### Nuprl Lemma : unshuffle-shuffle

`∀[T:Type]. ∀[ps:(T × T) List].  (unshuffle(shuffle(ps)) ~ ps)`

Proof

Definitions occuring in Statement :  unshuffle: `unshuffle(L)` shuffle: `shuffle(ps)` list: `T List` uall: `∀[x:A]. B[x]` product: `x:A × B[x]` 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)` concat: `concat(ll)` unshuffle: `unshuffle(L)` lt_int: `i <z j` ifthenelse: `if b then t else f fi ` btrue: `tt` cons: `[a / b]` colength: `colength(L)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` decidable: `Dec(P)` nil: `[]` it: `⋅` 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]` bool: `𝔹` unit: `Unit` uiff: `uiff(P;Q)` le: `A ≤ B` bfalse: `ff` bnot: `¬bb` assert: `↑b` pi1: `fst(t)` pi2: `snd(t)`
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 equal-wf-T-base nat_wf colength_wf_list less_than_transitivity1 less_than_irreflexivity list-cases map_nil_lemma reduce_nil_lemma length_of_nil_lemma reduce_tl_nil_lemma 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 reduce_hd_cons_lemma reduce_tl_cons_lemma lt_int_wf length_wf shuffle_wf bool_wf eqtt_to_assert assert_of_lt_int non_neg_length eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot list_wf
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 productEquality cumulativity applyEquality because_Cache unionElimination promote_hyp hypothesis_subsumption productElimination equalityTransitivity equalitySymmetry applyLambdaEquality dependent_set_memberEquality addEquality baseClosed instantiate imageElimination equalityElimination universeEquality

Latex:
\mforall{}[T:Type].  \mforall{}[ps:(T  \mtimes{}  T)  List].    (unshuffle(shuffle(ps))  \msim{}  ps)

Date html generated: 2017_04_17-AM-08_57_19
Last ObjectModification: 2017_02_27-PM-05_12_52

Theory : list_1

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