### Nuprl Lemma : l_sum-triangle-inequality

`∀[T:Type]. ∀[L:T List]. ∀[f,g:T ⟶ ℤ].`
`  (|l_sum(map(λa.f[a];L)) - l_sum(map(λa.g[a];L))| ≤ l_sum(map(λa.|f[a] - g[a]|;L)))`

Proof

Definitions occuring in Statement :  l_sum: `l_sum(L)` map: `map(f;as)` list: `T List` absval: `|i|` uall: `∀[x:A]. B[x]` so_apply: `x[s]` le: `A ≤ B` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` subtract: `n - m` int: `ℤ` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` prop: `ℙ` squash: `↓T` subtype_rel: `A ⊆r B` uimplies: `b supposing a` nat: `ℕ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` absval: `|i|` subtract: `n - m` false: `False` not: `¬A` implies: `P `` Q` sq_type: `SQType(T)` all: `∀x:A. B[x]` guard: `{T}` decidable: `Dec(P)` or: `P ∨ Q` uiff: `uiff(P;Q)` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` true: `True`
Lemmas referenced :  l_sum-triangle-inequality-general list_wf istype-universe le_wf squash_wf true_wf istype-int add-zero l_sum_wf map_wf absval_wf subtract_wf subtype_base_sq nat_wf set_subtype_base int_subtype_base istype-false minus-zero zero-add absval_pos istype-le decidable__equal_int add-is-int-iff full-omega-unsat intformnot_wf intformeq_wf itermVar_wf itermAdd_wf itermConstant_wf int_formula_prop_not_lemma istype-void int_formula_prop_eq_lemma int_term_value_var_lemma int_term_value_add_lemma int_term_value_constant_lemma int_formula_prop_wf false_wf
Rules used in proof :  cut introduction extract_by_obid sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  hypothesis sqequalHypSubstitution isectElimination thin hypothesisEquality Error :universeIsType,  instantiate universeEquality natural_numberEquality hyp_replacement equalitySymmetry sqequalRule applyEquality Error :lambdaEquality_alt,  imageElimination equalityTransitivity Error :inhabitedIsType,  because_Cache cumulativity independent_isectElimination intEquality independent_pairFormation Error :lambdaFormation_alt,  setElimination rename Error :dependent_set_memberEquality_alt,  dependent_functionElimination independent_functionElimination unionElimination pointwiseFunctionality promote_hyp productElimination baseApply closedConclusion baseClosed approximateComputation Error :dependent_pairFormation_alt,  int_eqEquality Error :isect_memberEquality_alt,  voidElimination imageMemberEquality Error :functionIsType

Latex:
\mforall{}[T:Type].  \mforall{}[L:T  List].  \mforall{}[f,g:T  {}\mrightarrow{}  \mBbbZ{}].
(|l\_sum(map(\mlambda{}a.f[a];L))  -  l\_sum(map(\mlambda{}a.g[a];L))|  \mleq{}  l\_sum(map(\mlambda{}a.|f[a]  -  g[a]|;L)))

Date html generated: 2019_06_20-PM-01_44_33
Last ObjectModification: 2018_10_18-PM-00_39_33

Theory : list_1

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