### Nuprl Lemma : select-unshuffle

`∀[T:Type]. ∀as:T List. ∀i:ℕ||unshuffle(as)||.  (unshuffle(as)[i] = <as[2 * i], as[(2 * i) + 1]> ∈ (T × T))`

Proof

Definitions occuring in Statement :  unshuffle: `unshuffle(L)` select: `L[n]` length: `||as||` list: `T List` int_seg: `{i..j-}` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` pair: `<a, b>` product: `x:A × B[x]` multiply: `n * m` add: `n + m` 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]` top: `Top` and: `P ∧ Q` prop: `ℙ` guard: `{T}` int_seg: `{i..j-}` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` le: `A ≤ B` less_than: `a < b` squash: `↓T` unshuffle: `unshuffle(L)` lt_int: `i <z j` ifthenelse: `if b then t else f fi ` btrue: `tt` cons: `[a / b]` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` uiff: `uiff(P;Q)` bfalse: `ff` bool: `𝔹` unit: `Unit` it: `⋅` select: `L[n]` subtract: `n - m` less_than': `less_than'(a;b)` nat_plus: `ℕ+` nequal: `a ≠ b ∈ T ` int_nzero: `ℤ-o`
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 int_seg_properties int_seg_wf subtract-1-ge-0 decidable__equal_int subtract_wf subtype_base_sq set_subtype_base int_subtype_base intformnot_wf intformeq_wf itermSubtract_wf int_formula_prop_not_lemma int_formula_prop_eq_lemma int_term_value_subtract_lemma decidable__le decidable__lt istype-le subtype_rel_self istype-nat non_neg_length length_wf unshuffle_wf itermAdd_wf int_term_value_add_lemma length_wf_nat list_wf istype-universe list-cases length_of_nil_lemma reduce_tl_nil_lemma product_subtype_list length_of_cons_lemma reduce_hd_cons_lemma reduce_tl_cons_lemma lt_int_wf cons_wf equal_wf length-unshuffle length-nil select_wf itermMultiply_wf int_term_value_mul_lemma iff_weakening_equal assert_wf bnot_wf not_wf less_than_wf istype-assert equal-wf-T-base bool_wf le_int_wf le_wf tl_wf add-is-int-iff false_wf bool_cases bool_subtype_base eqtt_to_assert assert_of_lt_int eqff_to_assert iff_transitivity iff_weakening_uiff assert_of_bnot uiff_transitivity assert_functionality_wrt_uiff bnot_of_lt_int assert_of_le_int rem_bounds_1 nequal_wf div_rem_sum le_weakening2 mul-distributes add-associates mul-commutes add-commutes add-swap select-cons-tl squash_wf true_wf select_cons_tl
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut Error :lambdaFormation_alt,  thin extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename intWeakElimination natural_numberEquality independent_isectElimination approximateComputation independent_functionElimination Error :dependent_pairFormation_alt,  Error :lambdaEquality_alt,  int_eqEquality dependent_functionElimination Error :isect_memberEquality_alt,  voidElimination sqequalRule independent_pairFormation Error :universeIsType,  axiomEquality Error :functionIsTypeImplies,  Error :inhabitedIsType,  productElimination unionElimination applyEquality instantiate because_Cache equalityTransitivity equalitySymmetry applyLambdaEquality Error :dependent_set_memberEquality_alt,  Error :productIsType,  hypothesis_subsumption imageElimination productEquality addEquality universeEquality promote_hyp independent_pairEquality multiplyEquality closedConclusion imageMemberEquality baseClosed Error :functionIsType,  pointwiseFunctionality baseApply cumulativity equalityElimination Error :equalityIsType1,  intEquality Error :equalityIsType4,  minusEquality

Latex:
\mforall{}[T:Type].  \mforall{}as:T  List.  \mforall{}i:\mBbbN{}||unshuffle(as)||.    (unshuffle(as)[i]  =  <as[2  *  i],  as[(2  *  i)  +  1]>)

Date html generated: 2019_06_20-PM-01_48_02
Last ObjectModification: 2018_10_18-PM-00_42_06

Theory : list_1

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