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Quelle rs_partition.pvs Sprache: PVS

rs_partition[T:TYPE+ from real]: THEORY
%----------------------------------------------------------------------------
%    Partitions for Riemann-Stieltjes Integration
%----------------------------------------------------------------------------
BEGIN

   ASSUMING
      IMPORTING deriv_domain_def[T]

      connected_domain : ASSUMPTION connected?[T]

      not_one_element : ASSUMPTION not_one_element?[T]

   ENDASSUMING

   IMPORTING finite_sets@finite_sets_minmax[real,<=],
         structures@sort_fseq[T,<=],
      structures@fseqs_ops[T]

   AUTO_REWRITE+ finseq_appl

   a,b,c,x:VAR T
   f,g,h: VAR [T -> real]
   n: VAR nat

   IMPORTING reals@intervals_real[T]

   bounded_on?(a,b,f): bool = (EXISTS (B: real):
                   (FORALL (x: closed_interval(a,b)): abs(f(x)) <= B))

% When the riemann integral was orginally defined, a partition was striclty increasing. Here, it is defined
% only as increasing.

   partition_pred?(a:T,b:{x:T|a<x})(fs:fseq): MACRO bool =
           (Let N = fs`length, xx = fs`seq IN
                      N > 1 AND xx(0) = a AND xx(N-1) = b AND
        increasing?(fs) AND
        (FORALL (i:below(N)): a <= xx(i) AND xx(i) <= b))

   partition(a:T,b:{x:T|a<x}): TYPE = (partition_pred?(a,b))

   partition_strictly_sort: LEMMA a<b IMPLIES FORALL (P:partition(a,b)): partition_pred?(a,b)(strictly_sort(P))

   width(a:T, b:{x:T|a<x}, P: partition(a,b)): posreal =
             LET xx = P`seq, N = P`length IN
                      max({ l: real | EXISTS (ii: below(N-1)):
                                         l = xx(ii+1) - xx(ii)})

   width_lem     : LEMMA (FORALL (a:T), (b:{x:T|a<x}),
                            (P: partition(a,b)),(ii: below(P`length-1)):
                         width(a,b,P) >= P(ii+1) - P(ii))

   width_lem_exists: LEMMA a<b IMPLIES FORALL (P:partition(a,b)):
          EXISTS (ii:below(P`length-1)): width(a,b,P) = P(ii+1)-P(ii)

   parts_order   : LEMMA  FORALL (P: partition(a,b), ii,jj: below(P`length)):
                              ii <= jj IMPLIES P`seq(ii) <= P`seq(jj)

   parts_disjoint: LEMMA FORALL (P: partition(a,b), ii,jj: below(P`length-1)):
                             P`seq(ii) < x AND x < P`seq(1 + ii) AND
                             P`seq(jj) < x AND x < P`seq(1 + jj)
                             IMPLIES
                                jj = ii

   in_part?(a:T, b:{x:T|a<x},P: partition(a,b),xx:T): MACRO bool =
                        EXISTS (ii: below(P`length-1)): xx = P`seq(ii)

   in_sect?(a:T, b:{x:T|a<x},P: partition(a,b),
            ii: below(P`length-1),xx:T): MACRO bool =
                      P`seq(ii) < xx AND xx < P`seq(ii+1)

   part_in       : LEMMA FORALL (P: partition(a,b)):
                             a < b AND a <= x AND x <= b IMPLIES
                                 (EXISTS (ii: below(P`length-1)):
                                       P`seq(ii) <= x AND x <= P`seq(ii+1))

   part_in_strict_left       : LEMMA FORALL (P: partition(a,b)):
                             a < b AND a < x AND x <= b IMPLIES
                                 (EXISTS (ii: below(P`length-1)):
                                       P`seq(ii) < x AND x <= P`seq(ii+1))

   part_not_in       : LEMMA a < b IMPLIES FORALL (P: partition(a,b)):
                                FORALL (ii,jj: below(P`length-1)):
                                  P`seq(ii) < x AND x < P`seq(ii+1)
                                IMPLIES x /= P`seq(jj)

   Prop: VAR [T -> bool]
   part_induction: LEMMA  (FORALL (P: partition(a,b)):
                              (FORALL ( x: closed_interval(a,b)):
                           LET xx = P`seq, N = P`length IN
                   (FORALL (ii : below(N-1)):
                           xx(ii) <= x AND x <= xx(ii+1) IMPLIES
                                Prop(x))
                  IMPLIES Prop(x) ) )

   IMPORTING reals@sigma_below, reals@sigma_upto

   eq_partition(a:T,b:{x:T|a<x},N: above(1)): partition(a,b) =
                  (# length := N,
                     seq := (LAMBDA (ii: nat): IF ii<N THEN a + ii*(b-a)/(N-1) ELSE default ENDIF) #)

   N: VAR above(1)

   len_eq_part  : LEMMA a < b IMPLIES length(eq_partition(a,b,N)) = N
   eq_part_lem_a: LEMMA a < b IMPLIES seq(eq_partition(a,b,N))(0) = a
   eq_part_lem_b: LEMMA a < b IMPLIES seq(eq_partition(a,b,N))(N-1) = b

   width_eq_part: LEMMA a < b IMPLIES
                            width(a,b,eq_partition(a,b,N)) = (b-a)/(N-1)

   delta: VAR posreal
   N_from_delta: LEMMA a < b IMPLIES
                          LET N = 2 + floor((b - a) / delta),
                              EP = eq_partition(a, b, N) IN
                          width(a, b, EP) < delta

   partition_exists: LEMMA a < b IMPLIES
                        (EXISTS (P: partition(a,b)): width(a,b,P) < delta)

   % Joining Partitions

   partjoin(a:T,b: {bb:T | a<bb}, c: {cc:T | b<cc})(Pab: partition(a,b),Pbc: partition(b,c)): partition(a,c) =
     concat(Pab,delete(0,Pbc))

   partjoin_def: LEMMA a<b AND b<c IMPLIES FORALL (Pab: partition(a,b),Pbc: partition(b,c)):
      LET Pac = partjoin(a,b,c)(Pab,Pbc) IN
   (n < Pab`length IMPLIES Pac`seq(n) = Pab`seq(n)) AND
   (Pab`length-1 <= n AND n<Pac`length IMPLIES Pac`seq(n) = Pbc`seq(n-Pab`length+1))

   partjoin_width: LEMMA a<b AND b<c IMPLIES FORALL (Pab: partition(a,b),Pbc: partition(b,c)):
        width(a,c,partjoin(a,b,c)(Pab,Pbc)) = max(width(a,b,Pab),width(b,c,Pbc))

   partition_union(a,(b|a<b))(P,Q: partition(a,b)): {PQ: partition(a,b) |
      (FORALL (x:T): member(x,PQ) IFF (member(x,P) OR member(x,Q))) AND
   strictly_increasing?(PQ)}

   partition_union_sym: LEMMA a<b IMPLIES FORALL (P,Q:partition(a,b)):
      partition_union(a,b)(P,Q) = partition_union(a,b)(Q,P)

   partition_union_unique: LEMMA a<b IMPLIES FORALL (P,Q,PQ:partition(a,b)):
         ((FORALL (x:T): member(x,PQ) IFF (member(x,P) OR member(x,Q))) AND
   strictly_increasing?(PQ))
      IFF
      PQ = partition_union(a,b)(P,Q)

   partition_union_width: LEMMA a<b IMPLIES FORALL (P,Q:partition(a,b)):
        width(a,b,partition_union(a,b)(P,Q)) <= min(width(a,b,P),width(a,b,Q))

   partition_strictly_sort_union: LEMMA a<b IMPLIES FORALL (P,Q:partition(a,b)):
         partition_union(a,b)(P,Q) = partition_union(a,b)(strictly_sort(P),Q)

   partition_union_is_strictly_sort: LEMMA a<b IMPLIES FORALL (P:partition(a,b)):
            partition_union(a,b)(P,P) = strictly_sort(P)

   partition_union_map(a,(b|a<b),P,Q:partition(a,b)): {pm:[below(P`length)->below(partition_union(a,b)(P,Q)`length)] |
               FORALL (ii:below(P`length)): P`seq(ii) = partition_union(a,b)(P,Q)`seq(pm(ii))}

   partition_union_map_unique: LEMMA a<b IMPLIES FORALL (P,Q:partition(a,b),pm:[below(P`length)->below(partition_union(a,b)(P,Q)`length)]):
          (FORALL (ii:below(P`length)): P`seq(ii) = partition_union(a,b)(P,Q)`seq(pm(ii)))
          IMPLIES
          pm = partition_union_map(a,b,P,Q)

   partition_union_map_increasing: LEMMA a<b IMPLIES FORALL (P,Q:partition(a,b),ii,jj:below(P`length)):
          strictly_increasing?(P) AND ii<jj IMPLIES partition_union_map(a,b,P,Q)(ii) < partition_union_map(a,b,P,Q)(jj)

   partition_union_strictly_sort_map_inv: LEMMA a<b IMPLIES FORALL (P:partition(a,b),ii:below(strictly_sort(P)`length)):
             partition_union_map(a,b,P,P)(strictly_sort_map(P)(ii)) = ii

   partition_union_map_inv(a,(b|a<b),P,Q:partition(a,b)): {pm:[below(partition_union(a,b)(P,Q)`length)->below(P`length)] |
      FORALL (jj:below(partition_union(a,b)(P,Q)`length)):
   P`seq(pm(jj)) <= partition_union(a,b)(P,Q)`seq(jj) AND (pm(jj) < P`length-1 IMPLIES partition_union(a,b)(P,Q)`seq(jj) < P`seq(pm(jj)+1))}

   partition_union_map_inv_def: LEMMA a<b IMPLIES FORALL (P,Q:partition(a,b)):
          LET PQ = partition_union(a,b)(P,Q),
        pum = partition_union_map(a,b,P,Q),
    puminv = partition_union_map_inv(a,b,P,Q)
       IN
       FORALL (ii:below(PQ`length),jj:below(P`length)):
    (pum(jj) <= ii AND (jj<P`length-1 IMPLIES ii < pum(jj+1)))
    IMPLIES
    puminv(ii) = jj

   partition_sort_inv_map: LEMMA a<b IMPLIES FORALL (P:partition(a,b)):
         LET SSP = partition_union(a,b)(P,P),
          sig = partition_union_map_inv(a,b,P,P)
      IN
          FORALL (ii:below(SSP`length)): SSP`seq(ii) = P`seq(sig(ii))

END rs_partition

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