### Nuprl Lemma : tuple_wf

`∀[L:Type List]. ∀[F:i:ℕ||L|| ⟶ L[i]]. ∀[n:{n:ℤ| n = ||L|| ∈ ℤ} ].  (tuple(n;i.F[i]) ∈ tuple-type(L))`

Proof

Definitions occuring in Statement :  tuple: `tuple(n;i.F[i])` tuple-type: `tuple-type(L)` select: `L[n]` length: `||as||` list: `T List` int_seg: `{i..j-}` uall: `∀[x:A]. B[x]` so_apply: `x[s]` member: `t ∈ T` set: `{x:A| B[x]} ` function: `x:A ⟶ B[x]` natural_number: `\$n` int: `ℤ` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` sq_type: `SQType(T)` all: `∀x:A. B[x]` implies: `P `` Q` guard: `{T}` tuple: `tuple(n;i.F[i])` nat: `ℕ` false: `False` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` top: `Top` and: `P ∧ Q` prop: `ℙ` le: `A ≤ B` int_seg: `{i..j-}` lelt: `i ≤ j < k` subtype_rel: `A ⊆r B` or: `P ∨ Q` select: `L[n]` nil: `[]` it: `⋅` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` upto: `upto(n)` from-upto: `[n, m)` ifthenelse: `if b then t else f fi ` lt_int: `i <z j` bfalse: `ff` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]` cons: `[a / b]` colength: `colength(L)` decidable: `Dec(P)` less_than: `a < b` squash: `↓T` so_lambda: `λ2x.t[x]` so_apply: `x[s]` less_than': `less_than'(a;b)` nat_plus: `ℕ+` true: `True` uiff: `uiff(P;Q)` assert: `↑b` btrue: `tt` subtract: `n - m` eq_int: `(i =z j)` bool: `𝔹` unit: `Unit` bnot: `¬bb` compose: `f o g` nequal: `a ≠ b ∈ T `
Lemmas referenced :  subtype_base_sq int_subtype_base nat_properties satisfiable-full-omega-tt 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 int_seg_wf length_wf select_wf int_seg_properties intformeq_wf int_formula_prop_eq_lemma equal-wf-T-base nat_wf colength_wf_list less_than_transitivity1 less_than_irreflexivity list_wf list-cases length_of_nil_lemma stuck-spread base_wf tupletype_nil_lemma map_nil_lemma list_ind_nil_lemma it_wf product_subtype_list spread_cons_lemma itermAdd_wf int_term_value_add_lemma decidable__le intformnot_wf int_formula_prop_not_lemma le_wf equal_wf decidable__lt subtract_wf itermSubtract_wf int_term_value_subtract_lemma set_subtype_base decidable__equal_int length_of_cons_lemma tupletype_cons_lemma upto_decomp2 add_nat_plus length_wf_nat nat_plus_wf nat_plus_properties add-is-int-iff false_wf map_cons_lemma list_ind_cons_lemma cons_wf non_neg_length set_wf equal-wf-base-T null-map null-upto decidable__assert null_wf null_nil_lemma lelt_wf null_cons_lemma bool_wf eqtt_to_assert assert_of_null eq_int_wf assert_of_eq_int btrue_wf not_assert_elim and_wf btrue_neq_bfalse eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot neg_assert_of_eq_int map-map nil_wf add-subtract-cancel add-member-int_seg2 subtype_rel-equal select-cons-tl
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut setElimination thin rename instantiate extract_by_obid sqequalHypSubstitution isectElimination cumulativity intEquality independent_isectElimination hypothesis dependent_functionElimination hypothesisEquality equalityTransitivity equalitySymmetry independent_functionElimination lambdaFormation sqequalRule intWeakElimination natural_numberEquality dependent_pairFormation lambdaEquality int_eqEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll axiomEquality functionEquality productElimination universeEquality because_Cache applyLambdaEquality applyEquality unionElimination baseClosed promote_hyp hypothesis_subsumption dependent_set_memberEquality addEquality imageElimination imageMemberEquality pointwiseFunctionality baseApply closedConclusion functionExtensionality equalityElimination independent_pairEquality

Latex:
\mforall{}[L:Type  List].  \mforall{}[F:i:\mBbbN{}||L||  {}\mrightarrow{}  L[i]].  \mforall{}[n:\{n:\mBbbZ{}|  n  =  ||L||\}  ].    (tuple(n;i.F[i])  \mmember{}  tuple-type(L))

Date html generated: 2017_04_17-AM-09_29_13
Last ObjectModification: 2017_02_27-PM-05_29_50

Theory : tuples

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