### Nuprl Lemma : select_fun_ap_wf

`∀[n:ℕ]. ∀[m:ℕn]. ∀[A:ℕn ⟶ Type]. ∀[g:∀[T:Type]. (funtype(n;A;T) ⟶ T)].  (select_fun_ap(g;n;m) ∈ A m)`

Proof

Definitions occuring in Statement :  select_fun_ap: `select_fun_ap(g;n;m)` funtype: `funtype(n;A;T)` int_seg: `{i..j-}` nat: `ℕ` uall: `∀[x:A]. B[x]` member: `t ∈ T` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` select_fun_ap: `select_fun_ap(g;n;m)` subtype_rel: `A ⊆r B` nat: `ℕ` int_seg: `{i..j-}` lelt: `i ≤ j < k` and: `P ∧ Q` ge: `i ≥ j ` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` implies: `P `` Q` not: `¬A` top: `Top` prop: `ℙ` le: `A ≤ B` less_than: `a < b` guard: `{T}` uiff: `uiff(P;Q)` less_than': `less_than'(a;b)` so_lambda: `λ2x.t[x]` so_apply: `x[s]` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` nequal: `a ≠ b ∈ T ` subtract: `n - m`
Lemmas referenced :  int_seg_wf funtype_wf funtype-split nat_properties decidable__lt satisfiable-full-omega-tt intformand_wf intformnot_wf intformless_wf itermVar_wf itermAdd_wf itermConstant_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_var_lemma int_term_value_add_lemma int_term_value_constant_lemma int_formula_prop_wf lelt_wf mk_lambdas_wf subtract_wf int_seg_properties decidable__le intformle_wf itermSubtract_wf int_formula_prop_le_lemma int_term_value_subtract_lemma le_wf add-member-int_seg1 int_seg_subtype_nat false_wf subtype_rel_dep_function int_seg_subtype subtype_rel_self funtype-unroll 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 subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int int_subtype_base zero-add subtype_rel_weakening ext-eq_weakening uall_wf nat_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule applyEquality hypothesisEquality lambdaEquality isectElimination functionExtensionality extract_by_obid sqequalHypSubstitution thin natural_numberEquality setElimination rename hypothesis equalityTransitivity equalitySymmetry isectEquality universeEquality cumulativity functionEquality because_Cache dependent_set_memberEquality productElimination independent_pairFormation dependent_functionElimination addEquality unionElimination independent_isectElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll lambdaFormation instantiate equalityElimination promote_hyp independent_functionElimination axiomEquality

Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[m:\mBbbN{}n].  \mforall{}[A:\mBbbN{}n  {}\mrightarrow{}  Type].  \mforall{}[g:\mforall{}[T:Type].  (funtype(n;A;T)  {}\mrightarrow{}  T)].
(select\_fun\_ap(g;n;m)  \mmember{}  A  m)

Date html generated: 2017_10_01-AM-08_40_11
Last ObjectModification: 2017_07_26-PM-04_27_56

Theory : untyped!computation

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