_lemma product_subtype_list null_cons_lemma last-lemma-sq pos_length iff_transitivity not_wf equal-wf-T-base assert_wf bnot_wf assert_of_null iff_weakening_uiff assert_of_bnot firstn_wf length_firstn zero-le-nat append_wf cons_wf last_wf nil_wf add_nat_wf add-is-int-iff itermAdd_wf int_term_value_add_lemma length_wf_nat
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalHypSubstitution pointwiseFunctionalityForEquality extract_by_obid isectElimination thin natural_numberEquality cumulativity hypothesisEquality sqequalRule lambdaEquality applyEquality functionExtensionality hypothesis because_Cache pertypeElimination productElimination equalityTransitivity equalitySymmetry lambdaFormation rename imageMemberEquality baseClosed dependent_functionElimination independent_functionElimination productEquality imageElimination dependent_set_memberEquality axiomEquality isect_memberEquality functionEquality universeEquality setElimination intWeakElimination independent_isectElimination dependent_pairFormation int_eqEquality intEquality voidElimination voidEquality independent_pairFormation computeAll independent_pairEquality addEquality unionElimination applyLambdaEquality hypothesis_subsumption promote_hyp impliesFunctionality pointwiseFunctionality baseApply closedConclusion

Latex:
\mforall{}[A:Type].  \mforall{}[f:A  {}\mrightarrow{}  \mBbbN{}].  \mforall{}[ba:bag(A)].    (bag-sum(ba;x.f[x])  \mmember{}  \mBbbN{})

Date html generated: 2017_10_01-AM-08_47_56
Last ObjectModification: 2017_07_26-PM-04_32_15

Theory : bags

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