### Nuprl Lemma : le-l_sum

`∀[T:Type]. ∀[f:T ⟶ ℕ]. ∀[L:T List]. ∀[t:T].  ((t ∈ L) `` ((f t) ≤ l_sum(map(f;L))))`

Proof

Definitions occuring in Statement :  l_sum: `l_sum(L)` l_member: `(x ∈ l)` map: `map(f;as)` list: `T List` nat: `ℕ` uall: `∀[x:A]. B[x]` le: `A ≤ B` implies: `P `` Q` apply: `f a` function: `x:A ⟶ B[x]` universe: `Type`
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` le: `A ≤ B` or: `P ∨ Q` cons: `[a / b]` decidable: `Dec(P)` colength: `colength(L)` nil: `[]` it: `⋅` guard: `{T}` 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)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` subtype_rel: `A ⊆r B` iff: `P `⇐⇒` Q` true: `True` rev_implies: `P `` Q` l_all: `(∀x∈L.P[x])` int_seg: `{i..j-}` lelt: `i ≤ j < k`
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 istype-less_than le_witness_for_triv list-cases null_nil_lemma btrue_wf member-implies-null-eq-bfalse nil_wf btrue_neq_bfalse l_member_wf product_subtype_list colength-cons-not-zero colength_wf_list decidable__le intformnot_wf int_formula_prop_not_lemma istype-le subtract-1-ge-0 subtype_base_sq intformeq_wf int_formula_prop_eq_lemma set_subtype_base int_subtype_base spread_cons_lemma decidable__equal_int subtract_wf itermSubtract_wf itermAdd_wf int_term_value_subtract_lemma int_term_value_add_lemma le_wf cons_member map_cons_lemma l_sum_cons_lemma squash_wf true_wf l_sum_wf map_wf subtype_rel_self iff_weakening_equal cons_wf istype-nat list_wf istype-universe l_sum_nonneg non_neg_length nat_wf map_length int_seg_properties select_wf length_wf int_seg_wf
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,  productElimination equalityTransitivity equalitySymmetry Error :functionIsTypeImplies,  Error :inhabitedIsType,  Error :isectIsTypeImplies,  unionElimination because_Cache promote_hyp hypothesis_subsumption Error :equalityIstype,  Error :dependent_set_memberEquality_alt,  instantiate applyLambdaEquality imageElimination baseApply closedConclusion baseClosed applyEquality intEquality sqequalBase addEquality functionExtensionality imageMemberEquality Error :functionIsType,  universeEquality

Latex:
\mforall{}[T:Type].  \mforall{}[f:T  {}\mrightarrow{}  \mBbbN{}].  \mforall{}[L:T  List].  \mforall{}[t:T].    ((t  \mmember{}  L)  {}\mRightarrow{}  ((f  t)  \mleq{}  l\_sum(map(f;L))))

Date html generated: 2019_06_20-PM-01_44_44
Last ObjectModification: 2019_02_23-PM-01_10_57

Theory : list_1

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