### Nuprl Lemma : split-tuple-append-tuple

`∀[L1,L2:Type List].`
`  ∀[x:tuple-type(L1)]. ∀[y:tuple-type(L2)].  (split-tuple(append-tuple(||L1||;||L2||;x;y);||L1||) ~ <x, y>) supposing 0 \000C< ||L2||`

Proof

Definitions occuring in Statement :  append-tuple: `append-tuple(n;m;x;y)` split-tuple: `split-tuple(x;n)` tuple-type: `tuple-type(L)` length: `||as||` list: `T List` less_than: `a < b` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` pair: `<a, b>` 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` subtype_rel: `A ⊆r B` or: `P ∨ Q` split-tuple: `split-tuple(x;n)` eq_int: `(i =z j)` append-tuple: `append-tuple(n;m;x;y)` le_int: `i ≤z j` lt_int: `i <z j` bnot: `¬bb` ifthenelse: `if b then t else f fi ` bfalse: `ff` subtract: `n - m` btrue: `tt` sq_type: `SQType(T)` guard: `{T}` cons: `[a / b]` colength: `colength(L)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` so_lambda: `λ2x.t[x]` so_apply: `x[s]` nil: `[]` it: `⋅` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` decidable: `Dec(P)` bool: `𝔹` unit: `Unit` uiff: `uiff(P;Q)` le: `A ≤ B` assert: `↑b` nequal: `a ≠ b ∈ T ` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` pi1: `fst(t)` pi2: `snd(t)` length: `||as||` list_ind: list_ind
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 less_than_wf tuple-type_wf length_wf nat_wf colength_wf_list int_subtype_base list_wf list-cases tupletype_nil_lemma length_of_nil_lemma subtype_base_sq unit_wf2 unit_subtype_base equal-unit it_wf product_subtype_list spread_cons_lemma intformeq_wf itermAdd_wf int_formula_prop_eq_lemma int_term_value_add_lemma set_subtype_base le_wf subtract-1-ge-0 decidable__equal_int subtract_wf intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma decidable__le tupletype_cons_lemma length_of_cons_lemma ifthenelse_wf null_wf eq_int_wf eqtt_to_assert assert_of_eq_int le_int_wf assert_of_le_int eqff_to_assert non_neg_length bool_cases_sqequal bool_wf bool_subtype_base assert-bnot neg_assert_of_eq_int subtype_rel-equal assert_of_null iff_weakening_uiff assert_wf equal-wf-T-base nequal-le-implies add-is-int-iff false_wf bnot_wf not_wf bool_cases iff_transitivity assert_of_bnot length_wf_nat add-subtract-cancel
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 instantiate universeEquality equalityTransitivity equalitySymmetry Error :inhabitedIsType,  Error :equalityIsType3,  applyEquality unionElimination cumulativity because_Cache promote_hyp hypothesis_subsumption productElimination applyLambdaEquality Error :equalityIsType4,  baseApply closedConclusion baseClosed intEquality Error :equalityIsType1,  imageElimination Error :dependent_set_memberEquality_alt,  productEquality addEquality equalityElimination pointwiseFunctionality hyp_replacement

Latex:
\mforall{}[L1,L2:Type  List].
\mforall{}[x:tuple-type(L1)].  \mforall{}[y:tuple-type(L2)].
(split-tuple(append-tuple(||L1||;||L2||;x;y);||L1||)  \msim{}  <x,  y>)
supposing  0  <  ||L2||

Date html generated: 2019_06_20-PM-02_03_40
Last ObjectModification: 2018_09_30-PM-02_47_08

Theory : tuples

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