### Nuprl Lemma : select-shorten-tuple

`∀[n,m:ℕ]. ∀[L:Type List].  ∀[x:tuple-type(L)]. (shorten-tuple(x;n).m ~ x.n + m) supposing n + m < ||L||`

Proof

Definitions occuring in Statement :  shorten-tuple: `shorten-tuple(x;n)` select-tuple: `x.n` tuple-type: `tuple-type(L)` length: `||as||` list: `T List` nat: `ℕ` less_than: `a < b` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` subtract: `n - m` add: `n + m` universe: `Type` sqequal: `s ~ t`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` 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]` all: `∀x:A. B[x]` top: `Top` and: `P ∧ Q` prop: `ℙ` shorten-tuple: `shorten-tuple(x;n)` select-tuple: `x.n` 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` squash: `↓T` decidable: `Dec(P)` or: `P ∨ Q` true: `True` sq_type: `SQType(T)` guard: `{T}` bool: `𝔹` unit: `Unit` it: `⋅` uiff: `uiff(P;Q)` subtype_rel: `A ⊆r B` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` so_lambda: `λ2x.t[x]` so_apply: `x[s]` assert: `↑b` nequal: `a ≠ b ∈ T ` less_than: `a < b` le: `A ≤ B` cons: `[a / b]` pi2: `snd(t)`
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 subtype_base_sq bool_wf bool_subtype_base eq_int_wf squash_wf true_wf decidable__equal_int intformnot_wf intformeq_wf itermAdd_wf int_formula_prop_not_lemma int_formula_prop_eq_lemma int_term_value_add_lemma subtract_wf length_wf eqtt_to_assert assert_of_eq_int eqff_to_assert minus-zero add-zero equal_wf istype-universe eq_int_eq_true btrue_wf subtype_rel_self iff_weakening_equal int_subtype_base length_wf_nat set_subtype_base le_wf bfalse_wf bnot_wf assert_elim btrue_neq_bfalse bool_cases_sqequal assert-bnot neg_assert_of_eq_int non_neg_length itermSubtract_wf int_term_value_subtract_lemma tuple-type_wf list_wf istype-nat subtract-1-ge-0 equal-wf-base assert_wf equal-wf-T-base le_int_wf lt_int_wf less_than_wf not_wf istype-assert list-cases length_of_nil_lemma tupletype_nil_lemma product_subtype_list length_of_cons_lemma tupletype_cons_lemma null_nil_lemma null_cons_lemma decidable__lt add-is-int-iff false_wf add-subtract-cancel general_arith_equation1 minus-add minus-minus minus-one-mul add-swap add-commutes add-associates subtype_rel-equal nat_wf base_wf sqeq-copath5 uiff_transitivity iff_transitivity iff_weakening_uiff assert_of_bnot assert_of_le_int assert_functionality_wrt_uiff bnot_of_le_int assert_of_lt_int
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis setElimination rename intWeakElimination Error :lambdaFormation_alt,  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 :isectIsTypeImplies,  Error :inhabitedIsType,  Error :functionIsTypeImplies,  instantiate cumulativity applyEquality imageElimination because_Cache equalityTransitivity equalitySymmetry unionElimination imageMemberEquality baseClosed universeEquality equalityElimination productElimination Error :equalityIsType3,  Error :dependent_set_memberEquality_alt,  Error :productIsType,  Error :equalityIsType4,  baseApply closedConclusion intEquality applyLambdaEquality Error :equalityIsType2,  promote_hyp Error :equalityIsType1,  addEquality Error :equalityIstype,  sqequalBase Error :functionIsType,  hypothesis_subsumption pointwiseFunctionality multiplyEquality minusEquality

Latex:
\mforall{}[n,m:\mBbbN{}].  \mforall{}[L:Type  List].
\mforall{}[x:tuple-type(L)].  (shorten-tuple(x;n).m  \msim{}  x.n  +  m)  supposing  n  +  m  <  ||L||

Date html generated: 2019_06_20-PM-02_03_46
Last ObjectModification: 2018_11_22-AM-11_23_35

Theory : tuples

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