Nuprl Lemma : partition-split-cons-mesh

∀[I:Interval]
  ∀[a:ℝ]. ∀[bs:ℝ List].
    (partitions(I;[a / bs])
    ⇒ (partitions([left-endpoint(I), a];[])
       ∧ partitions([a, right-endpoint(I)];bs)
       ∧ (left-endpoint(I) ≤ a)
       ∧ (a ≤ right-endpoint(I))
       ∧ (partition-mesh([left-endpoint(I), a];[]) ≤ partition-mesh(I;[a / bs]))
       ∧ (partition-mesh([a, right-endpoint(I)];bs) ≤ partition-mesh(I;[a / bs])))) 
  supposing icompact(I)


Proof



Definitions occuring in Statement :  partition-mesh: partition-mesh(I;p),  partitions: partitions(I;p),  icompact: icompact(I),  rccint: [l, u],  right-endpoint: right-endpoint(I),  left-endpoint: left-endpoint(I),  interval: Interval,  rleq: x ≤ y,  real: ℝ,  cons: [a / b],  nil: [],  list: T List,  uimplies: b supposing a,  uall: ∀[x:A]. B[x],  implies: P ⇒ Q,  and: P ∧ Q
Definitions unfolded in proof :  uall: ∀[x:A]. B[x],  uimplies: b supposing a,  implies: P ⇒ Q,  member: t ∈ T,  and: P ∧ Q,  cand: A c∧ B,  partitions: partitions(I;p),  all: ∀x:A. B[x],  top: Top,  select: L[n],  cons: [a / b],  nat_plus: ℕ+,  less_than: a < b,  squash: ↓T,  less_than': less_than'(a;b),  true: True,  prop: ℙ,  guard: {T},  decidable: Dec(P),  or: P ∨ Q,  false: False,  uiff: uiff(P;Q),  satisfiable_int_formula: satisfiable_int_formula(fmla),  exists: ∃x:A. B[x],  not: ¬A,  iff: P ⇐⇒ Q,  subtype_rel: A ⊆r B,  icompact: icompact(I),  sorted-by: sorted-by(R;L),  int_seg: {i..j-},  lelt: i ≤ j < k,  ge: i ≥ j ,  le: A ≤ B,  last: last(L),  subtract: n - m,  partition: partition(I),  full-partition: full-partition(I;p),  append: as @ bs,  so_lambda: so_lambda(x,y,z.t[x; y; z]),  so_apply: x[s1;s2;s3],  rbetween: x≤y≤z,  left-endpoint: left-endpoint(I),  i-length: |I|,  endpoints: endpoints(I),  sq_stable: SqStable(P),  partition-mesh: partition-mesh(I;p),  rleq: x ≤ y,  rnonneg: rnonneg(x),  sq_type: SQType(T),  rccint: [l, u],  outl: outl(x),  pi1: fst(t)
Lemmas referenced :  partition-split-cons,  length_of_cons_lemma,  add_nat_plus,  length_wf_nat,  real_wf,  less_than_wf,  nat_plus_wf,  nat_plus_properties,  decidable__lt,  add-is-int-iff,  satisfiable-full-omega-tt,  intformand_wf,  intformnot_wf,  intformless_wf,  itermConstant_wf,  itermVar_wf,  itermAdd_wf,  intformeq_wf,  int_formula_prop_and_lemma,  int_formula_prop_not_lemma,  int_formula_prop_less_lemma,  int_term_value_constant_lemma,  int_term_value_var_lemma,  int_term_value_add_lemma,  int_formula_prop_eq_lemma,  int_formula_prop_wf,  false_wf,  equal_wf,  partitions_wf,  cons_wf,  list_wf,  icompact_wf,  interval_wf,  frs-non-dec-sorted-by,  rleq_transitivity,  last_wf,  assert_elim,  null_wf3,  subtype_rel_list,  top_wf,  null_cons_lemma,  bfalse_wf,  btrue_neq_bfalse,  assert_wf,  right-endpoint_wf,  non_neg_length,  decidable__le,  intformle_wf,  int_formula_prop_le_lemma,  length_wf,  lelt_wf,  add-subtract-cancel,  list-cases,  length_of_nil_lemma,  product_subtype_list,  rleq_weakening_equal,  partition-mesh-nil,  adjacent-full-partition-points,  list_ind_cons_lemma,  length-append,  subtract_wf,  itermSubtract_wf,  int_term_value_subtract_lemma,  append_wf,  nil_wf,  right_endpoint_rccint_lemma,  left_endpoint_rccint_lemma,  sq_stable__rleq,  partition-mesh_wf,  rccint_wf,  rccint-icompact,  frs-mesh-bound,  full-partition_wf,  frs-mesh_wf,  less_than'_wf,  rsub_wf,  select_wf,  int_seg_properties,  subtract-is-int-iff,  int_seg_wf,  partition-mesh-nonneg,  add-member-int_seg2,  select-cons-tl,  subtype_base_sq,  int_subtype_base,  decidable__equal_int
Rules used in proof :  cut,  introduction,  extract_by_obid,  sqequalSubstitution,  sqequalTransitivity,  computationStep,  sqequalReflexivity,  isect_memberFormation,  lambdaFormation,  hypothesis,  sqequalHypSubstitution,  isectElimination,  thin,  hypothesisEquality,  independent_isectElimination,  independent_functionElimination,  productElimination,  independent_pairFormation,  sqequalRule,  dependent_functionElimination,  isect_memberEquality,  voidElimination,  voidEquality,  dependent_set_memberEquality,  natural_numberEquality,  imageMemberEquality,  baseClosed,  equalityTransitivity,  equalitySymmetry,  applyLambdaEquality,  setElimination,  rename,  unionElimination,  pointwiseFunctionality,  promote_hyp,  baseApply,  closedConclusion,  dependent_pairFormation,  lambdaEquality,  int_eqEquality,  intEquality,  computeAll,  because_Cache,  addLevel,  applyEquality,  levelHypothesis,  addEquality,  hypothesis_subsumption,  imageElimination,  independent_pairEquality,  minusEquality,  axiomEquality,  instantiate,  cumulativity
Latex:
\mforall{}[I:Interval]
    \mforall{}[a:\mBbbR{}].  \mforall{}[bs:\mBbbR{}  List].
        (partitions(I;[a  /  bs])
        {}\mRightarrow{}  (partitions([left-endpoint(I),  a];[])
              \mwedge{}  partitions([a,  right-endpoint(I)];bs)
              \mwedge{}  (left-endpoint(I)  \mleq{}  a)
              \mwedge{}  (a  \mleq{}  right-endpoint(I))
              \mwedge{}  (partition-mesh([left-endpoint(I),  a];[])  \mleq{}  partition-mesh(I;[a  /  bs]))
              \mwedge{}  (partition-mesh([a,  right-endpoint(I)];bs)  \mleq{}  partition-mesh(I;[a  /  bs])))) 
    supposing  icompact(I)


Date html generated: 2017_10_03-AM-09_41_52
Last ObjectModification: 2017_07_28-AM-07_56_51

Theory : reals


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