### Nuprl Lemma : summand-le-l_sum

`∀[T:Type]. ∀[L:T List]. ∀[f:{x:T| (x ∈ L)}  ⟶ ℤ].`
`  ∀x:{x:T| (x ∈ L)} . (f[x] ≤ l_sum(map(f;L))) supposing ∀x:{x:T| (x ∈ L)} . (0 ≤ f[x])`

Proof

Definitions occuring in Statement :  l_sum: `l_sum(L)` l_member: `(x ∈ l)` map: `map(f;as)` list: `T List` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s]` le: `A ≤ B` all: `∀x:A. B[x]` set: `{x:A| B[x]} ` function: `x:A ⟶ B[x]` natural_number: `\$n` int: `ℤ` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` all: `∀x:A. B[x]` squash: `↓T` prop: `ℙ` so_apply: `x[s]` true: `True` subtype_rel: `A ⊆r B` guard: `{T}` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` implies: `P `` Q` so_lambda: `λ2x.t[x]` int_seg: `{i..j-}` lelt: `i ≤ j < k` le: `A ≤ B` less_than: `a < b` decidable: `Dec(P)` or: `P ∨ Q` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` sq_stable: `SqStable(P)` l_member: `(x ∈ l)` cand: `A c∧ B` nat: `ℕ` ge: `i ≥ j ` label: `...\$L... t` sq_type: `SQType(T)`
Lemmas referenced :  le_wf squash_wf true_wf istype-int l_sum-sum subtype_rel_self iff_weakening_equal sq_stable__le l_member_wf sum_wf length_wf_nat select_wf int_seg_properties decidable__le full-omega-unsat intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf decidable__lt length_wf intformless_wf int_formula_prop_less_lemma select_member int_seg_wf summand-le-sum nat_properties istype-le istype-less_than subtype_base_sq int_subtype_base equal_wf le_witness_for_triv list_wf istype-universe
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut lambdaFormation_alt applyEquality thin lambdaEquality_alt sqequalHypSubstitution imageElimination extract_by_obid isectElimination hypothesisEquality equalityTransitivity hypothesis equalitySymmetry universeIsType inhabitedIsType natural_numberEquality sqequalRule imageMemberEquality baseClosed instantiate because_Cache independent_isectElimination productElimination independent_functionElimination setElimination rename dependent_set_memberEquality_alt dependent_functionElimination unionElimination approximateComputation dependent_pairFormation_alt int_eqEquality Error :memTop,  independent_pairFormation voidElimination productIsType cumulativity intEquality equalityIstype setIsType functionIsTypeImplies functionIsType isect_memberEquality_alt isectIsTypeImplies universeEquality

Latex:
\mforall{}[T:Type].  \mforall{}[L:T  List].  \mforall{}[f:\{x:T|  (x  \mmember{}  L)\}    {}\mrightarrow{}  \mBbbZ{}].
\mforall{}x:\{x:T|  (x  \mmember{}  L)\}  .  (f[x]  \mleq{}  l\_sum(map(f;L)))  supposing  \mforall{}x:\{x:T|  (x  \mmember{}  L)\}  .  (0  \mleq{}  f[x])

Date html generated: 2020_05_19-PM-09_45_56
Last ObjectModification: 2020_01_23-PM-00_48_26

Theory : list_1

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