### Nuprl Lemma : l_sum_as_reduce_general

`∀[T:Type]. ∀[f:T ⟶ ℤ]. ∀[L:T List]. ∀[x:ℤ].  (reduce(λa,s. (s + f[a]);x;L) ~ l_sum(map(λx.f[x];L)) + x)`

Proof

Definitions occuring in Statement :  l_sum: `l_sum(L)` map: `map(f;as)` reduce: `reduce(f;k;as)` list: `T List` uall: `∀[x:A]. B[x]` so_apply: `x[s]` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` add: `n + m` int: `ℤ` universe: `Type` sqequal: `s ~ 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` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` top: `Top` and: `P ∧ Q` prop: `ℙ` subtype_rel: `A ⊆r B` guard: `{T}` or: `P ∨ Q` cons: `[a / b]` colength: `colength(L)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` decidable: `Dec(P)` nil: `[]` it: `⋅` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)`
Lemmas referenced :  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 equal-wf-T-base nat_wf colength_wf_list less_than_transitivity1 less_than_irreflexivity list-cases reduce_nil_lemma map_nil_lemma l_sum_nil_lemma zero-add product_subtype_list spread_cons_lemma intformeq_wf itermAdd_wf int_formula_prop_eq_lemma int_term_value_add_lemma decidable__le intformnot_wf int_formula_prop_not_lemma le_wf equal_wf subtract_wf itermSubtract_wf int_term_value_subtract_lemma subtype_base_sq set_subtype_base int_subtype_base decidable__equal_int reduce_cons_lemma map_cons_lemma l_sum_cons_lemma l_sum_wf map_wf list_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut thin lambdaFormation extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename intWeakElimination natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll independent_functionElimination sqequalAxiom cumulativity applyEquality because_Cache unionElimination promote_hyp hypothesis_subsumption productElimination equalityTransitivity equalitySymmetry applyLambdaEquality dependent_set_memberEquality addEquality baseClosed instantiate imageElimination functionExtensionality functionEquality universeEquality

Latex:
\mforall{}[T:Type].  \mforall{}[f:T  {}\mrightarrow{}  \mBbbZ{}].  \mforall{}[L:T  List].  \mforall{}[x:\mBbbZ{}].    (reduce(\mlambda{}a,s.  (s  +  f[a]);x;L)  \msim{}  l\_sum(map(\mlambda{}x.f[x];L))  \000C+  x)

Date html generated: 2017_04_17-AM-08_40_18
Last ObjectModification: 2017_02_27-PM-04_59_35

Theory : list_1

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