### Nuprl Lemma : length-filter-lsum

`∀[T:Type]. ∀[L:T List]. ∀[P:{x:T| (x ∈ L)}  ⟶ 𝔹].  (||filter(P;L)|| = Σ(if P x then 1 else 0 fi  | x ∈ L) ∈ ℤ)`

Proof

Definitions occuring in Statement :  lsum: `Σ(f[x] | x ∈ L)` l_member: `(x ∈ l)` length: `||as||` filter: `filter(P;l)` list: `T List` ifthenelse: `if b then t else f fi ` bool: `𝔹` uall: `∀[x:A]. B[x]` set: `{x:A| B[x]} ` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n` 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` or: `P ∨ Q` so_lambda: `λ2x.t[x]` so_apply: `x[s]` cons: `[a / b]` le: `A ≤ B` less_than': `less_than'(a;b)` colength: `colength(L)` nil: `[]` it: `⋅` guard: `{T}` sq_type: `SQType(T)` less_than: `a < b` squash: `↓T` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` decidable: `Dec(P)` subtype_rel: `A ⊆r B` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` istype: `istype(T)` l_member: `(x ∈ l)` select: `L[n]` cand: `A c∧ B` nat_plus: `ℕ+` uiff: `uiff(P;Q)` bool: `𝔹` unit: `Unit` btrue: `tt` ifthenelse: `if b then t else f fi ` bfalse: `ff` bnot: `¬bb` assert: `↑b`
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 list-cases filter_nil_lemma lsum_nil_lemma length_of_nil_lemma l_member_wf nil_wf bool_wf product_subtype_list colength-cons-not-zero colength_wf_list 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 intformnot_wf itermSubtract_wf itermAdd_wf int_formula_prop_not_lemma int_term_value_subtract_lemma int_term_value_add_lemma decidable__le le_wf filter_cons_lemma lsum_cons_lemma subtype_rel_dep_function cons_wf subtype_rel_sets_simple cons_member length_of_cons_lemma add_nat_plus length_wf_nat decidable__lt nat_plus_properties add-is-int-iff false_wf length_wf select_wf eqtt_to_assert eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot istype-nat list_wf istype-universe
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut thin lambdaFormation_alt extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename intWeakElimination natural_numberEquality independent_isectElimination approximateComputation independent_functionElimination dependent_pairFormation_alt lambdaEquality_alt int_eqEquality dependent_functionElimination isect_memberEquality_alt voidElimination sqequalRule independent_pairFormation universeIsType axiomEquality isectIsTypeImplies inhabitedIsType functionIsTypeImplies unionElimination functionIsType setIsType promote_hyp hypothesis_subsumption productElimination equalityIstype because_Cache dependent_set_memberEquality_alt instantiate equalityTransitivity equalitySymmetry applyLambdaEquality imageElimination baseApply closedConclusion baseClosed applyEquality intEquality sqequalBase setEquality inrFormation_alt pointwiseFunctionality productIsType equalityElimination cumulativity universeEquality

Latex:
\mforall{}[T:Type].  \mforall{}[L:T  List].  \mforall{}[P:\{x:T|  (x  \mmember{}  L)\}    {}\mrightarrow{}  \mBbbB{}].
(||filter(P;L)||  =  \mSigma{}(if  P  x  then  1  else  0  fi    |  x  \mmember{}  L))

Date html generated: 2020_05_19-PM-09_48_12
Last ObjectModification: 2019_11_12-PM-11_40_03

Theory : list_1

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