### Nuprl Lemma : filter-index_wf

`∀[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[L:T List].  (filter-index(P;L) ∈ i:{i:ℕ||L||| ↑(P L[i])}  ⟶ {j:ℕ||filter(P;L)||| filter(P;L)[\000Cj] = L[i] ∈ T} )`

Proof

Definitions occuring in Statement :  filter-index: `filter-index(P;L)` select: `L[n]` length: `||as||` filter: `filter(P;l)` list: `T List` int_seg: `{i..j-}` assert: `↑b` bool: `𝔹` uall: `∀[x:A]. B[x]` member: `t ∈ T` set: `{x:A| B[x]} ` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n` 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]` and: `P ∧ Q` prop: `ℙ` or: `P ∨ Q` select: `L[n]` nil: `[]` it: `⋅` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` cons: `[a / b]` le: `A ≤ B` less_than': `less_than'(a;b)` colength: `colength(L)` guard: `{T}` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` less_than: `a < b` squash: `↓T` decidable: `Dec(P)` subtype_rel: `A ⊆r B` int_seg: `{i..j-}` lelt: `i ≤ j < k` filter-index: `filter-index(P;L)` so_lambda: so_lambda3 so_apply: `x[s1;s2;s3]` eq_int: `(i =z j)` subtract: `n - m` ifthenelse: `if b then t else f fi ` btrue: `tt` istype: `istype(T)` nat_plus: `ℕ+` uiff: `uiff(P;Q)` bool: `𝔹` unit: `Unit` bfalse: `ff` bnot: `¬bb` assert: `↑b` nequal: `a ≠ b ∈ T ` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  void_wf nat_properties full-omega-unsat intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf istype-int 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 istype-less_than list-cases length_of_nil_lemma stuck-spread istype-base filter_nil_lemma int_seg_wf product_subtype_list colength-cons-not-zero colength_wf_list istype-void 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 length_of_cons_lemma filter_cons_lemma length_wf istype-nat istype-assert select_wf int_seg_properties decidable__lt list_wf bool_wf istype-universe list_ind_cons_lemma istype-false add_nat_plus length_wf_nat filter_wf5 subtype_rel_dep_function l_member_wf nat_plus_properties add-is-int-iff false_wf cons_wf non_neg_length assert_functionality_wrt_uiff select_cons_tl eq_int_wf eqtt_to_assert assert_of_eq_int eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot neg_assert_of_eq_int add-member-int_seg2 ifthenelse_wf select_cons_tl_sq2 int_seg_subtype_nat equal_wf squash_wf true_wf subtype_rel_self iff_weakening_equal select-cons-tl equal-wf-T-base assert_wf bnot_wf not_wf add-zero uiff_transitivity assert_of_bnot
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut functionExtensionality voidElimination thin instantiate extract_by_obid hypothesis functionExtensionality_alt setElimination rename lambdaFormation_alt sqequalHypSubstitution isectElimination hypothesisEquality sqequalRule intWeakElimination natural_numberEquality independent_isectElimination approximateComputation independent_functionElimination dependent_pairFormation_alt lambdaEquality_alt int_eqEquality dependent_functionElimination Error :memTop,  independent_pairFormation universeIsType axiomEquality equalityTransitivity equalitySymmetry functionIsTypeImplies inhabitedIsType unionElimination baseClosed promote_hyp hypothesis_subsumption productElimination equalityIstype because_Cache dependent_set_memberEquality_alt applyLambdaEquality imageElimination baseApply closedConclusion applyEquality intEquality sqequalBase addEquality setIsType isect_memberEquality_alt isectIsTypeImplies functionIsType universeEquality cumulativity setEquality pointwiseFunctionality productIsType equalityElimination minusEquality imageMemberEquality

Latex:
\mforall{}[T:Type].  \mforall{}[P:T  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[L:T  List].
(filter-index(P;L)  \mmember{}  i:\{i:\mBbbN{}||L|||  \muparrow{}(P  L[i])\}    {}\mrightarrow{}  \{j:\mBbbN{}||filter(P;L)|||  filter(P;L)[j]  =  L[i]\}  )

Date html generated: 2020_05_19-PM-09_42_29
Last ObjectModification: 2019_12_31-PM-00_12_36

Theory : list_1

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