src/HOL/ex/Primrec.thy
 author nipkow Sun May 13 07:11:21 2007 +0200 (2007-05-13) changeset 22944 1d471b8dec4e parent 22283 26140713540b child 23776 2215169c93fa permissions -rw-r--r--
Got rid of listsp
```     1 (*  Title:      HOL/ex/Primrec.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     4     Copyright   1997  University of Cambridge
```
```     5
```
```     6 Primitive Recursive Functions.  Demonstrates recursive definitions,
```
```     7 the TFL package.
```
```     8 *)
```
```     9
```
```    10 header {* Primitive Recursive Functions *}
```
```    11
```
```    12 theory Primrec imports Main begin
```
```    13
```
```    14 text {*
```
```    15   Proof adopted from
```
```    16
```
```    17   Nora Szasz, A Machine Checked Proof that Ackermann's Function is not
```
```    18   Primitive Recursive, In: Huet \& Plotkin, eds., Logical Environments
```
```    19   (CUP, 1993), 317-338.
```
```    20
```
```    21   See also E. Mendelson, Introduction to Mathematical Logic.  (Van
```
```    22   Nostrand, 1964), page 250, exercise 11.
```
```    23   \medskip
```
```    24 *}
```
```    25
```
```    26 consts ack :: "nat * nat => nat"
```
```    27 recdef ack  "less_than <*lex*> less_than"
```
```    28   "ack (0, n) =  Suc n"
```
```    29   "ack (Suc m, 0) = ack (m, 1)"
```
```    30   "ack (Suc m, Suc n) = ack (m, ack (Suc m, n))"
```
```    31
```
```    32 consts list_add :: "nat list => nat"
```
```    33 primrec
```
```    34   "list_add [] = 0"
```
```    35   "list_add (m # ms) = m + list_add ms"
```
```    36
```
```    37 consts zeroHd :: "nat list => nat"
```
```    38 primrec
```
```    39   "zeroHd [] = 0"
```
```    40   "zeroHd (m # ms) = m"
```
```    41
```
```    42
```
```    43 text {* The set of primitive recursive functions of type @{typ "nat list => nat"}. *}
```
```    44
```
```    45 definition
```
```    46   SC :: "nat list => nat" where
```
```    47   "SC l = Suc (zeroHd l)"
```
```    48
```
```    49 definition
```
```    50   CONSTANT :: "nat => nat list => nat" where
```
```    51   "CONSTANT k l = k"
```
```    52
```
```    53 definition
```
```    54   PROJ :: "nat => nat list => nat" where
```
```    55   "PROJ i l = zeroHd (drop i l)"
```
```    56
```
```    57 definition
```
```    58   COMP :: "(nat list => nat) => (nat list => nat) list => nat list => nat" where
```
```    59   "COMP g fs l = g (map (\<lambda>f. f l) fs)"
```
```    60
```
```    61 definition
```
```    62   PREC :: "(nat list => nat) => (nat list => nat) => nat list => nat" where
```
```    63   "PREC f g l =
```
```    64     (case l of
```
```    65       [] => 0
```
```    66     | x # l' => nat_rec (f l') (\<lambda>y r. g (r # y # l')) x)"
```
```    67   -- {* Note that @{term g} is applied first to @{term "PREC f g y"} and then to @{term y}! *}
```
```    68
```
```    69 inductive2 PRIMREC :: "(nat list => nat) => bool"
```
```    70   where
```
```    71     SC: "PRIMREC SC"
```
```    72   | CONSTANT: "PRIMREC (CONSTANT k)"
```
```    73   | PROJ: "PRIMREC (PROJ i)"
```
```    74   | COMP: "PRIMREC g ==> \<forall>f \<in> set fs. PRIMREC f ==> PRIMREC (COMP g fs)"
```
```    75   | PREC: "PRIMREC f ==> PRIMREC g ==> PRIMREC (PREC f g)"
```
```    76
```
```    77
```
```    78 text {* Useful special cases of evaluation *}
```
```    79
```
```    80 lemma SC [simp]: "SC (x # l) = Suc x"
```
```    81   apply (simp add: SC_def)
```
```    82   done
```
```    83
```
```    84 lemma CONSTANT [simp]: "CONSTANT k l = k"
```
```    85   apply (simp add: CONSTANT_def)
```
```    86   done
```
```    87
```
```    88 lemma PROJ_0 [simp]: "PROJ 0 (x # l) = x"
```
```    89   apply (simp add: PROJ_def)
```
```    90   done
```
```    91
```
```    92 lemma COMP_1 [simp]: "COMP g [f] l = g [f l]"
```
```    93   apply (simp add: COMP_def)
```
```    94   done
```
```    95
```
```    96 lemma PREC_0 [simp]: "PREC f g (0 # l) = f l"
```
```    97   apply (simp add: PREC_def)
```
```    98   done
```
```    99
```
```   100 lemma PREC_Suc [simp]: "PREC f g (Suc x # l) = g (PREC f g (x # l) # x # l)"
```
```   101   apply (simp add: PREC_def)
```
```   102   done
```
```   103
```
```   104
```
```   105 text {* PROPERTY A 4 *}
```
```   106
```
```   107 lemma less_ack2 [iff]: "j < ack (i, j)"
```
```   108   apply (induct i j rule: ack.induct)
```
```   109     apply simp_all
```
```   110   done
```
```   111
```
```   112
```
```   113 text {* PROPERTY A 5-, the single-step lemma *}
```
```   114
```
```   115 lemma ack_less_ack_Suc2 [iff]: "ack(i, j) < ack (i, Suc j)"
```
```   116   apply (induct i j rule: ack.induct)
```
```   117     apply simp_all
```
```   118   done
```
```   119
```
```   120
```
```   121 text {* PROPERTY A 5, monotonicity for @{text "<"} *}
```
```   122
```
```   123 lemma ack_less_mono2: "j < k ==> ack (i, j) < ack (i, k)"
```
```   124   apply (induct i k rule: ack.induct)
```
```   125     apply simp_all
```
```   126   apply (blast elim!: less_SucE intro: less_trans)
```
```   127   done
```
```   128
```
```   129
```
```   130 text {* PROPERTY A 5', monotonicity for @{text \<le>} *}
```
```   131
```
```   132 lemma ack_le_mono2: "j \<le> k ==> ack (i, j) \<le> ack (i, k)"
```
```   133   apply (simp add: order_le_less)
```
```   134   apply (blast intro: ack_less_mono2)
```
```   135   done
```
```   136
```
```   137
```
```   138 text {* PROPERTY A 6 *}
```
```   139
```
```   140 lemma ack2_le_ack1 [iff]: "ack (i, Suc j) \<le> ack (Suc i, j)"
```
```   141   apply (induct j)
```
```   142    apply simp_all
```
```   143   apply (blast intro: ack_le_mono2 less_ack2 [THEN Suc_leI] le_trans)
```
```   144   done
```
```   145
```
```   146
```
```   147 text {* PROPERTY A 7-, the single-step lemma *}
```
```   148
```
```   149 lemma ack_less_ack_Suc1 [iff]: "ack (i, j) < ack (Suc i, j)"
```
```   150   apply (blast intro: ack_less_mono2 less_le_trans)
```
```   151   done
```
```   152
```
```   153
```
```   154 text {* PROPERTY A 4'? Extra lemma needed for @{term CONSTANT} case, constant functions *}
```
```   155
```
```   156 lemma less_ack1 [iff]: "i < ack (i, j)"
```
```   157   apply (induct i)
```
```   158    apply simp_all
```
```   159   apply (blast intro: Suc_leI le_less_trans)
```
```   160   done
```
```   161
```
```   162
```
```   163 text {* PROPERTY A 8 *}
```
```   164
```
```   165 lemma ack_1 [simp]: "ack (Suc 0, j) = j + 2"
```
```   166   apply (induct j)
```
```   167    apply simp_all
```
```   168   done
```
```   169
```
```   170
```
```   171 text {* PROPERTY A 9.  The unary @{text 1} and @{text 2} in @{term
```
```   172   ack} is essential for the rewriting. *}
```
```   173
```
```   174 lemma ack_2 [simp]: "ack (Suc (Suc 0), j) = 2 * j + 3"
```
```   175   apply (induct j)
```
```   176    apply simp_all
```
```   177   done
```
```   178
```
```   179
```
```   180 text {* PROPERTY A 7, monotonicity for @{text "<"} [not clear why
```
```   181   @{thm [source] ack_1} is now needed first!] *}
```
```   182
```
```   183 lemma ack_less_mono1_aux: "ack (i, k) < ack (Suc (i +i'), k)"
```
```   184   apply (induct i k rule: ack.induct)
```
```   185     apply simp_all
```
```   186    prefer 2
```
```   187    apply (blast intro: less_trans ack_less_mono2)
```
```   188   apply (induct_tac i' n rule: ack.induct)
```
```   189     apply simp_all
```
```   190   apply (blast intro: Suc_leI [THEN le_less_trans] ack_less_mono2)
```
```   191   done
```
```   192
```
```   193 lemma ack_less_mono1: "i < j ==> ack (i, k) < ack (j, k)"
```
```   194   apply (drule less_imp_Suc_add)
```
```   195   apply (blast intro!: ack_less_mono1_aux)
```
```   196   done
```
```   197
```
```   198
```
```   199 text {* PROPERTY A 7', monotonicity for @{text "\<le>"} *}
```
```   200
```
```   201 lemma ack_le_mono1: "i \<le> j ==> ack (i, k) \<le> ack (j, k)"
```
```   202   apply (simp add: order_le_less)
```
```   203   apply (blast intro: ack_less_mono1)
```
```   204   done
```
```   205
```
```   206
```
```   207 text {* PROPERTY A 10 *}
```
```   208
```
```   209 lemma ack_nest_bound: "ack(i1, ack (i2, j)) < ack (2 + (i1 + i2), j)"
```
```   210   apply (simp add: numerals)
```
```   211   apply (rule ack2_le_ack1 [THEN [2] less_le_trans])
```
```   212   apply simp
```
```   213   apply (rule le_add1 [THEN ack_le_mono1, THEN le_less_trans])
```
```   214   apply (rule ack_less_mono1 [THEN ack_less_mono2])
```
```   215   apply (simp add: le_imp_less_Suc le_add2)
```
```   216   done
```
```   217
```
```   218
```
```   219 text {* PROPERTY A 11 *}
```
```   220
```
```   221 lemma ack_add_bound: "ack (i1, j) + ack (i2, j) < ack (4 + (i1 + i2), j)"
```
```   222   apply (rule_tac j = "ack (Suc (Suc 0), ack (i1 + i2, j))" in less_trans)
```
```   223    prefer 2
```
```   224    apply (rule ack_nest_bound [THEN less_le_trans])
```
```   225    apply (simp add: Suc3_eq_add_3)
```
```   226   apply simp
```
```   227   apply (cut_tac i = i1 and m1 = i2 and k = j in le_add1 [THEN ack_le_mono1])
```
```   228   apply (cut_tac i = "i2" and m1 = i1 and k = j in le_add2 [THEN ack_le_mono1])
```
```   229   apply auto
```
```   230   done
```
```   231
```
```   232
```
```   233 text {* PROPERTY A 12.  Article uses existential quantifier but the ALF proof
```
```   234   used @{text "k + 4"}.  Quantified version must be nested @{text
```
```   235   "\<exists>k'. \<forall>i j. ..."} *}
```
```   236
```
```   237 lemma ack_add_bound2: "i < ack (k, j) ==> i + j < ack (4 + k, j)"
```
```   238   apply (rule_tac j = "ack (k, j) + ack (0, j)" in less_trans)
```
```   239    prefer 2
```
```   240    apply (rule ack_add_bound [THEN less_le_trans])
```
```   241    apply simp
```
```   242   apply (rule add_less_mono less_ack2 | assumption)+
```
```   243   done
```
```   244
```
```   245
```
```   246
```
```   247 text {* Inductive definition of the @{term PR} functions *}
```
```   248
```
```   249 text {* MAIN RESULT *}
```
```   250
```
```   251 lemma SC_case: "SC l < ack (1, list_add l)"
```
```   252   apply (unfold SC_def)
```
```   253   apply (induct l)
```
```   254   apply (simp_all add: le_add1 le_imp_less_Suc)
```
```   255   done
```
```   256
```
```   257 lemma CONSTANT_case: "CONSTANT k l < ack (k, list_add l)"
```
```   258   apply simp
```
```   259   done
```
```   260
```
```   261 lemma PROJ_case [rule_format]: "\<forall>i. PROJ i l < ack (0, list_add l)"
```
```   262   apply (simp add: PROJ_def)
```
```   263   apply (induct l)
```
```   264    apply simp_all
```
```   265   apply (rule allI)
```
```   266   apply (case_tac i)
```
```   267   apply (simp (no_asm_simp) add: le_add1 le_imp_less_Suc)
```
```   268   apply (simp (no_asm_simp))
```
```   269   apply (blast intro: less_le_trans intro!: le_add2)
```
```   270   done
```
```   271
```
```   272
```
```   273 text {* @{term COMP} case *}
```
```   274
```
```   275 lemma COMP_map_aux: "\<forall>f \<in> set fs. PRIMREC f \<and> (\<exists>kf. \<forall>l. f l < ack (kf, list_add l))
```
```   276   ==> \<exists>k. \<forall>l. list_add (map (\<lambda>f. f l) fs) < ack (k, list_add l)"
```
```   277   apply (induct fs)
```
```   278   apply (rule_tac x = 0 in exI)
```
```   279    apply simp
```
```   280   apply simp
```
```   281   apply (blast intro: add_less_mono ack_add_bound less_trans)
```
```   282   done
```
```   283
```
```   284 lemma COMP_case:
```
```   285   "\<forall>l. g l < ack (kg, list_add l) ==>
```
```   286   \<forall>f \<in> set fs. PRIMREC f \<and> (\<exists>kf. \<forall>l. f l < ack(kf, list_add l))
```
```   287   ==> \<exists>k. \<forall>l. COMP g fs  l < ack(k, list_add l)"
```
```   288   apply (unfold COMP_def)
```
```   289     --{*Now, if meson tolerated map, we could finish with
```
```   290   @{text "(drule COMP_map_aux, meson ack_less_mono2 ack_nest_bound less_trans)"} *}
```
```   291   apply (erule COMP_map_aux [THEN exE])
```
```   292   apply (rule exI)
```
```   293   apply (rule allI)
```
```   294   apply (drule spec)+
```
```   295   apply (erule less_trans)
```
```   296   apply (blast intro: ack_less_mono2 ack_nest_bound less_trans)
```
```   297   done
```
```   298
```
```   299
```
```   300 text {* @{term PREC} case *}
```
```   301
```
```   302 lemma PREC_case_aux:
```
```   303   "\<forall>l. f l + list_add l < ack (kf, list_add l) ==>
```
```   304     \<forall>l. g l + list_add l < ack (kg, list_add l) ==>
```
```   305     PREC f g l + list_add l < ack (Suc (kf + kg), list_add l)"
```
```   306   apply (unfold PREC_def)
```
```   307   apply (case_tac l)
```
```   308    apply simp_all
```
```   309    apply (blast intro: less_trans)
```
```   310   apply (erule ssubst) -- {* get rid of the needless assumption *}
```
```   311   apply (induct_tac a)
```
```   312    apply simp_all
```
```   313    txt {* base case *}
```
```   314    apply (blast intro: le_add1 [THEN le_imp_less_Suc, THEN ack_less_mono1] less_trans)
```
```   315   txt {* induction step *}
```
```   316   apply (rule Suc_leI [THEN le_less_trans])
```
```   317    apply (rule le_refl [THEN add_le_mono, THEN le_less_trans])
```
```   318     prefer 2
```
```   319     apply (erule spec)
```
```   320    apply (simp add: le_add2)
```
```   321   txt {* final part of the simplification *}
```
```   322   apply simp
```
```   323   apply (rule le_add2 [THEN ack_le_mono1, THEN le_less_trans])
```
```   324   apply (erule ack_less_mono2)
```
```   325   done
```
```   326
```
```   327 lemma PREC_case:
```
```   328   "\<forall>l. f l < ack (kf, list_add l) ==>
```
```   329     \<forall>l. g l < ack (kg, list_add l) ==>
```
```   330     \<exists>k. \<forall>l. PREC f g l < ack (k, list_add l)"
```
```   331   apply (rule exI)
```
```   332   apply (rule allI)
```
```   333   apply (rule le_less_trans [OF le_add1 PREC_case_aux])
```
```   334    apply (blast intro: ack_add_bound2)+
```
```   335   done
```
```   336
```
```   337 lemma ack_bounds_PRIMREC: "PRIMREC f ==> \<exists>k. \<forall>l. f l < ack (k, list_add l)"
```
```   338   apply (erule PRIMREC.induct)
```
```   339       apply (blast intro: SC_case CONSTANT_case PROJ_case COMP_case PREC_case)+
```
```   340   done
```
```   341
```
```   342 lemma ack_not_PRIMREC: "\<not> PRIMREC (\<lambda>l. case l of [] => 0 | x # l' => ack (x, x))"
```
```   343   apply (rule notI)
```
```   344   apply (erule ack_bounds_PRIMREC [THEN exE])
```
```   345   apply (rule less_irrefl)
```
```   346   apply (drule_tac x = "[x]" in spec)
```
```   347   apply simp
```
```   348   done
```
```   349
```
```   350 end
```