### Nuprl Lemma : l_sum-sum

`∀[T:Type]. ∀[L:T List]. ∀[f:{x:T| (x ∈ L)}  ⟶ ℤ].  (l_sum(map(f;L)) = Σ(f L[i] | i < ||L||) ∈ ℤ)`

Proof

Definitions occuring in Statement :  l_sum: `l_sum(L)` sum: `Σ(f[x] | x < k)` l_member: `(x ∈ l)` select: `L[n]` length: `||as||` map: `map(f;as)` list: `T List` uall: `∀[x:A]. B[x]` set: `{x:A| B[x]} ` apply: `f a` function: `x:A ⟶ B[x]` int: `ℤ` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` 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]` top: `Top` and: `P ∧ Q` prop: `ℙ` guard: `{T}` or: `P ∨ Q` select: `L[n]` nil: `[]` it: `⋅` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` sum: `Σ(f[x] | x < k)` sum_aux: `sum_aux(k;v;i;x.f[x])` cons: `[a / b]` le: `A ≤ B` less_than': `less_than'(a;b)` colength: `colength(L)` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` less_than: `a < b` squash: `↓T` decidable: `Dec(P)` subtype_rel: `A ⊆r B` l_sum: `l_sum(L)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` uiff: `uiff(P;Q)` int_seg: `{i..j-}` lelt: `i ≤ j < k` true: `True`
Lemmas referenced :  nat_properties full-omega-unsat intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf istype-int int_formula_prop_and_lemma istype-void 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 intformeq_wf int_formula_prop_eq_lemma list-cases map_nil_lemma length_of_nil_lemma stuck-spread istype-base product_subtype_list colength-cons-not-zero colength_wf_list istype-false le_wf istype-universe l_member_wf subtract-1-ge-0 subtype_base_sq set_subtype_base int_subtype_base spread_cons_lemma decidable__equal_int subtract_wf intformnot_wf itermSubtract_wf itermAdd_wf int_formula_prop_not_lemma int_term_value_subtract_lemma int_term_value_add_lemma decidable__le map_cons_lemma length_of_cons_lemma nat_wf list_wf nil_wf reduce_nil_lemma subtype_rel_dep_function cons_wf subtype_rel_sets cons_member reduce_cons_lemma sum_split length_wf add_nat_wf length_wf_nat add-is-int-iff false_wf select_wf int_seg_properties non_neg_length decidable__lt select_member int_seg_wf satisfiable-full-omega-tt list-subtype sum-as-primrec primrec1_lemma sum_wf squash_wf true_wf select-cons-tl add-subtract-cancel
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut thin Error :lambdaFormation_alt,  extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename intWeakElimination natural_numberEquality independent_isectElimination approximateComputation independent_functionElimination Error :dependent_pairFormation_alt,  Error :lambdaEquality_alt,  int_eqEquality dependent_functionElimination Error :isect_memberEquality_alt,  voidElimination sqequalRule independent_pairFormation Error :universeIsType,  axiomEquality equalityTransitivity equalitySymmetry applyLambdaEquality Error :functionIsTypeImplies,  Error :inhabitedIsType,  unionElimination baseClosed promote_hyp hypothesis_subsumption productElimination Error :equalityIsType1,  because_Cache Error :dependent_set_memberEquality_alt,  Error :functionIsType,  Error :setIsType,  instantiate imageElimination Error :equalityIsType4,  baseApply closedConclusion applyEquality intEquality universeEquality cumulativity setEquality functionEquality isect_memberFormation voidEquality isect_memberEquality Error :inrFormation_alt,  addEquality pointwiseFunctionality Error :productIsType,  computeAll dependent_pairFormation functionExtensionality lambdaEquality lambdaFormation dependent_set_memberEquality inlFormation imageMemberEquality

Latex:
\mforall{}[T:Type].  \mforall{}[L:T  List].  \mforall{}[f:\{x:T|  (x  \mmember{}  L)\}    {}\mrightarrow{}  \mBbbZ{}].    (l\_sum(map(f;L))  =  \mSigma{}(f  L[i]  |  i  <  ||L||))

Date html generated: 2019_06_20-PM-01_43_48
Last ObjectModification: 2018_10_06-PM-11_56_02

Theory : list_1

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