### Nuprl Lemma : split-tuple_wf

`∀[L:Type List]. ∀[x:tuple-type(L)]. ∀[n:ℕ||L||].  (split-tuple(x;n) ∈ tuple-type(firstn(n;L)) × tuple-type(nth_tl(n;L)))`

Proof

Definitions occuring in Statement :  split-tuple: `split-tuple(x;n)` tuple-type: `tuple-type(L)` firstn: `firstn(n;as)` length: `||as||` nth_tl: `nth_tl(n;as)` list: `T List` int_seg: `{i..j-}` uall: `∀[x:A]. B[x]` member: `t ∈ T` product: `x:A × B[x]` natural_number: `\$n` universe: `Type`
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` or: `P ∨ Q` firstn: `firstn(n;as)` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]` int_seg: `{i..j-}` lelt: `i ≤ j < k` cons: `[a / b]` decidable: `Dec(P)` colength: `colength(L)` nil: `[]` it: `⋅` guard: `{T}` 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)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` subtype_rel: `A ⊆r B` split-tuple: `split-tuple(x;n)` bool: `𝔹` unit: `Unit` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` nth_tl: `nth_tl(n;as)` le_int: `i ≤z j` lt_int: `i <z j` bnot: `¬bb` bfalse: `ff` assert: `↑b` nequal: `a ≠ b ∈ T ` pi1: `fst(t)` pi2: `snd(t)` null: `null(as)` list_ind: list_ind tuple-type: `tuple-type(L)` subtract: `n - m` tl: `tl(l)` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` int_iseg: `{i...j}` cand: `A c∧ B`
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 list-cases length_of_nil_lemma tupletype_nil_lemma list_ind_nil_lemma nth_tl_nil int_seg_properties int_seg_wf length_wf nil_wf tuple-type_wf product_subtype_list colength-cons-not-zero istype-nat colength_wf_list decidable__le intformnot_wf int_formula_prop_not_lemma istype-le list_wf subtract-1-ge-0 subtype_base_sq intformeq_wf int_formula_prop_eq_lemma set_subtype_base int_subtype_base spread_cons_lemma decidable__equal_int subtract_wf itermSubtract_wf itermAdd_wf int_term_value_subtract_lemma int_term_value_add_lemma le_wf length_of_cons_lemma tupletype_cons_lemma eq_int_wf eqtt_to_assert assert_of_eq_int first0 cons_wf subtype_rel_list top_wf it_wf eqff_to_assert bool_cases_sqequal bool_wf bool_subtype_base assert-bnot neg_assert_of_eq_int null_cons_lemma subtype_rel_self decidable__lt add-is-int-iff false_wf reduce_tl_cons_lemma le_int_wf assert_of_le_int iff_weakening_uiff assert_wf list_ind_cons_lemma lt_int_wf assert_of_lt_int null_wf firstn_wf assert_of_null length_firstn equal-wf-T-base less_than_wf
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 sqequalRule intWeakElimination natural_numberEquality independent_isectElimination approximateComputation independent_functionElimination Error :dependent_pairFormation_alt,  Error :lambdaEquality_alt,  int_eqEquality dependent_functionElimination Error :isect_memberEquality_alt,  voidElimination independent_pairFormation Error :universeIsType,  axiomEquality equalityTransitivity equalitySymmetry Error :isectIsTypeImplies,  Error :inhabitedIsType,  Error :functionIsTypeImplies,  universeEquality instantiate unionElimination productElimination voidEquality promote_hyp hypothesis_subsumption Error :equalityIstype,  Error :dependent_set_memberEquality_alt,  because_Cache applyLambdaEquality imageElimination baseApply closedConclusion baseClosed applyEquality intEquality sqequalBase equalityElimination cumulativity independent_pairEquality addEquality isect_memberEquality pointwiseFunctionality Error :productIsType

Latex:
\mforall{}[L:Type  List].  \mforall{}[x:tuple-type(L)].  \mforall{}[n:\mBbbN{}||L||].
(split-tuple(x;n)  \mmember{}  tuple-type(firstn(n;L))  \mtimes{}  tuple-type(nth\_tl(n;L)))

Date html generated: 2019_06_20-PM-02_03_31
Last ObjectModification: 2018_12_30-PM-10_15_01

Theory : tuples

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