### Nuprl Lemma : permutation-filter

`∀[A:Type]. ∀L1,L2:A List.  (permutation(A;L1;L2) `` (∀p:A ⟶ 𝔹. permutation({a:A| ↑(p a)} ;filter(p;L1);filter(p;L2))))`

Proof

Definitions occuring in Statement :  permutation: `permutation(T;L1;L2)` filter: `filter(P;l)` list: `T List` assert: `↑b` bool: `𝔹` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` set: `{x:A| B[x]} ` apply: `f a` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` member: `t ∈ T` so_lambda: `λ2x.t[x]` implies: `P `` Q` prop: `ℙ` so_apply: `x[s]` top: `Top` or: `P ∨ Q` cons: `[a / b]` uimplies: `b supposing a` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` and: `P ∧ Q` ifthenelse: `if b then t else f fi ` ge: `i ≥ j ` le: `A ≤ B` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` bfalse: `ff` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` cand: `A c∧ B`
Lemmas referenced :  list_induction all_wf list_wf permutation_wf bool_wf assert_wf filter_type filter_nil_lemma nil_wf filter_cons_lemma cons_wf list-cases product_subtype_list permutation_weakening permutation-length length_of_nil_lemma length_of_cons_lemma eqtt_to_assert non_neg_length satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformeq_wf itermAdd_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_eq_lemma int_term_value_add_lemma int_formula_prop_wf eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot permutation-cons filter_append_sq append_wf length_wf length-append exists_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut thin introduction extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality sqequalRule lambdaEquality cumulativity hypothesis functionEquality setEquality applyEquality functionExtensionality independent_functionElimination dependent_functionElimination isect_memberEquality voidElimination voidEquality because_Cache rename universeEquality unionElimination promote_hyp hypothesis_subsumption productElimination independent_isectElimination equalityElimination equalityTransitivity equalitySymmetry natural_numberEquality dependent_pairFormation int_eqEquality intEquality independent_pairFormation computeAll instantiate hyp_replacement applyLambdaEquality dependent_set_memberEquality productEquality

Latex:
\mforall{}[A:Type]
\mforall{}L1,L2:A  List.
(permutation(A;L1;L2)  {}\mRightarrow{}  (\mforall{}p:A  {}\mrightarrow{}  \mBbbB{}.  permutation(\{a:A|  \muparrow{}(p  a)\}  ;filter(p;L1);filter(p;L2))))

Date html generated: 2017_04_17-AM-08_24_46
Last ObjectModification: 2017_02_27-PM-04_46_50

Theory : list_1

Home Index