Nuprl Lemma : select-as-reduce

[n:ℕ]. ∀[L:Top List].
  L[||L|| 1] 
  outr(reduce(λu,x. case of inl(i) => if (i =z n) then inr u  else inl (i 1) fi  inr(u) => x;inl 0;L)) 
  supposing n < ||L||


Definitions occuring in Statement :  select: L[n] length: ||as|| reduce: reduce(f;k;as) list: List nat: outr: outr(x) ifthenelse: if then else fi  eq_int: (i =z j) less_than: a < b uimplies: supposing a uall: [x:A]. B[x] top: Top lambda: λx.A[x] decide: case of inl(x) => s[x] inr(y) => t[y] inr: inr  inl: inl x subtract: m add: m natural_number: $n sqequal: t
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T all: x:A. B[x] nat: implies:  Q false: False ge: i ≥  uimplies: supposing a satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] not: ¬A top: Top and: P ∧ Q prop: subtype_rel: A ⊆B guard: {T} or: P ∨ Q select: L[n] nil: [] it: so_lambda: λ2y.t[x; y] so_apply: x[s1;s2] cons: [a b] colength: colength(L) decidable: Dec(P) so_lambda: λ2x.t[x] so_apply: x[s] sq_type: SQType(T) less_than: a < b squash: T less_than': less_than'(a;b) bool: 𝔹 unit: Unit btrue: tt uiff: uiff(P;Q) ifthenelse: if then else fi  bfalse: ff bnot: ¬bb assert: b le: A ≤ B iff: ⇐⇒ Q rev_implies:  Q int_seg: {i..j-} lelt: i ≤ j < k outr: outr(x)
Lemmas referenced :  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 equal-wf-T-base nat_wf colength_wf_list top_wf less_than_transitivity1 less_than_irreflexivity list-cases reduce_nil_lemma length_of_nil_lemma stuck-spread base_wf product_subtype_list spread_cons_lemma intformeq_wf itermAdd_wf int_formula_prop_eq_lemma int_term_value_add_lemma decidable__le intformnot_wf int_formula_prop_not_lemma le_wf equal_wf subtract_wf itermSubtract_wf int_term_value_subtract_lemma subtype_base_sq set_subtype_base int_subtype_base decidable__equal_int reduce_cons_lemma length_of_cons_lemma list_wf length_wf le_int_wf bool_wf assert_wf lt_int_wf bnot_wf eqtt_to_assert assert_of_le_int eq_int_wf assert_of_eq_int eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot neg_assert_of_eq_int add-is-int-iff false_wf not_wf uiff_transitivity assert_functionality_wrt_uiff bnot_of_le_int assert_of_lt_int iff_transitivity iff_weakening_uiff assert_of_bnot general_arith_equation1 select_cons_tl_sq decidable__lt lelt_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaFormation thin extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename intWeakElimination natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll independent_functionElimination sqequalAxiom applyEquality because_Cache unionElimination baseClosed promote_hyp hypothesis_subsumption productElimination equalityTransitivity equalitySymmetry applyLambdaEquality dependent_set_memberEquality addEquality instantiate cumulativity imageElimination equalityElimination pointwiseFunctionality baseApply closedConclusion impliesFunctionality

\mforall{}[n:\mBbbN{}].  \mforall{}[L:Top  List].
    L[||L||  -  n  +  1]  \msim{}  outr(reduce(\mlambda{}u,x.  case  x
                                                                              of  inl(i)  =>
                                                                              if  (i  =\msubz{}  n)  then  inr  u    else  inl  (i  +  1)  fi 
                                                                              |  inr(u)  =>
                                                                              x;inl  0;L)) 
    supposing  n  <  ||L||

Date html generated: 2018_05_21-PM-06_51_51
Last ObjectModification: 2017_07_26-PM-04_58_00

Theory : general

Home Index