### Nuprl Lemma : select-shuffle

`∀[T:Type]. ∀[ps:(T × T) List].`
`  ∀i:ℕ||shuffle(ps)||. (shuffle(ps)[i] ~ if (i rem 2 =z 0) then fst(ps[i ÷ 2]) else snd(ps[i ÷ 2]) fi )`

Proof

Definitions occuring in Statement :  shuffle: `shuffle(ps)` select: `L[n]` length: `||as||` list: `T List` int_seg: `{i..j-}` ifthenelse: `if b then t else f fi ` eq_int: `(i =z j)` uall: `∀[x:A]. B[x]` pi1: `fst(t)` pi2: `snd(t)` all: `∀x:A. B[x]` product: `x:A × B[x]` remainder: `n rem m` divide: `n ÷ m` 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` shuffle: `shuffle(ps)` select: `L[n]` nil: `[]` it: `⋅` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` concat: `concat(ll)` int_seg: `{i..j-}` lelt: `i ≤ j < k` cons: `[a / b]` colength: `colength(L)` decidable: `Dec(P)` 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)` append: `as @ bs` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]` eq_int: `(i =z j)` ifthenelse: `if b then t else f fi ` btrue: `tt` subtract: `n - m` bfalse: `ff` uiff: `uiff(P;Q)` bool: `𝔹` unit: `Unit` bnot: `¬bb` assert: `↑b` le: `A ≤ B` nat_plus: `ℕ+` true: `True` nequal: `a ≠ b ∈ T ` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` int_nzero: `ℤ-o`
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 int_seg_wf length_wf shuffle_wf equal-wf-T-base nat_wf colength_wf_list less_than_transitivity1 less_than_irreflexivity list-cases map_nil_lemma stuck-spread base_wf reduce_nil_lemma length_of_nil_lemma int_seg_properties 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 map_cons_lemma reduce_cons_lemma list_ind_cons_lemma list_ind_nil_lemma length_of_cons_lemma decidable__lt add-is-int-iff false_wf lelt_wf list_wf select-cons le_int_wf bool_wf eqtt_to_assert assert_of_le_int eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot squash_wf true_wf eq_int_wf rem_rec_case int_seg_subtype_nat equal-wf-base iff_weakening_equal div_rec_case div_rem_sum nequal_wf rem_bounds_1 itermMultiply_wf int_term_value_mul_lemma
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 productEquality applyEquality because_Cache unionElimination baseClosed productElimination promote_hyp hypothesis_subsumption equalityTransitivity equalitySymmetry applyLambdaEquality dependent_set_memberEquality addEquality instantiate imageElimination pointwiseFunctionality baseApply closedConclusion universeEquality equalityElimination imageMemberEquality remainderEquality addLevel divideEquality

Latex:
\mforall{}[T:Type].  \mforall{}[ps:(T  \mtimes{}  T)  List].
\mforall{}i:\mBbbN{}||shuffle(ps)||
(shuffle(ps)[i]  \msim{}  if  (i  rem  2  =\msubz{}  0)  then  fst(ps[i  \mdiv{}  2])  else  snd(ps[i  \mdiv{}  2])  fi  )

Date html generated: 2017_04_17-AM-08_56_12
Last ObjectModification: 2017_02_27-PM-05_14_37

Theory : list_1

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