src/HOL/ex/Numeral.thy
author huffman
Mon, 16 Feb 2009 13:08:21 -0800
changeset 29943 922b931fd2eb
parent 29942 31069b8d39df
child 29944 ca43d393c2f1
permissions -rw-r--r--
datatype num = One | Dig0 num | Dig1 num
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
     1
(*  Title:      HOL/ex/Numeral.thy
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
     2
    ID:         $Id$
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
     3
    Author:     Florian Haftmann
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
     4
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
     5
An experimental alternative numeral representation.
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
     6
*)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
     7
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
     8
theory Numeral
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
     9
imports Int Inductive
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
    10
begin
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
    11
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
    12
subsection {* The @{text num} type *}
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
    13
29943
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    14
datatype num = One | Dig0 num | Dig1 num
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    15
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    16
text {* Increment function for type @{typ num} *}
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    17
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    18
primrec
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    19
  inc :: "num \<Rightarrow> num"
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    20
where
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    21
  "inc One = Dig0 One"
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    22
| "inc (Dig0 x) = Dig1 x"
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    23
| "inc (Dig1 x) = Dig0 (inc x)"
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    24
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    25
text {* Converting between type @{typ num} and type @{typ nat} *}
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    26
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    27
primrec
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    28
  nat_of_num :: "num \<Rightarrow> nat"
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    29
where
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    30
  "nat_of_num One = Suc 0"
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    31
| "nat_of_num (Dig0 x) = nat_of_num x + nat_of_num x"
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    32
| "nat_of_num (Dig1 x) = Suc (nat_of_num x + nat_of_num x)"
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    33
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    34
primrec
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    35
  num_of_nat :: "nat \<Rightarrow> num"
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    36
where
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    37
  "num_of_nat 0 = One"
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    38
| "num_of_nat (Suc n) = (if 0 < n then inc (num_of_nat n) else One)"
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    39
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    40
lemma nat_of_num_gt_0: "0 < nat_of_num x"
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    41
  by (induct x) simp_all
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    42
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    43
lemma nat_of_num_neq_0: " nat_of_num x \<noteq> 0"
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    44
  by (induct x) simp_all
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    45
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    46
lemma nat_of_num_inc: "nat_of_num (inc x) = Suc (nat_of_num x)"
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    47
  by (induct x) simp_all
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    48
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    49
lemma num_of_nat_double:
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    50
  "0 < n \<Longrightarrow> num_of_nat (n + n) = Dig0 (num_of_nat n)"
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    51
  by (induct n) simp_all
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    52
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
    53
text {*
29943
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    54
  Type @{typ num} is isomorphic to the strictly positive
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
    55
  natural numbers.
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
    56
*}
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
    57
29943
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    58
lemma nat_of_num_inverse: "num_of_nat (nat_of_num x) = x"
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    59
  by (induct x) (simp_all add: num_of_nat_double nat_of_num_gt_0)
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
    60
29943
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    61
lemma num_of_nat_inverse: "0 < n \<Longrightarrow> nat_of_num (num_of_nat n) = n"
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    62
  by (induct n) (simp_all add: nat_of_num_inc)
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
    63
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
    64
lemma num_eq_iff: "x = y \<longleftrightarrow> nat_of_num x = nat_of_num y"
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
    65
  apply safe
29943
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    66
  apply (drule arg_cong [where f=num_of_nat])
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
    67
  apply (simp add: nat_of_num_inverse)
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
    68
  done
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
    69
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
    70
instantiation num :: "semiring"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
    71
begin
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
    72
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
    73
definition plus_num :: "num \<Rightarrow> num \<Rightarrow> num" where
28562
4e74209f113e `code func` now just `code`
haftmann
parents: 28367
diff changeset
    74
  [code del]: "m + n = num_of_nat (nat_of_num m + nat_of_num n)"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
    75
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
    76
definition times_num :: "num \<Rightarrow> num \<Rightarrow> num" where
28562
4e74209f113e `code func` now just `code`
haftmann
parents: 28367
diff changeset
    77
  [code del]: "m * n = num_of_nat (nat_of_num m * nat_of_num n)"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
    78
29943
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    79
lemma nat_of_num_add: "nat_of_num (x + y) = nat_of_num x + nat_of_num y"
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    80
  unfolding plus_num_def
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    81
  by (intro num_of_nat_inverse add_pos_pos nat_of_num_gt_0)
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
    82
29943
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    83
lemma nat_of_num_mult: "nat_of_num (x * y) = nat_of_num x * nat_of_num y"
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    84
  unfolding times_num_def
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    85
  by (intro num_of_nat_inverse mult_pos_pos nat_of_num_gt_0)
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
    86
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
    87
instance proof
29943
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    88
qed (simp_all add: num_eq_iff nat_of_num_add nat_of_num_mult nat_distrib)
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
    89
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
    90
end
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
    91
29943
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    92
lemma num_induct [case_names One inc]:
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    93
  fixes P :: "num \<Rightarrow> bool"
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    94
  assumes One: "P One"
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    95
    and inc: "\<And>x. P x \<Longrightarrow> P (inc x)"
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    96
  shows "P x"
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    97
proof -
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    98
  obtain n where n: "Suc n = nat_of_num x"
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
    99
    by (cases "nat_of_num x", simp_all add: nat_of_num_neq_0)
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   100
  have "P (num_of_nat (Suc n))"
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   101
  proof (induct n)
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   102
    case 0 show ?case using One by simp
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   103
  next
29943
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   104
    case (Suc n)
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   105
    then have "P (inc (num_of_nat (Suc n)))" by (rule inc)
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   106
    then show "P (num_of_nat (Suc (Suc n)))" by simp
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   107
  qed
29943
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   108
  with n show "P x"
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   109
    by (simp add: nat_of_num_inverse)
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   110
qed
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   111
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   112
text {*
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   113
  From now on, there are two possible models for @{typ num}:
29943
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   114
  as positive naturals (rule @{text "num_induct"})
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   115
  and as digit representation (rules @{text "num.induct"}, @{text "num.cases"}).
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   116
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   117
  It is not entirely clear in which context it is better to use
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   118
  the one or the other, or whether the construction should be reversed.
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   119
*}
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   120
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   121
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   122
subsection {* Binary numerals *}
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   123
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   124
text {*
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   125
  We embed binary representations into a generic algebraic
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   126
  structure using @{text of_num}
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   127
*}
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   128
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   129
ML {*
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   130
structure DigSimps =
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   131
  NamedThmsFun(val name = "numeral"; val description = "Simplification rules for numerals")
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   132
*}
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   133
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   134
setup DigSimps.setup
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   135
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   136
class semiring_numeral = semiring + monoid_mult
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   137
begin
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   138
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   139
primrec of_num :: "num \<Rightarrow> 'a" where
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   140
  of_num_one [numeral]: "of_num One = 1"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   141
  | "of_num (Dig0 n) = of_num n + of_num n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   142
  | "of_num (Dig1 n) = of_num n + of_num n + 1"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   143
29943
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   144
lemma of_num_inc: "of_num (inc x) = of_num x + 1"
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   145
  by (induct x) (simp_all add: add_ac)
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   146
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   147
declare of_num.simps [simp del]
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   148
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   149
end
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   150
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   151
text {*
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   152
  ML stuff and syntax.
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   153
*}
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   154
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   155
ML {*
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   156
fun mk_num 1 = @{term One}
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   157
  | mk_num k =
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   158
      let
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   159
        val (l, b) = Integer.div_mod k 2;
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   160
        val bit = (if b = 0 then @{term Dig0} else @{term Dig1});
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   161
      in bit $ (mk_num l) end;
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   162
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   163
fun dest_num @{term One} = 1
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   164
  | dest_num (@{term Dig0} $ n) = 2 * dest_num n
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   165
  | dest_num (@{term Dig1} $ n) = 2 * dest_num n + 1;
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   166
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   167
(*FIXME these have to gain proper context via morphisms phi*)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   168
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   169
fun mk_numeral T k = Const (@{const_name of_num}, @{typ num} --> T)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   170
  $ mk_num k
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   171
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   172
fun dest_numeral (Const (@{const_name of_num}, Type ("fun", [@{typ num}, T])) $ t) =
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   173
  (T, dest_num t)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   174
*}
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   175
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   176
syntax
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   177
  "_Numerals" :: "xnum \<Rightarrow> 'a"    ("_")
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   178
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   179
parse_translation {*
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   180
let
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   181
  fun num_of_int n = if n > 0 then case IntInf.quotRem (n, 2)
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   182
     of (0, 1) => Const (@{const_name One}, dummyT)
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   183
      | (n, 0) => Const (@{const_name Dig0}, dummyT) $ num_of_int n
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   184
      | (n, 1) => Const (@{const_name Dig1}, dummyT) $ num_of_int n
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   185
    else raise Match;
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   186
  fun numeral_tr [Free (num, _)] =
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   187
        let
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   188
          val {leading_zeros, value, ...} = Syntax.read_xnum num;
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   189
          val _ = leading_zeros = 0 andalso value > 0
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   190
            orelse error ("Bad numeral: " ^ num);
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   191
        in Const (@{const_name of_num}, @{typ num} --> dummyT) $ num_of_int value end
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   192
    | numeral_tr ts = raise TERM ("numeral_tr", ts);
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   193
in [("_Numerals", numeral_tr)] end
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   194
*}
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   195
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   196
typed_print_translation {*
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   197
let
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   198
  fun dig b n = b + 2 * n; 
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   199
  fun int_of_num' (Const (@{const_syntax Dig0}, _) $ n) =
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   200
        dig 0 (int_of_num' n)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   201
    | int_of_num' (Const (@{const_syntax Dig1}, _) $ n) =
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   202
        dig 1 (int_of_num' n)
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   203
    | int_of_num' (Const (@{const_syntax One}, _)) = 1;
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   204
  fun num_tr' show_sorts T [n] =
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   205
    let
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   206
      val k = int_of_num' n;
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   207
      val t' = Syntax.const "_Numerals" $ Syntax.free ("#" ^ string_of_int k);
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   208
    in case T
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   209
     of Type ("fun", [_, T']) =>
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   210
         if not (! show_types) andalso can Term.dest_Type T' then t'
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   211
         else Syntax.const Syntax.constrainC $ t' $ Syntax.term_of_typ show_sorts T'
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   212
      | T' => if T' = dummyT then t' else raise Match
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   213
    end;
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   214
in [(@{const_syntax of_num}, num_tr')] end
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   215
*}
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   216
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   217
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   218
subsection {* Numeral operations *}
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   219
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   220
text {*
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   221
  First, addition and multiplication on digits.
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   222
*}
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   223
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   224
lemma Dig_plus [numeral, simp, code]:
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   225
  "One + One = Dig0 One"
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   226
  "One + Dig0 m = Dig1 m"
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   227
  "One + Dig1 m = Dig0 (m + One)"
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   228
  "Dig0 n + One = Dig1 n"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   229
  "Dig0 n + Dig0 m = Dig0 (n + m)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   230
  "Dig0 n + Dig1 m = Dig1 (n + m)"
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   231
  "Dig1 n + One = Dig0 (n + One)"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   232
  "Dig1 n + Dig0 m = Dig1 (n + m)"
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   233
  "Dig1 n + Dig1 m = Dig0 (n + m + One)"
29943
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   234
  by (simp_all add: num_eq_iff nat_of_num_add)
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   235
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   236
lemma Dig_times [numeral, simp, code]:
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   237
  "One * One = One"
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   238
  "One * Dig0 n = Dig0 n"
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   239
  "One * Dig1 n = Dig1 n"
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   240
  "Dig0 n * One = Dig0 n"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   241
  "Dig0 n * Dig0 m = Dig0 (n * Dig0 m)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   242
  "Dig0 n * Dig1 m = Dig0 (n * Dig1 m)"
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   243
  "Dig1 n * One = Dig1 n"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   244
  "Dig1 n * Dig0 m = Dig0 (n * Dig0 m + m)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   245
  "Dig1 n * Dig1 m = Dig1 (n * Dig1 m + m)"
29943
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   246
  by (simp_all add: num_eq_iff nat_of_num_add nat_of_num_mult
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   247
                    left_distrib right_distrib)
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   248
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   249
text {*
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   250
  @{const of_num} is a morphism.
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   251
*}
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   252
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   253
context semiring_numeral
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   254
begin
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   255
29943
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   256
abbreviation "Num1 \<equiv> of_num One"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   257
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   258
text {*
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   259
  Alas, there is still the duplication of @{term 1},
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   260
  thought the duplicated @{term 0} has disappeared.
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   261
  We could get rid of it by replacing the constructor
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   262
  @{term 1} in @{typ num} by two constructors
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   263
  @{text two} and @{text three}, resulting in a further
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   264
  blow-up.  But it could be worth the effort.
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   265
*}
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   266
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   267
lemma of_num_plus_one [numeral]:
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   268
  "of_num n + 1 = of_num (n + One)"
29943
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   269
  by (rule sym, induct n) (simp_all add: of_num.simps add_ac)
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   270
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   271
lemma of_num_one_plus [numeral]:
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   272
  "1 + of_num n = of_num (n + One)"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   273
  unfolding of_num_plus_one [symmetric] add_commute ..
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   274
29943
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   275
text {* Rules for addition in the One/inc view *}
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   276
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   277
lemma add_One: "x + One = inc x"
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   278
  by (simp add: num_eq_iff nat_of_num_add nat_of_num_inc)
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   279
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   280
lemma add_inc: "x + inc y = inc (x + y)"
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   281
  by (simp add: num_eq_iff nat_of_num_add nat_of_num_inc)
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   282
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   283
text {* Rules for multiplication in the One/inc view *}
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   284
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   285
lemma mult_One: "x * One = x"
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   286
  by (simp add: num_eq_iff nat_of_num_mult)
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   287
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   288
lemma mult_inc: "x * inc y = x * y + x"
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   289
  by (simp add: num_eq_iff nat_of_num_mult nat_of_num_add nat_of_num_inc)
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   290
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   291
lemma of_num_plus [numeral]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   292
  "of_num m + of_num n = of_num (m + n)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   293
  by (induct n rule: num_induct)
29943
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   294
     (simp_all add: add_One add_inc of_num_one of_num_inc add_ac)
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   295
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   296
lemma of_num_times_one [numeral]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   297
  "of_num n * 1 = of_num n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   298
  by simp
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   299
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   300
lemma of_num_one_times [numeral]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   301
  "1 * of_num n = of_num n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   302
  by simp
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   303
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   304
lemma of_num_times [numeral]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   305
  "of_num m * of_num n = of_num (m * n)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   306
  by (induct n rule: num_induct)
29943
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   307
    (simp_all add: of_num_plus [symmetric] mult_One mult_inc
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   308
    semiring_class.right_distrib right_distrib of_num_one of_num_inc)
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   309
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   310
end
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   311
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   312
text {*
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   313
  Structures with a @{term 0}.
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   314
*}
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   315
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   316
context semiring_1
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   317
begin
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   318
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   319
subclass semiring_numeral ..
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   320
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   321
lemma of_nat_of_num [numeral]: "of_nat (of_num n) = of_num n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   322
  by (induct n)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   323
    (simp_all add: semiring_numeral_class.of_num.simps of_num.simps add_ac)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   324
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   325
declare of_nat_1 [numeral]
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   326
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   327
lemma Dig_plus_zero [numeral]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   328
  "0 + 1 = 1"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   329
  "0 + of_num n = of_num n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   330
  "1 + 0 = 1"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   331
  "of_num n + 0 = of_num n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   332
  by simp_all
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   333
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   334
lemma Dig_times_zero [numeral]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   335
  "0 * 1 = 0"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   336
  "0 * of_num n = 0"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   337
  "1 * 0 = 0"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   338
  "of_num n * 0 = 0"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   339
  by simp_all
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   340
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   341
end
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   342
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   343
lemma nat_of_num_of_num: "nat_of_num = of_num"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   344
proof
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   345
  fix n
29943
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   346
  have "of_num n = nat_of_num n"
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   347
    by (induct n) (simp_all add: of_num.simps)
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   348
  then show "nat_of_num n = of_num n" by simp
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   349
qed
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   350
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   351
text {*
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   352
  Equality.
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   353
*}
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   354
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   355
context semiring_char_0
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   356
begin
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   357
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   358
lemma of_num_eq_iff [numeral]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   359
  "of_num m = of_num n \<longleftrightarrow> m = n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   360
  unfolding of_nat_of_num [symmetric] nat_of_num_of_num [symmetric]
29943
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   361
    of_nat_eq_iff num_eq_iff ..
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   362
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   363
lemma of_num_eq_one_iff [numeral]:
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   364
  "of_num n = 1 \<longleftrightarrow> n = One"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   365
proof -
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   366
  have "of_num n = of_num One \<longleftrightarrow> n = One" unfolding of_num_eq_iff ..
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   367
  then show ?thesis by (simp add: of_num_one)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   368
qed
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   369
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   370
lemma one_eq_of_num_iff [numeral]:
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   371
  "1 = of_num n \<longleftrightarrow> n = One"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   372
  unfolding of_num_eq_one_iff [symmetric] by auto
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   373
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   374
end
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   375
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   376
text {*
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   377
  Comparisons.  Could be perhaps more general than here.
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   378
*}
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   379
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   380
lemma (in ordered_semidom) of_num_pos: "0 < of_num n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   381
proof -
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   382
  have "(0::nat) < of_num n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   383
    by (induct n) (simp_all add: semiring_numeral_class.of_num.simps)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   384
  then have "of_nat 0 \<noteq> of_nat (of_num n)" 
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   385
    by (cases n) (simp_all only: semiring_numeral_class.of_num.simps of_nat_eq_iff)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   386
  then have "0 \<noteq> of_num n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   387
    by (simp add: of_nat_of_num)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   388
  moreover have "0 \<le> of_nat (of_num n)" by simp
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   389
  ultimately show ?thesis by (simp add: of_nat_of_num)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   390
qed
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   391
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   392
instantiation num :: linorder
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   393
begin
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   394
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   395
definition less_eq_num :: "num \<Rightarrow> num \<Rightarrow> bool" where
28562
4e74209f113e `code func` now just `code`
haftmann
parents: 28367
diff changeset
   396
  [code del]: "m \<le> n \<longleftrightarrow> nat_of_num m \<le> nat_of_num n"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   397
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   398
definition less_num :: "num \<Rightarrow> num \<Rightarrow> bool" where
28562
4e74209f113e `code func` now just `code`
haftmann
parents: 28367
diff changeset
   399
  [code del]: "m < n \<longleftrightarrow> nat_of_num m < nat_of_num n"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   400
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   401
instance proof
29943
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   402
qed (auto simp add: less_eq_num_def less_num_def num_eq_iff)
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   403
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   404
end
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   405
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   406
lemma less_eq_num_code [numeral, simp, code]:
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   407
  "One \<le> n \<longleftrightarrow> True"
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   408
  "Dig0 m \<le> One \<longleftrightarrow> False"
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   409
  "Dig1 m \<le> One \<longleftrightarrow> False"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   410
  "Dig0 m \<le> Dig0 n \<longleftrightarrow> m \<le> n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   411
  "Dig0 m \<le> Dig1 n \<longleftrightarrow> m \<le> n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   412
  "Dig1 m \<le> Dig1 n \<longleftrightarrow> m \<le> n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   413
  "Dig1 m \<le> Dig0 n \<longleftrightarrow> m < n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   414
  using of_num_pos [of n, where ?'a = nat] of_num_pos [of m, where ?'a = nat]
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   415
  by (auto simp add: less_eq_num_def less_num_def nat_of_num_of_num of_num.simps)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   416
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   417
lemma less_num_code [numeral, simp, code]:
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   418
  "m < One \<longleftrightarrow> False"
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   419
  "One < One \<longleftrightarrow> False"
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   420
  "One < Dig0 n \<longleftrightarrow> True"
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   421
  "One < Dig1 n \<longleftrightarrow> True"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   422
  "Dig0 m < Dig0 n \<longleftrightarrow> m < n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   423
  "Dig0 m < Dig1 n \<longleftrightarrow> m \<le> n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   424
  "Dig1 m < Dig1 n \<longleftrightarrow> m < n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   425
  "Dig1 m < Dig0 n \<longleftrightarrow> m < n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   426
  using of_num_pos [of n, where ?'a = nat] of_num_pos [of m, where ?'a = nat]
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   427
  by (auto simp add: less_eq_num_def less_num_def nat_of_num_of_num of_num.simps)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   428
  
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   429
context ordered_semidom
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   430
begin
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   431
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   432
lemma of_num_less_eq_iff [numeral]: "of_num m \<le> of_num n \<longleftrightarrow> m \<le> n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   433
proof -
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   434
  have "of_nat (of_num m) \<le> of_nat (of_num n) \<longleftrightarrow> m \<le> n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   435
    unfolding less_eq_num_def nat_of_num_of_num of_nat_le_iff ..
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   436
  then show ?thesis by (simp add: of_nat_of_num)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   437
qed
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   438
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   439
lemma of_num_less_eq_one_iff [numeral]: "of_num n \<le> 1 \<longleftrightarrow> n = One"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   440
proof -
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   441
  have "of_num n \<le> of_num One \<longleftrightarrow> n = One"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   442
    by (cases n) (simp_all add: of_num_less_eq_iff)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   443
  then show ?thesis by (simp add: of_num_one)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   444
qed
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   445
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   446
lemma one_less_eq_of_num_iff [numeral]: "1 \<le> of_num n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   447
proof -
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   448
  have "of_num One \<le> of_num n"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   449
    by (cases n) (simp_all add: of_num_less_eq_iff)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   450
  then show ?thesis by (simp add: of_num_one)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   451
qed
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   452
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   453
lemma of_num_less_iff [numeral]: "of_num m < of_num n \<longleftrightarrow> m < n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   454
proof -
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   455
  have "of_nat (of_num m) < of_nat (of_num n) \<longleftrightarrow> m < n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   456
    unfolding less_num_def nat_of_num_of_num of_nat_less_iff ..
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   457
  then show ?thesis by (simp add: of_nat_of_num)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   458
qed
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   459
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   460
lemma of_num_less_one_iff [numeral]: "\<not> of_num n < 1"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   461
proof -
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   462
  have "\<not> of_num n < of_num One"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   463
    by (cases n) (simp_all add: of_num_less_iff)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   464
  then show ?thesis by (simp add: of_num_one)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   465
qed
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   466
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   467
lemma one_less_of_num_iff [numeral]: "1 < of_num n \<longleftrightarrow> n \<noteq> One"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   468
proof -
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   469
  have "of_num One < of_num n \<longleftrightarrow> n \<noteq> One"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   470
    by (cases n) (simp_all add: of_num_less_iff)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   471
  then show ?thesis by (simp add: of_num_one)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   472
qed
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   473
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   474
end
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   475
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   476
text {*
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   477
  Structures with subtraction @{term "op -"}.
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   478
*}
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   479
29941
b951d80774d5 replace dec with double-and-decrement function
huffman
parents: 29667
diff changeset
   480
text {* A double-and-decrement function *}
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   481
29941
b951d80774d5 replace dec with double-and-decrement function
huffman
parents: 29667
diff changeset
   482
primrec DigM :: "num \<Rightarrow> num" where
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   483
  "DigM One = One"
29941
b951d80774d5 replace dec with double-and-decrement function
huffman
parents: 29667
diff changeset
   484
  | "DigM (Dig0 n) = Dig1 (DigM n)"
b951d80774d5 replace dec with double-and-decrement function
huffman
parents: 29667
diff changeset
   485
  | "DigM (Dig1 n) = Dig1 (Dig0 n)"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   486
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   487
lemma DigM_plus_one: "DigM n + One = Dig0 n"
29941
b951d80774d5 replace dec with double-and-decrement function
huffman
parents: 29667
diff changeset
   488
  by (induct n) simp_all
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   489
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   490
lemma one_plus_DigM: "One + DigM n = Dig0 n"
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   491
  unfolding add_commute [of One] DigM_plus_one ..
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   492
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   493
class semiring_minus = semiring + minus + zero +
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   494
  assumes minus_inverts_plus1: "a + b = c \<Longrightarrow> c - b = a"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   495
  assumes minus_minus_zero_inverts_plus1: "a + b = c \<Longrightarrow> b - c = 0 - a"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   496
begin
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   497
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   498
lemma minus_inverts_plus2: "a + b = c \<Longrightarrow> c - a = b"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   499
  by (simp add: add_ac minus_inverts_plus1 [of b a])
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   500
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   501
lemma minus_minus_zero_inverts_plus2: "a + b = c \<Longrightarrow> a - c = 0 - b"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   502
  by (simp add: add_ac minus_minus_zero_inverts_plus1 [of b a])
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   503
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   504
end
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   505
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   506
class semiring_1_minus = semiring_1 + semiring_minus
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   507
begin
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   508
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   509
lemma Dig_of_num_pos:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   510
  assumes "k + n = m"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   511
  shows "of_num m - of_num n = of_num k"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   512
  using assms by (simp add: of_num_plus minus_inverts_plus1)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   513
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   514
lemma Dig_of_num_zero:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   515
  shows "of_num n - of_num n = 0"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   516
  by (rule minus_inverts_plus1) simp
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   517
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   518
lemma Dig_of_num_neg:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   519
  assumes "k + m = n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   520
  shows "of_num m - of_num n = 0 - of_num k"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   521
  by (rule minus_minus_zero_inverts_plus1) (simp add: of_num_plus assms)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   522
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   523
lemmas Dig_plus_eval =
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   524
  of_num_plus of_num_eq_iff Dig_plus refl [of One, THEN eqTrueI] num.inject
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   525
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   526
simproc_setup numeral_minus ("of_num m - of_num n") = {*
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   527
  let
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   528
    (*TODO proper implicit use of morphism via pattern antiquotations*)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   529
    fun cdest_of_num ct = (snd o split_last o snd o Drule.strip_comb) ct;
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   530
    fun cdest_minus ct = case (rev o snd o Drule.strip_comb) ct of [n, m] => (m, n);
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   531
    fun attach_num ct = (dest_num (Thm.term_of ct), ct);
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   532
    fun cdifference t = (pairself (attach_num o cdest_of_num) o cdest_minus) t;
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   533
    val simplify = MetaSimplifier.rewrite false (map mk_meta_eq @{thms Dig_plus_eval});
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   534
    fun cert ck cl cj = @{thm eqTrueE} OF [@{thm meta_eq_to_obj_eq} OF [simplify (Drule.list_comb (@{cterm "op = :: num \<Rightarrow> _"},
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   535
      [Drule.list_comb (@{cterm "op + :: num \<Rightarrow> _"}, [ck, cl]), cj]))]];
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   536
  in fn phi => fn _ => fn ct => case try cdifference ct
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   537
   of NONE => (NONE)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   538
    | SOME ((k, ck), (l, cl)) => SOME (let val j = k - l in if j = 0
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   539
        then MetaSimplifier.rewrite false [mk_meta_eq (Morphism.thm phi @{thm Dig_of_num_zero})] ct
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   540
        else mk_meta_eq (let
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   541
          val cj = Thm.cterm_of (Thm.theory_of_cterm ct) (mk_num (abs j));
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   542
        in
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   543
          (if j > 0 then (Morphism.thm phi @{thm Dig_of_num_pos}) OF [cert cj cl ck]
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   544
          else (Morphism.thm phi @{thm Dig_of_num_neg}) OF [cert cj ck cl])
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   545
        end) end)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   546
  end
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   547
*}
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   548
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   549
lemma Dig_of_num_minus_zero [numeral]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   550
  "of_num n - 0 = of_num n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   551
  by (simp add: minus_inverts_plus1)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   552
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   553
lemma Dig_one_minus_zero [numeral]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   554
  "1 - 0 = 1"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   555
  by (simp add: minus_inverts_plus1)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   556
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   557
lemma Dig_one_minus_one [numeral]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   558
  "1 - 1 = 0"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   559
  by (simp add: minus_inverts_plus1)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   560
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   561
lemma Dig_of_num_minus_one [numeral]:
29941
b951d80774d5 replace dec with double-and-decrement function
huffman
parents: 29667
diff changeset
   562
  "of_num (Dig0 n) - 1 = of_num (DigM n)"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   563
  "of_num (Dig1 n) - 1 = of_num (Dig0 n)"
29941
b951d80774d5 replace dec with double-and-decrement function
huffman
parents: 29667
diff changeset
   564
  by (auto intro: minus_inverts_plus1 simp add: DigM_plus_one of_num.simps of_num_plus_one)
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   565
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   566
lemma Dig_one_minus_of_num [numeral]:
29941
b951d80774d5 replace dec with double-and-decrement function
huffman
parents: 29667
diff changeset
   567
  "1 - of_num (Dig0 n) = 0 - of_num (DigM n)"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   568
  "1 - of_num (Dig1 n) = 0 - of_num (Dig0 n)"
29941
b951d80774d5 replace dec with double-and-decrement function
huffman
parents: 29667
diff changeset
   569
  by (auto intro: minus_minus_zero_inverts_plus1 simp add: DigM_plus_one of_num.simps of_num_plus_one)
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   570
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   571
end
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   572
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   573
context ring_1
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   574
begin
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   575
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   576
subclass semiring_1_minus
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 28823
diff changeset
   577
  proof qed (simp_all add: algebra_simps)
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   578
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   579
lemma Dig_zero_minus_of_num [numeral]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   580
  "0 - of_num n = - of_num n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   581
  by simp
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   582
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   583
lemma Dig_zero_minus_one [numeral]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   584
  "0 - 1 = - 1"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   585
  by simp
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   586
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   587
lemma Dig_uminus_uminus [numeral]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   588
  "- (- of_num n) = of_num n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   589
  by simp
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   590
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   591
lemma Dig_plus_uminus [numeral]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   592
  "of_num m + - of_num n = of_num m - of_num n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   593
  "- of_num m + of_num n = of_num n - of_num m"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   594
  "- of_num m + - of_num n = - (of_num m + of_num n)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   595
  "of_num m - - of_num n = of_num m + of_num n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   596
  "- of_num m - of_num n = - (of_num m + of_num n)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   597
  "- of_num m - - of_num n = of_num n - of_num m"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   598
  by (simp_all add: diff_minus add_commute)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   599
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   600
lemma Dig_times_uminus [numeral]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   601
  "- of_num n * of_num m = - (of_num n * of_num m)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   602
  "of_num n * - of_num m = - (of_num n * of_num m)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   603
  "- of_num n * - of_num m = of_num n * of_num m"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   604
  by (simp_all add: minus_mult_left [symmetric] minus_mult_right [symmetric])
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   605
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   606
lemma of_int_of_num [numeral]: "of_int (of_num n) = of_num n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   607
by (induct n)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   608
  (simp_all only: of_num.simps semiring_numeral_class.of_num.simps of_int_add, simp_all)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   609
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   610
declare of_int_1 [numeral]
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   611
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   612
end
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   613
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   614
text {*
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   615
  Greetings to @{typ nat}.
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   616
*}
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   617
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   618
instance nat :: semiring_1_minus proof qed simp_all
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   619
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   620
lemma Suc_of_num [numeral]: "Suc (of_num n) = of_num (n + One)"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   621
  unfolding of_num_plus_one [symmetric] by simp
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   622
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   623
lemma nat_number:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   624
  "1 = Suc 0"
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   625
  "of_num One = Suc 0"
29941
b951d80774d5 replace dec with double-and-decrement function
huffman
parents: 29667
diff changeset
   626
  "of_num (Dig0 n) = Suc (of_num (DigM n))"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   627
  "of_num (Dig1 n) = Suc (of_num (Dig0 n))"
29941
b951d80774d5 replace dec with double-and-decrement function
huffman
parents: 29667
diff changeset
   628
  by (simp_all add: of_num.simps DigM_plus_one Suc_of_num)
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   629
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   630
declare diff_0_eq_0 [numeral]
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   631
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   632
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   633
subsection {* Code generator setup for @{typ int} *}
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   634
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   635
definition Pls :: "num \<Rightarrow> int" where
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   636
  [simp, code post]: "Pls n = of_num n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   637
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   638
definition Mns :: "num \<Rightarrow> int" where
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   639
  [simp, code post]: "Mns n = - of_num n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   640
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   641
code_datatype "0::int" Pls Mns
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   642
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   643
lemmas [code inline] = Pls_def [symmetric] Mns_def [symmetric]
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   644
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   645
definition sub :: "num \<Rightarrow> num \<Rightarrow> int" where
28562
4e74209f113e `code func` now just `code`
haftmann
parents: 28367
diff changeset
   646
  [simp, code del]: "sub m n = (of_num m - of_num n)"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   647
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   648
definition dup :: "int \<Rightarrow> int" where
28562
4e74209f113e `code func` now just `code`
haftmann
parents: 28367
diff changeset
   649
  [code del]: "dup k = 2 * k"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   650
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   651
lemma Dig_sub [code]:
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   652
  "sub One One = 0"
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   653
  "sub (Dig0 m) One = of_num (DigM m)"
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   654
  "sub (Dig1 m) One = of_num (Dig0 m)"
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   655
  "sub One (Dig0 n) = - of_num (DigM n)"
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   656
  "sub One (Dig1 n) = - of_num (Dig0 n)"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   657
  "sub (Dig0 m) (Dig0 n) = dup (sub m n)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   658
  "sub (Dig1 m) (Dig1 n) = dup (sub m n)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   659
  "sub (Dig1 m) (Dig0 n) = dup (sub m n) + 1"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   660
  "sub (Dig0 m) (Dig1 n) = dup (sub m n) - 1"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 28823
diff changeset
   661
  apply (simp_all add: dup_def algebra_simps)
29941
b951d80774d5 replace dec with double-and-decrement function
huffman
parents: 29667
diff changeset
   662
  apply (simp_all add: of_num_plus one_plus_DigM)[4]
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   663
  apply (simp_all add: of_num.simps)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   664
  done
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   665
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   666
lemma dup_code [code]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   667
  "dup 0 = 0"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   668
  "dup (Pls n) = Pls (Dig0 n)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   669
  "dup (Mns n) = Mns (Dig0 n)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   670
  by (simp_all add: dup_def of_num.simps)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   671
  
28562
4e74209f113e `code func` now just `code`
haftmann
parents: 28367
diff changeset
   672
lemma [code, code del]:
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   673
  "(1 :: int) = 1"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   674
  "(op + :: int \<Rightarrow> int \<Rightarrow> int) = op +"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   675
  "(uminus :: int \<Rightarrow> int) = uminus"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   676
  "(op - :: int \<Rightarrow> int \<Rightarrow> int) = op -"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   677
  "(op * :: int \<Rightarrow> int \<Rightarrow> int) = op *"
28367
10ea34297962 op = vs. eq
haftmann
parents: 28053
diff changeset
   678
  "(eq_class.eq :: int \<Rightarrow> int \<Rightarrow> bool) = eq_class.eq"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   679
  "(op \<le> :: int \<Rightarrow> int \<Rightarrow> bool) = op \<le>"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   680
  "(op < :: int \<Rightarrow> int \<Rightarrow> bool) = op <"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   681
  by rule+
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   682
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   683
lemma one_int_code [code]:
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   684
  "1 = Pls One"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   685
  by (simp add: of_num_one)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   686
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   687
lemma plus_int_code [code]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   688
  "k + 0 = (k::int)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   689
  "0 + l = (l::int)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   690
  "Pls m + Pls n = Pls (m + n)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   691
  "Pls m - Pls n = sub m n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   692
  "Mns m + Mns n = Mns (m + n)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   693
  "Mns m - Mns n = sub n m"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   694
  by (simp_all add: of_num_plus [symmetric])
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   695
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   696
lemma uminus_int_code [code]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   697
  "uminus 0 = (0::int)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   698
  "uminus (Pls m) = Mns m"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   699
  "uminus (Mns m) = Pls m"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   700
  by simp_all
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   701
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   702
lemma minus_int_code [code]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   703
  "k - 0 = (k::int)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   704
  "0 - l = uminus (l::int)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   705
  "Pls m - Pls n = sub m n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   706
  "Pls m - Mns n = Pls (m + n)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   707
  "Mns m - Pls n = Mns (m + n)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   708
  "Mns m - Mns n = sub n m"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   709
  by (simp_all add: of_num_plus [symmetric])
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   710
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   711
lemma times_int_code [code]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   712
  "k * 0 = (0::int)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   713
  "0 * l = (0::int)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   714
  "Pls m * Pls n = Pls (m * n)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   715
  "Pls m * Mns n = Mns (m * n)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   716
  "Mns m * Pls n = Mns (m * n)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   717
  "Mns m * Mns n = Pls (m * n)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   718
  by (simp_all add: of_num_times [symmetric])
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   719
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   720
lemma eq_int_code [code]:
28367
10ea34297962 op = vs. eq
haftmann
parents: 28053
diff changeset
   721
  "eq_class.eq 0 (0::int) \<longleftrightarrow> True"
10ea34297962 op = vs. eq
haftmann
parents: 28053
diff changeset
   722
  "eq_class.eq 0 (Pls l) \<longleftrightarrow> False"
10ea34297962 op = vs. eq
haftmann
parents: 28053
diff changeset
   723
  "eq_class.eq 0 (Mns l) \<longleftrightarrow> False"
10ea34297962 op = vs. eq
haftmann
parents: 28053
diff changeset
   724
  "eq_class.eq (Pls k) 0 \<longleftrightarrow> False"
10ea34297962 op = vs. eq
haftmann
parents: 28053
diff changeset
   725
  "eq_class.eq (Pls k) (Pls l) \<longleftrightarrow> eq_class.eq k l"
10ea34297962 op = vs. eq
haftmann
parents: 28053
diff changeset
   726
  "eq_class.eq (Pls k) (Mns l) \<longleftrightarrow> False"
10ea34297962 op = vs. eq
haftmann
parents: 28053
diff changeset
   727
  "eq_class.eq (Mns k) 0 \<longleftrightarrow> False"
10ea34297962 op = vs. eq
haftmann
parents: 28053
diff changeset
   728
  "eq_class.eq (Mns k) (Pls l) \<longleftrightarrow> False"
10ea34297962 op = vs. eq
haftmann
parents: 28053
diff changeset
   729
  "eq_class.eq (Mns k) (Mns l) \<longleftrightarrow> eq_class.eq k l"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   730
  using of_num_pos [of l, where ?'a = int] of_num_pos [of k, where ?'a = int]
28367
10ea34297962 op = vs. eq
haftmann
parents: 28053
diff changeset
   731
  by (simp_all add: of_num_eq_iff eq)
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   732
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   733
lemma less_eq_int_code [code]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   734
  "0 \<le> (0::int) \<longleftrightarrow> True"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   735
  "0 \<le> Pls l \<longleftrightarrow> True"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   736
  "0 \<le> Mns l \<longleftrightarrow> False"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   737
  "Pls k \<le> 0 \<longleftrightarrow> False"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   738
  "Pls k \<le> Pls l \<longleftrightarrow> k \<le> l"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   739
  "Pls k \<le> Mns l \<longleftrightarrow> False"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   740
  "Mns k \<le> 0 \<longleftrightarrow> True"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   741
  "Mns k \<le> Pls l \<longleftrightarrow> True"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   742
  "Mns k \<le> Mns l \<longleftrightarrow> l \<le> k"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   743
  using of_num_pos [of l, where ?'a = int] of_num_pos [of k, where ?'a = int]
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   744
  by (simp_all add: of_num_less_eq_iff)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   745
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   746
lemma less_int_code [code]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   747
  "0 < (0::int) \<longleftrightarrow> False"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   748
  "0 < Pls l \<longleftrightarrow> True"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   749
  "0 < Mns l \<longleftrightarrow> False"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   750
  "Pls k < 0 \<longleftrightarrow> False"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   751
  "Pls k < Pls l \<longleftrightarrow> k < l"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   752
  "Pls k < Mns l \<longleftrightarrow> False"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   753
  "Mns k < 0 \<longleftrightarrow> True"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   754
  "Mns k < Pls l \<longleftrightarrow> True"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   755
  "Mns k < Mns l \<longleftrightarrow> l < k"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   756
  using of_num_pos [of l, where ?'a = int] of_num_pos [of k, where ?'a = int]
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   757
  by (simp_all add: of_num_less_iff)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   758
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   759
lemma [code inline del]: "(0::int) \<equiv> Numeral0" by simp
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   760
lemma [code inline del]: "(1::int) \<equiv> Numeral1" by simp
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   761
declare zero_is_num_zero [code inline del]
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   762
declare one_is_num_one [code inline del]
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   763
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   764
hide (open) const sub dup
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   765
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   766
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   767
subsection {* Numeral equations as default simplification rules *}
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   768
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   769
text {* TODO.  Be more precise here with respect to subsumed facts. *}
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   770
declare (in semiring_numeral) numeral [simp]
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   771
declare (in semiring_1) numeral [simp]
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   772
declare (in semiring_char_0) numeral [simp]
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   773
declare (in ring_1) numeral [simp]
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   774
thm numeral
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   775
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   776
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   777
text {* Toy examples *}
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   778
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   779
definition "bar \<longleftrightarrow> #4 * #2 + #7 = (#8 :: nat) \<and> #4 * #2 + #7 \<ge> (#8 :: int) - #3"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   780
code_thms bar
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   781
export_code bar in Haskell file -
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   782
export_code bar in OCaml module_name Foo file -
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   783
ML {* @{code bar} *}
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   784
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   785
end