src/HOL/Library/Nat_Infinity.thy
 author nipkow Sun Oct 21 14:53:44 2007 +0200 (2007-10-21) changeset 25134 3d4953e88449 parent 25112 98824cc791c0 child 25594 43c718438f9f permissions -rw-r--r--
Eliminated most of the neq0_conv occurrences. As a result, many
theorems had to be rephrased with ~= 0 instead of > 0.
```     1 (*  Title:      HOL/Library/Nat_Infinity.thy
```
```     2     ID:         \$Id\$
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```     3     Author:     David von Oheimb, TU Muenchen
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```     4 *)
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```     5
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```     6 header {* Natural numbers with infinity *}
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```     7
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```     8 theory Nat_Infinity
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```     9 imports Main
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```    10 begin
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```    11
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```    12 subsection "Definitions"
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```    13
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```    14 text {*
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```    15   We extend the standard natural numbers by a special value indicating
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```    16   infinity.  This includes extending the ordering relations @{term "op
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```    17   <"} and @{term "op \<le>"}.
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```    18 *}
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```    19
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```    20 datatype inat = Fin nat | Infty
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```    21
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```    22 notation (xsymbols)
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```    23   Infty  ("\<infinity>")
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```    24
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```    25 notation (HTML output)
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```    26   Infty  ("\<infinity>")
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```    27
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```    28 instance inat :: "{ord, zero}" ..
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```    29
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```    30 definition
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```    31   iSuc :: "inat => inat" where
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```    32   "iSuc i = (case i of Fin n => Fin (Suc n) | \<infinity> => \<infinity>)"
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```    33
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```    34 defs (overloaded)
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```    35   Zero_inat_def: "0 == Fin 0"
```
```    36   iless_def: "m < n ==
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```    37     case m of Fin m1 => (case n of Fin n1 => m1 < n1 | \<infinity> => True)
```
```    38     | \<infinity>  => False"
```
```    39   ile_def: "(m::inat) \<le> n == \<not> (n < m)"
```
```    40
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```    41 lemmas inat_defs = Zero_inat_def iSuc_def iless_def ile_def
```
```    42 lemmas inat_splits = inat.split inat.split_asm
```
```    43
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```    44 text {*
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```    45   Below is a not quite complete set of theorems.  Use the method
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```    46   @{text "(simp add: inat_defs split:inat_splits, arith?)"} to prove
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```    47   new theorems or solve arithmetic subgoals involving @{typ inat} on
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```    48   the fly.
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```    49 *}
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```    50
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```    51 subsection "Constructors"
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```    52
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```    53 lemma Fin_0: "Fin 0 = 0"
```
```    54 by (simp add: inat_defs split:inat_splits)
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```    55
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```    56 lemma Infty_ne_i0 [simp]: "\<infinity> \<noteq> 0"
```
```    57 by (simp add: inat_defs split:inat_splits)
```
```    58
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```    59 lemma i0_ne_Infty [simp]: "0 \<noteq> \<infinity>"
```
```    60 by (simp add: inat_defs split:inat_splits)
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```    61
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```    62 lemma iSuc_Fin [simp]: "iSuc (Fin n) = Fin (Suc n)"
```
```    63 by (simp add: inat_defs split:inat_splits)
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```    64
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```    65 lemma iSuc_Infty [simp]: "iSuc \<infinity> = \<infinity>"
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```    66 by (simp add: inat_defs split:inat_splits)
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```    67
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```    68 lemma iSuc_ne_0 [simp]: "iSuc n \<noteq> 0"
```
```    69 by (simp add: inat_defs split:inat_splits)
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```    70
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```    71 lemma iSuc_inject [simp]: "(iSuc x = iSuc y) = (x = y)"
```
```    72 by (simp add: inat_defs split:inat_splits)
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```    73
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```    74
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```    75 subsection "Ordering relations"
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```    76
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```    77 lemma Infty_ilessE [elim!]: "\<infinity> < Fin m ==> R"
```
```    78 by (simp add: inat_defs split:inat_splits)
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```    79
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```    80 lemma iless_linear: "m < n \<or> m = n \<or> n < (m::inat)"
```
```    81 by (simp add: inat_defs split:inat_splits, arith)
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```    82
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```    83 lemma iless_not_refl [simp]: "\<not> n < (n::inat)"
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```    84 by (simp add: inat_defs split:inat_splits)
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```    85
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```    86 lemma iless_trans: "i < j ==> j < k ==> i < (k::inat)"
```
```    87 by (simp add: inat_defs split:inat_splits)
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```    88
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```    89 lemma iless_not_sym: "n < m ==> \<not> m < (n::inat)"
```
```    90 by (simp add: inat_defs split:inat_splits)
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```    91
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```    92 lemma Fin_iless_mono [simp]: "(Fin n < Fin m) = (n < m)"
```
```    93 by (simp add: inat_defs split:inat_splits)
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```    94
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```    95 lemma Fin_iless_Infty [simp]: "Fin n < \<infinity>"
```
```    96 by (simp add: inat_defs split:inat_splits)
```
```    97
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```    98 lemma Infty_eq [simp]: "(n < \<infinity>) = (n \<noteq> \<infinity>)"
```
```    99 by (simp add: inat_defs split:inat_splits)
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```   100
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```   101 lemma i0_eq [simp]: "((0::inat) < n) = (n \<noteq> 0)"
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```   102 by (fastsimp simp: inat_defs split:inat_splits)
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```   103
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```   104 lemma i0_iless_iSuc [simp]: "0 < iSuc n"
```
```   105 by (simp add: inat_defs split:inat_splits)
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```   106
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```   107 lemma not_ilessi0 [simp]: "\<not> n < (0::inat)"
```
```   108 by (simp add: inat_defs split:inat_splits)
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```   109
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```   110 lemma Fin_iless: "n < Fin m ==> \<exists>k. n = Fin k"
```
```   111 by (simp add: inat_defs split:inat_splits)
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```   112
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```   113 lemma iSuc_mono [simp]: "(iSuc n < iSuc m) = (n < m)"
```
```   114 by (simp add: inat_defs split:inat_splits)
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```   115
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```   116
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```   117
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```   118 lemma ile_def2: "(m \<le> n) = (m < n \<or> m = (n::inat))"
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```   119 by (simp add: inat_defs split:inat_splits, arith)
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```   120
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```   121 lemma ile_refl [simp]: "n \<le> (n::inat)"
```
```   122 by (simp add: inat_defs split:inat_splits)
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```   123
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```   124 lemma ile_trans: "i \<le> j ==> j \<le> k ==> i \<le> (k::inat)"
```
```   125 by (simp add: inat_defs split:inat_splits)
```
```   126
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```   127 lemma ile_iless_trans: "i \<le> j ==> j < k ==> i < (k::inat)"
```
```   128 by (simp add: inat_defs split:inat_splits)
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```   129
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```   130 lemma iless_ile_trans: "i < j ==> j \<le> k ==> i < (k::inat)"
```
```   131 by (simp add: inat_defs split:inat_splits)
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```   132
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```   133 lemma Infty_ub [simp]: "n \<le> \<infinity>"
```
```   134 by (simp add: inat_defs split:inat_splits)
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```   135
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```   136 lemma i0_lb [simp]: "(0::inat) \<le> n"
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```   137 by (simp add: inat_defs split:inat_splits)
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```   138
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```   139 lemma Infty_ileE [elim!]: "\<infinity> \<le> Fin m ==> R"
```
```   140 by (simp add: inat_defs split:inat_splits)
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```   141
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```   142 lemma Fin_ile_mono [simp]: "(Fin n \<le> Fin m) = (n \<le> m)"
```
```   143 by (simp add: inat_defs split:inat_splits, arith)
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```   144
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```   145 lemma ilessI1: "n \<le> m ==> n \<noteq> m ==> n < (m::inat)"
```
```   146 by (simp add: inat_defs split:inat_splits)
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```   147
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```   148 lemma ileI1: "m < n ==> iSuc m \<le> n"
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```   149 by (simp add: inat_defs split:inat_splits)
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```   150
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```   151 lemma Suc_ile_eq: "(Fin (Suc m) \<le> n) = (Fin m < n)"
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```   152 by (simp add: inat_defs split:inat_splits, arith)
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```   153
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```   154 lemma iSuc_ile_mono [simp]: "(iSuc n \<le> iSuc m) = (n \<le> m)"
```
```   155 by (simp add: inat_defs split:inat_splits)
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```   156
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```   157 lemma iless_Suc_eq [simp]: "(Fin m < iSuc n) = (Fin m \<le> n)"
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```   158 by (simp add: inat_defs split:inat_splits, arith)
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```   159
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```   160 lemma not_iSuc_ilei0 [simp]: "\<not> iSuc n \<le> 0"
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```   161 by (simp add: inat_defs split:inat_splits)
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```   162
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```   163 lemma ile_iSuc [simp]: "n \<le> iSuc n"
```
```   164 by (simp add: inat_defs split:inat_splits)
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```   165
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```   166 lemma Fin_ile: "n \<le> Fin m ==> \<exists>k. n = Fin k"
```
```   167 by (simp add: inat_defs split:inat_splits)
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```   168
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```   169 lemma chain_incr: "\<forall>i. \<exists>j. Y i < Y j ==> \<exists>j. Fin k < Y j"
```
```   170 apply (induct_tac k)
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```   171  apply (simp (no_asm) only: Fin_0)
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```   172  apply (fast intro: ile_iless_trans i0_lb)
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```   173 apply (erule exE)
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```   174 apply (drule spec)
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```   175 apply (erule exE)
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```   176 apply (drule ileI1)
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```   177 apply (rule iSuc_Fin [THEN subst])
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```   178 apply (rule exI)
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```   179 apply (erule (1) ile_iless_trans)
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```   180 done
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```   181
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```   182 end
```