src/HOL/ex/Primrec.thy
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```     1 (*  Title:      HOL/ex/Primrec.thy
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```     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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```     3     Copyright   1997  University of Cambridge
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```     4
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```     5 Ackermann's Function and the
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```     6 Primitive Recursive Functions.
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```     7 *)
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```     8
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```     9 section \<open>Primitive Recursive Functions\<close>
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```    10
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```    11 theory Primrec imports Main begin
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```    12
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```    13 text \<open>
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```    14   Proof adopted from
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```    15
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```    16   Nora Szasz, A Machine Checked Proof that Ackermann's Function is not
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```    17   Primitive Recursive, In: Huet \& Plotkin, eds., Logical Environments
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```    18   (CUP, 1993), 317-338.
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```    19
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```    20   See also E. Mendelson, Introduction to Mathematical Logic.  (Van
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```    21   Nostrand, 1964), page 250, exercise 11.
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```    22   \medskip
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```    23 \<close>
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```    24
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```    25
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```    26 subsection\<open>Ackermann's Function\<close>
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```    27
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```    28 fun ack :: "nat => nat => nat" where
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```    29 "ack 0 n =  Suc n" |
```
```    30 "ack (Suc m) 0 = ack m 1" |
```
```    31 "ack (Suc m) (Suc n) = ack m (ack (Suc m) n)"
```
```    32
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```    33
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```    34 text \<open>PROPERTY A 4\<close>
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```    35
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```    36 lemma less_ack2 [iff]: "j < ack i j"
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```    37 by (induct i j rule: ack.induct) simp_all
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```    38
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```    39
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```    40 text \<open>PROPERTY A 5-, the single-step lemma\<close>
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```    41
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```    42 lemma ack_less_ack_Suc2 [iff]: "ack i j < ack i (Suc j)"
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```    43 by (induct i j rule: ack.induct) simp_all
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```    44
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```    45
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```    46 text \<open>PROPERTY A 5, monotonicity for \<open><\<close>\<close>
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```    47
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```    48 lemma ack_less_mono2: "j < k ==> ack i j < ack i k"
```
```    49 using lift_Suc_mono_less[where f = "ack i"]
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```    50 by (metis ack_less_ack_Suc2)
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```    51
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```    52
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```    53 text \<open>PROPERTY A 5', monotonicity for \<open>\<le>\<close>\<close>
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```    54
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```    55 lemma ack_le_mono2: "j \<le> k ==> ack i j \<le> ack i k"
```
```    56 apply (simp add: order_le_less)
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```    57 apply (blast intro: ack_less_mono2)
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```    58 done
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```    59
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```    60
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```    61 text \<open>PROPERTY A 6\<close>
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```    62
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```    63 lemma ack2_le_ack1 [iff]: "ack i (Suc j) \<le> ack (Suc i) j"
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```    64 proof (induct j)
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```    65   case 0 show ?case by simp
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```    66 next
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```    67   case (Suc j) show ?case
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```    68     by (auto intro!: ack_le_mono2)
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```    69       (metis Suc Suc_leI Suc_lessI less_ack2 linorder_not_less)
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```    70 qed
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```    71
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```    72
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```    73 text \<open>PROPERTY A 7-, the single-step lemma\<close>
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```    74
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```    75 lemma ack_less_ack_Suc1 [iff]: "ack i j < ack (Suc i) j"
```
```    76 by (blast intro: ack_less_mono2 less_le_trans)
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```    77
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```    78
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```    79 text \<open>PROPERTY A 4'? Extra lemma needed for @{term CONSTANT} case, constant functions\<close>
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```    80
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```    81 lemma less_ack1 [iff]: "i < ack i j"
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```    82 apply (induct i)
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```    83  apply simp_all
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```    84 apply (blast intro: Suc_leI le_less_trans)
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```    85 done
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```    86
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```    87
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```    88 text \<open>PROPERTY A 8\<close>
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```    89
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```    90 lemma ack_1 [simp]: "ack (Suc 0) j = j + 2"
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```    91 by (induct j) simp_all
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```    92
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```    93
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```    94 text \<open>PROPERTY A 9.  The unary \<open>1\<close> and \<open>2\<close> in @{term
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```    95   ack} is essential for the rewriting.\<close>
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```    96
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```    97 lemma ack_2 [simp]: "ack (Suc (Suc 0)) j = 2 * j + 3"
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```    98 by (induct j) simp_all
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```    99
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```   100
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```   101 text \<open>PROPERTY A 7, monotonicity for \<open><\<close> [not clear why
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```   102   @{thm [source] ack_1} is now needed first!]\<close>
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```   103
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```   104 lemma ack_less_mono1_aux: "ack i k < ack (Suc (i +i')) k"
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```   105 proof (induct i k rule: ack.induct)
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```   106   case (1 n) show ?case
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```   107     by (simp, metis ack_less_ack_Suc1 less_ack2 less_trans_Suc)
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```   108 next
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```   109   case (2 m) thus ?case by simp
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```   110 next
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```   111   case (3 m n) thus ?case
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```   112     by (simp, blast intro: less_trans ack_less_mono2)
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```   113 qed
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```   114
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```   115 lemma ack_less_mono1: "i < j ==> ack i k < ack j k"
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```   116 apply (drule less_imp_Suc_add)
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```   117 apply (blast intro!: ack_less_mono1_aux)
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```   118 done
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```   119
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```   120
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```   121 text \<open>PROPERTY A 7', monotonicity for \<open>\<le>\<close>\<close>
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```   122
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```   123 lemma ack_le_mono1: "i \<le> j ==> ack i k \<le> ack j k"
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```   124 apply (simp add: order_le_less)
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```   125 apply (blast intro: ack_less_mono1)
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```   126 done
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```   127
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```   128
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```   129 text \<open>PROPERTY A 10\<close>
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```   130
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```   131 lemma ack_nest_bound: "ack i1 (ack i2 j) < ack (2 + (i1 + i2)) j"
```
```   132 apply (simp add: numerals)
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```   133 apply (rule ack2_le_ack1 [THEN [2] less_le_trans])
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```   134 apply simp
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```   135 apply (rule le_add1 [THEN ack_le_mono1, THEN le_less_trans])
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```   136 apply (rule ack_less_mono1 [THEN ack_less_mono2])
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```   137 apply (simp add: le_imp_less_Suc le_add2)
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```   138 done
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```   139
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```   140
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```   141 text \<open>PROPERTY A 11\<close>
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```   142
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```   143 lemma ack_add_bound: "ack i1 j + ack i2 j < ack (4 + (i1 + i2)) j"
```
```   144 apply (rule less_trans [of _ "ack (Suc (Suc 0)) (ack (i1 + i2) j)"])
```
```   145  prefer 2
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```   146  apply (rule ack_nest_bound [THEN less_le_trans])
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```   147  apply (simp add: Suc3_eq_add_3)
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```   148 apply simp
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```   149 apply (cut_tac i = i1 and m1 = i2 and k = j in le_add1 [THEN ack_le_mono1])
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```   150 apply (cut_tac i = "i2" and m1 = i1 and k = j in le_add2 [THEN ack_le_mono1])
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```   151 apply auto
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```   152 done
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```   153
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```   154
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```   155 text \<open>PROPERTY A 12.  Article uses existential quantifier but the ALF proof
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```   156   used \<open>k + 4\<close>.  Quantified version must be nested \<open>\<exists>k'. \<forall>i j. ...\<close>\<close>
```
```   157
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```   158 lemma ack_add_bound2: "i < ack k j ==> i + j < ack (4 + k) j"
```
```   159 apply (rule less_trans [of _ "ack k j + ack 0 j"])
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```   160  apply (blast intro: add_less_mono)
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```   161 apply (rule ack_add_bound [THEN less_le_trans])
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```   162 apply simp
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```   163 done
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```   164
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```   165
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```   166 subsection\<open>Primitive Recursive Functions\<close>
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```   167
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```   168 primrec hd0 :: "nat list => nat" where
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```   169 "hd0 [] = 0" |
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```   170 "hd0 (m # ms) = m"
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```   171
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```   172
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```   173 text \<open>Inductive definition of the set of primitive recursive functions of type @{typ "nat list => nat"}.\<close>
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```   174
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```   175 definition SC :: "nat list => nat" where
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```   176 "SC l = Suc (hd0 l)"
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```   177
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```   178 definition CONSTANT :: "nat => nat list => nat" where
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```   179 "CONSTANT k l = k"
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```   180
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```   181 definition PROJ :: "nat => nat list => nat" where
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```   182 "PROJ i l = hd0 (drop i l)"
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```   183
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```   184 definition
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```   185 COMP :: "(nat list => nat) => (nat list => nat) list => nat list => nat"
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```   186 where "COMP g fs l = g (map (\<lambda>f. f l) fs)"
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```   187
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```   188 definition PREC :: "(nat list => nat) => (nat list => nat) => nat list => nat"
```
```   189 where
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```   190   "PREC f g l =
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```   191     (case l of
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```   192       [] => 0
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```   193     | x # l' => rec_nat (f l') (\<lambda>y r. g (r # y # l')) x)"
```
```   194   \<comment> \<open>Note that @{term g} is applied first to @{term "PREC f g y"} and then to @{term y}!\<close>
```
```   195
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```   196 inductive PRIMREC :: "(nat list => nat) => bool" where
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```   197 SC: "PRIMREC SC" |
```
```   198 CONSTANT: "PRIMREC (CONSTANT k)" |
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```   199 PROJ: "PRIMREC (PROJ i)" |
```
```   200 COMP: "PRIMREC g ==> \<forall>f \<in> set fs. PRIMREC f ==> PRIMREC (COMP g fs)" |
```
```   201 PREC: "PRIMREC f ==> PRIMREC g ==> PRIMREC (PREC f g)"
```
```   202
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```   203
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```   204 text \<open>Useful special cases of evaluation\<close>
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```   205
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```   206 lemma SC [simp]: "SC (x # l) = Suc x"
```
```   207 by (simp add: SC_def)
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```   208
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```   209 lemma CONSTANT [simp]: "CONSTANT k l = k"
```
```   210 by (simp add: CONSTANT_def)
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```   211
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```   212 lemma PROJ_0 [simp]: "PROJ 0 (x # l) = x"
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```   213 by (simp add: PROJ_def)
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```   214
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```   215 lemma COMP_1 [simp]: "COMP g [f] l = g [f l]"
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```   216 by (simp add: COMP_def)
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```   217
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```   218 lemma PREC_0 [simp]: "PREC f g (0 # l) = f l"
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```   219 by (simp add: PREC_def)
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```   220
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```   221 lemma PREC_Suc [simp]: "PREC f g (Suc x # l) = g (PREC f g (x # l) # x # l)"
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```   222 by (simp add: PREC_def)
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```   223
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```   224
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```   225 text \<open>MAIN RESULT\<close>
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```   226
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```   227 lemma SC_case: "SC l < ack 1 (listsum l)"
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```   228 apply (unfold SC_def)
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```   229 apply (induct l)
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```   230 apply (simp_all add: le_add1 le_imp_less_Suc)
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```   231 done
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```   232
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```   233 lemma CONSTANT_case: "CONSTANT k l < ack k (listsum l)"
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```   234 by simp
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```   235
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```   236 lemma PROJ_case: "PROJ i l < ack 0 (listsum l)"
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```   237 apply (simp add: PROJ_def)
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```   238 apply (induct l arbitrary:i)
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```   239  apply (auto simp add: drop_Cons split: nat.split)
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```   240 apply (blast intro: less_le_trans le_add2)
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```   241 done
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```   242
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```   243
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```   244 text \<open>@{term COMP} case\<close>
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```   245
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```   246 lemma COMP_map_aux: "\<forall>f \<in> set fs. PRIMREC f \<and> (\<exists>kf. \<forall>l. f l < ack kf (listsum l))
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```   247   ==> \<exists>k. \<forall>l. listsum (map (\<lambda>f. f l) fs) < ack k (listsum l)"
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```   248 apply (induct fs)
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```   249  apply (rule_tac x = 0 in exI)
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```   250  apply simp
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```   251 apply simp
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```   252 apply (blast intro: add_less_mono ack_add_bound less_trans)
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```   253 done
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```   254
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```   255 lemma COMP_case:
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```   256   "\<forall>l. g l < ack kg (listsum l) ==>
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```   257   \<forall>f \<in> set fs. PRIMREC f \<and> (\<exists>kf. \<forall>l. f l < ack kf (listsum l))
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```   258   ==> \<exists>k. \<forall>l. COMP g fs  l < ack k (listsum l)"
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```   259 apply (unfold COMP_def)
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```   260 apply (drule COMP_map_aux)
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```   261 apply (meson ack_less_mono2 ack_nest_bound less_trans)
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```   262 done
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```   263
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```   264
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```   265 text \<open>@{term PREC} case\<close>
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```   266
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```   267 lemma PREC_case_aux:
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```   268   "\<forall>l. f l + listsum l < ack kf (listsum l) ==>
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```   269     \<forall>l. g l + listsum l < ack kg (listsum l) ==>
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```   270     PREC f g l + listsum l < ack (Suc (kf + kg)) (listsum l)"
```
```   271 apply (unfold PREC_def)
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```   272 apply (case_tac l)
```
```   273  apply simp_all
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```   274  apply (blast intro: less_trans)
```
```   275 apply (erule ssubst) \<comment> \<open>get rid of the needless assumption\<close>
```
```   276 apply (induct_tac a)
```
```   277  apply simp_all
```
```   278  txt \<open>base case\<close>
```
```   279  apply (blast intro: le_add1 [THEN le_imp_less_Suc, THEN ack_less_mono1] less_trans)
```
```   280 txt \<open>induction step\<close>
```
```   281 apply (rule Suc_leI [THEN le_less_trans])
```
```   282  apply (rule le_refl [THEN add_le_mono, THEN le_less_trans])
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```   283   prefer 2
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```   284   apply (erule spec)
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```   285  apply (simp add: le_add2)
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```   286 txt \<open>final part of the simplification\<close>
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```   287 apply simp
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```   288 apply (rule le_add2 [THEN ack_le_mono1, THEN le_less_trans])
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```   289 apply (erule ack_less_mono2)
```
```   290 done
```
```   291
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```   292 lemma PREC_case:
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```   293   "\<forall>l. f l < ack kf (listsum l) ==>
```
```   294     \<forall>l. g l < ack kg (listsum l) ==>
```
```   295     \<exists>k. \<forall>l. PREC f g l < ack k (listsum l)"
```
```   296 by (metis le_less_trans [OF le_add1 PREC_case_aux] ack_add_bound2)
```
```   297
```
```   298 lemma ack_bounds_PRIMREC: "PRIMREC f ==> \<exists>k. \<forall>l. f l < ack k (listsum l)"
```
```   299 apply (erule PRIMREC.induct)
```
```   300     apply (blast intro: SC_case CONSTANT_case PROJ_case COMP_case PREC_case)+
```
```   301 done
```
```   302
```
```   303 theorem ack_not_PRIMREC:
```
```   304   "\<not> PRIMREC (\<lambda>l. case l of [] => 0 | x # l' => ack x x)"
```
```   305 apply (rule notI)
```
```   306 apply (erule ack_bounds_PRIMREC [THEN exE])
```
```   307 apply (rule less_irrefl [THEN notE])
```
```   308 apply (drule_tac x = "[x]" in spec)
```
```   309 apply simp
```
```   310 done
```
```   311
```
```   312 end
```