### Nuprl Lemma : ap-tuple-as-tuple

`∀[n:ℕ]. ∀[f,t:n-tuple(n)].  (ap-tuple(n;f;t) ~ tuple(n;i.f.i t.i))`

Proof

Definitions occuring in Statement :  select-tuple: `x.n` ap-tuple: `ap-tuple(len;f;t)` tuple: `tuple(n;i.F[i])` n-tuple: `n-tuple(n)` nat: `ℕ` uall: `∀[x:A]. B[x]` apply: `f a` 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` le: `A ≤ B` less_than': `less_than'(a;b)` so_lambda: `λ2x.t[x]` so_apply: `x[s]` ap-tuple: `ap-tuple(len;f;t)` eq_int: `(i =z j)` subtract: `n - m` ifthenelse: `if b then t else f fi ` btrue: `tt` sq_type: `SQType(T)` guard: `{T}` decidable: `Dec(P)` or: `P ∨ Q` bool: `𝔹` unit: `Unit` it: `⋅` uiff: `uiff(P;Q)` bfalse: `ff` bnot: `¬bb` assert: `↑b` select-tuple: `x.n` nequal: `a ≠ b ∈ T ` pi2: `snd(t)` tuple: `tuple(n;i.F[i])` pi1: `fst(t)` subtype_rel: `A ⊆r B` cons: `[a / b]` colength: `colength(L)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` nil: `[]` less_than: `a < b` squash: `↓T` int_seg: `{i..j-}` lelt: `i ≤ j < k`
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 n-tuple_wf n-tuple-decomp false_wf le_wf tuple-decomp subtype_base_sq unit_wf2 unit_subtype_base equal-unit it_wf decidable__le subtract_wf intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma eq_int_wf bool_wf eqtt_to_assert assert_of_eq_int intformeq_wf int_formula_prop_eq_lemma eqff_to_assert equal_wf bool_cases_sqequal bool_subtype_base assert-bnot neg_assert_of_eq_int nat_wf upto_wf list_wf int_seg_wf equal-wf-T-base colength_wf_list int_subtype_base list-cases map_nil_lemma product_subtype_list spread_cons_lemma itermAdd_wf int_term_value_add_lemma set_subtype_base decidable__equal_int map_cons_lemma int_seg_properties add-subtract-cancel
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis setElimination rename intWeakElimination lambdaFormation natural_numberEquality independent_isectElimination approximateComputation independent_functionElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation sqequalAxiom dependent_set_memberEquality because_Cache instantiate cumulativity equalityTransitivity equalitySymmetry unionElimination equalityElimination productElimination promote_hyp applyEquality hypothesis_subsumption applyLambdaEquality addEquality baseClosed imageElimination

Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[f,t:n-tuple(n)].    (ap-tuple(n;f;t)  \msim{}  tuple(n;i.f.i  t.i))

Date html generated: 2018_05_21-PM-00_52_22
Last ObjectModification: 2018_05_19-AM-06_40_40

Theory : tuples

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