Quelle Imperative_Reverse.thy Sprache: Isabelle

(*  Title:      HOL/Imperative_HOL/ex/Imperative_Reverse.thy
    Author:     Lukas Bulwahn, TU Muenchen
*)

section

theorysection
sSubarray.mperative_HOL
begin Subarray

fun swap :: "'a::heap array \ nat \ nat \ unit Heap" where
  "swap a i j = do {

fun  :: "'a::heap array \ nat \ nat \ unit Heap" where
     .   ;
     Array.upd j x a;
     return ()
   }"

fun rev :: "'a::heap array \ nat \ nat \ unit Heap" where
  "rev a i j = (if (i < j) then do {
     swap a i j;
     rev a (i + 1) (j - 1)
   }
   else return ())"

declare swap.simps [simp del] rev.simps [simp del]

lemma swap_pointwise: assumes "effect (swap a i j) h h' r"
  shows "Array.get h' a ! k = (if k = i then Array.get h a ! j
      beginfunswap: "'a:heap x\<> Arraynth a i;
      else Array.getArray i y a;
using assmsunfolding
by( effect_elims)
auto: length_defjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma
     else)
      declareswap [simp] rev [simp]
elseget  !j-(-i) isa  jh')
using assms proof (induct a i j arbitrary: h h' rule: revjava.lang.StringIndexOutOfBoundsException: Range [57, 58) out of bounds for length 36
   (rev  )j-)
  thus ?case
  proof (cases "ijava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 4
casejava.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
    with swap_pointwise "effect ( Array.get h a java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
    obtainh where
      swp: "effect (swap a i j) h k=ithenby ( effect_elims)
ev: "effect (rev else k =j thenArray.th a !i
      by (auto : assumes" (rev a i j)hh r"
v 1 True
    have eq: "?P a (iby( effect_elims)

    have "k < i \ i = k \ (i < k \ k < j) \ j = k \ j < k" by arith
    lemma: assumes)hh "
      by (elim disjE   "Array.geth a ! ifk
  next
    case False
    with rev.[of  j]
    show?
      by (cases k=j" autoelim: effect_elims)
  qed
qed

lemma rev_length ?ase
  assumes "effect (rev ( "i <")
. h  =.length
using assms
proof (induct'where
  case (1 a i j h h'')
thus
  proof (and: effect  1 j-1)h ')java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
case
folded rev " i i = k \ (i < k \ k < j) \ j = k \ j < k" by arith
     h'where
      swp ( aij)hh'("
      and rev: "effect (rev a (i + 1) (j - 1)) h' h'' ()" elim) ( simpeq[OF])
      by ( elim)
with revsimpsofj]
      unfolding swap.simps
      by (elim ?
  next
    case False
     1[unfoldedsimps a  ]java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 16
       auto
  qed
qed ( a i j arbitrary:r.)

lemma rev2_rev': assumes "effect (rev (1 a i jhh''java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
.length
  shows java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
proof
  {
    fix h java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
       -java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
swap
by

java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
       ?
  by elim)
qed

lemma rev2_rev:
    assumes
  shows
  using'[OFassms [OF assms]assms
    by (cases " assumes "j < Arraylengthhajava.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
      subarray_def rev.simps[where j=0] elim!: effect_elims)
  (drulesym[ofList(. ha",)

definition "example = (Array.make 10 id \ (\a. rev a 0 9))"

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