### Nuprl Lemma : length-unshuffle

`∀[T:Type]. ∀[L:T List].  (||unshuffle(L)|| ~ ||L|| ÷ 2)`

Proof

Definitions occuring in Statement :  unshuffle: `unshuffle(L)` length: `||as||` list: `T List` uall: `∀[x:A]. B[x]` 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` 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` le: `A ≤ B` decidable: `Dec(P)` or: `P ∨ Q` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` less_than: `a < b` squash: `↓T` unshuffle: `unshuffle(L)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` less_than': `less_than'(a;b)` bfalse: `ff` true: `True` cons: `[a / b]` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` divide: `n ÷ m` nat_plus: `ℕ+` int_nzero: `ℤ-o` nequal: `a ≠ b ∈ T ` subtract: `n - m`
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 non_neg_length length_wf nat_wf le_wf lt_int_wf uiff_transitivity equal-wf-T-base bool_wf assert_wf less_than_wf eqtt_to_assert assert_of_lt_int length_of_nil_lemma istype-false le_int_wf bnot_wf eqff_to_assert assert_functionality_wrt_uiff bnot_of_lt_int assert_of_le_int length_of_cons_lemma tl_wf list-cases reduce_tl_nil_lemma product_subtype_list reduce_tl_cons_lemma length_tl add-is-int-iff itermAdd_wf int_term_value_add_lemma false_wf iff_weakening_equal add_nat_wf divide_wf equal_wf divide_wfa nequal_wf div_rec_case add-associates add-swap add-commutes zero-add istype-nat length_wf_nat list_wf istype-universe
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut thin Error :lambdaFormation_alt,  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,  axiomSqEquality Error :functionIsTypeImplies,  Error :inhabitedIsType,  productElimination unionElimination applyEquality instantiate because_Cache equalityTransitivity equalitySymmetry applyLambdaEquality Error :dependent_set_memberEquality_alt,  Error :productIsType,  hypothesis_subsumption imageElimination cumulativity intEquality equalityElimination baseClosed promote_hyp pointwiseFunctionality baseApply closedConclusion addEquality imageMemberEquality Error :equalityIstype,  sqequalBase Error :isectIsTypeImplies,  universeEquality

Latex:
\mforall{}[T:Type].  \mforall{}[L:T  List].    (||unshuffle(L)||  \msim{}  ||L||  \mdiv{}  2)

Date html generated: 2019_06_20-PM-01_47_34
Last ObjectModification: 2019_03_06-AM-10_30_07

Theory : list_1

Home Index