### Nuprl Lemma : cons_filter2

`∀[T:Type]. ∀[x:T]. ∀[L:T List]. ∀[P:ℕ||L|| + 1 ⟶ 𝔹].`
`  (filter2(P;[x / L]) = if P 0 then [x / filter2(λi.(P (i + 1));L)] else filter2(λi.(P (i + 1));L) fi  ∈ (T List))`

Proof

Definitions occuring in Statement :  filter2: `filter2(P;L)` length: `||as||` cons: `[a / b]` list: `T List` int_seg: `{i..j-}` ifthenelse: `if b then t else f fi ` bool: `𝔹` uall: `∀[x:A]. B[x]` apply: `f a` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` filter2: `filter2(P;L)` all: `∀x:A. B[x]` top: `Top` int_seg: `{i..j-}` lelt: `i ≤ j < k` and: `P ∧ Q` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` implies: `P `` Q` prop: `ℙ` nat_plus: `ℕ+` less_than: `a < b` squash: `↓T` true: `True` guard: `{T}` decidable: `Dec(P)` or: `P ∨ Q` uiff: `uiff(P;Q)` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` bfalse: `ff` nat: `ℕ` subtract: `n - m`
Lemmas referenced :  reduce2_cons_lemma int_seg_wf length_wf bool_wf list_wf false_wf add_nat_plus length_wf_nat less_than_wf nat_plus_wf nat_plus_properties decidable__lt add-is-int-iff satisfiable-full-omega-tt intformand_wf intformnot_wf intformless_wf itermConstant_wf itermVar_wf itermAdd_wf intformeq_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_var_lemma int_term_value_add_lemma int_formula_prop_eq_lemma int_formula_prop_wf equal_wf lelt_wf equal-wf-T-base assert_wf bnot_wf not_wf eqtt_to_assert uiff_transitivity eqff_to_assert assert_of_bnot cons_wf squash_wf true_wf reduce2_shift nil_wf le_wf add-member-int_seg2 decidable__le subtract_wf intformle_wf itermSubtract_wf int_formula_prop_le_lemma int_term_value_subtract_lemma reduce2_wf nat_wf decidable__equal_int add-associates add-swap add-commutes zero-add
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule extract_by_obid sqequalHypSubstitution dependent_functionElimination thin isect_memberEquality voidElimination voidEquality hypothesis functionEquality isectElimination natural_numberEquality addEquality cumulativity hypothesisEquality axiomEquality because_Cache universeEquality applyEquality dependent_set_memberEquality independent_pairFormation lambdaFormation imageMemberEquality baseClosed equalityTransitivity equalitySymmetry applyLambdaEquality setElimination rename unionElimination pointwiseFunctionality promote_hyp baseApply closedConclusion productElimination independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality computeAll independent_functionElimination equalityElimination imageElimination functionExtensionality hyp_replacement

Latex:
\mforall{}[T:Type].  \mforall{}[x:T].  \mforall{}[L:T  List].  \mforall{}[P:\mBbbN{}||L||  +  1  {}\mrightarrow{}  \mBbbB{}].
(filter2(P;[x  /  L])
=  if  P  0  then  [x  /  filter2(\mlambda{}i.(P  (i  +  1));L)]  else  filter2(\mlambda{}i.(P  (i  +  1));L)  fi  )

Date html generated: 2017_10_01-AM-08_35_09
Last ObjectModification: 2017_07_26-PM-04_25_43

Theory : list!

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