### Nuprl Lemma : unshuffle-map

`∀[f:ℕ ⟶ Top]. ∀[m:ℕ].  (unshuffle(map(f;upto(2 * m))) ~ map(λi.<f (2 * i), f ((2 * i) + 1)>;upto(m)))`

Proof

Definitions occuring in Statement :  unshuffle: `unshuffle(L)` upto: `upto(n)` map: `map(f;as)` nat: `ℕ` uall: `∀[x:A]. B[x]` top: `Top` apply: `f a` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` pair: `<a, b>` multiply: `n * m` add: `n + m` natural_number: `\$n` sqequal: `s ~ t`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` upto: `upto(n)` 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` int_seg: `{i..j-}` from-upto: `[n, m)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` lelt: `i ≤ j < k` shuffle: `shuffle(ps)` concat: `concat(ll)` decidable: `Dec(P)` pi1: `fst(t)` pi2: `snd(t)` le: `A ≤ B` less_than': `less_than'(a;b)` so_lambda: `λ2x.t[x]` so_apply: `x[s]` squash: `↓T` subtype_rel: `A ⊆r B` true: `True` append: `as @ bs` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]` has-value: `(a)↓` subtract: `n - m` cand: `A c∧ B`
Lemmas referenced :  nat_properties full-omega-unsat 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 le_wf subtract_wf int_seg_wf lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int map_cons_lemma eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot int_seg_properties itermMultiply_wf itermSubtract_wf int_term_value_mul_lemma int_term_value_subtract_lemma map_nil_lemma reduce_nil_lemma decidable__le intformnot_wf int_formula_prop_not_lemma false_wf decidable__lt itermAdd_wf int_term_value_add_lemma lelt_wf list_wf list_subtype_base set_subtype_base int_subtype_base from-upto_wf squash_wf true_wf mul-commutes unshuffle-shuffle top_wf map_wf upto_wf nat_wf reduce_cons_lemma list_ind_cons_lemma list_ind_nil_lemma value-type-has-value int-value-type add-member-int_seg2 add-subtract-cancel decidable__equal_int intformeq_wf int_formula_prop_eq_lemma subtype_rel_list_set
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 approximateComputation independent_functionElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation sqequalAxiom addEquality because_Cache multiplyEquality unionElimination equalityElimination equalityTransitivity equalitySymmetry productElimination promote_hyp instantiate cumulativity dependent_set_memberEquality setEquality productEquality applyEquality imageElimination imageMemberEquality baseClosed independent_pairEquality functionEquality callbyvalueReduce

Latex:
\mforall{}[f:\mBbbN{}  {}\mrightarrow{}  Top].  \mforall{}[m:\mBbbN{}].
(unshuffle(map(f;upto(2  *  m)))  \msim{}  map(\mlambda{}i.<f  (2  *  i),  f  ((2  *  i)  +  1)>upto(m)))

Date html generated: 2018_05_21-PM-00_44_48
Last ObjectModification: 2018_05_19-AM-06_49_28

Theory : list_1

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