### Nuprl Lemma : deq-member-length-filter

`∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[L:A List]. ∀[x:A].  (x ∈b L ~ 0 <z ||filter(λy.(eq y x);L)||)`

Proof

Definitions occuring in Statement :  length: `||as||` deq-member: `x ∈b L` filter: `filter(P;l)` list: `T List` deq: `EqDecider(T)` lt_int: `i <z j` uall: `∀[x:A]. B[x]` apply: `f a` lambda: `λx.A[x]` natural_number: `\$n` 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` lt_int: `i <z j` 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)` deq: `EqDecider(T)` bool: `𝔹` unit: `Unit` btrue: `tt` uiff: `uiff(P;Q)` eqof: `eqof(d)` ifthenelse: `if b then t else f fi ` iff: `P `⇐⇒` Q` nat_plus: `ℕ+` true: `True` rev_implies: `P `` Q` assert: `↑b` bfalse: `ff` bnot: `¬bb` bor: `p ∨bq`
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 deq_member_nil_lemma filter_nil_lemma length_of_nil_lemma 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 deq_member_cons_lemma filter_cons_lemma bool_wf eqtt_to_assert safe-assert-deq testxxx_lemma length_of_cons_lemma bool_subtype_base iff_imp_equal_bool btrue_wf lt_int_wf length_wf filter_wf5 l_member_wf add_nat_plus length_wf_nat eqof_wf nat_plus_wf nat_plus_properties decidable__lt add-is-int-iff false_wf true_wf assert_of_lt_int assert_wf iff_wf eqff_to_assert bool_cases_sqequal assert-bnot list_wf deq_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 equalityElimination setEquality imageMemberEquality pointwiseFunctionality baseApply closedConclusion addLevel impliesFunctionality universeEquality

Latex:
\mforall{}[A:Type].  \mforall{}[eq:EqDecider(A)].  \mforall{}[L:A  List].  \mforall{}[x:A].    (x  \mmember{}\msubb{}  L  \msim{}  0  <z  ||filter(\mlambda{}y.(eq  y  x);L)||)

Date html generated: 2017_09_29-PM-06_04_23
Last ObjectModification: 2017_07_26-PM-02_53_03

Theory : decidable!equality

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