src/HOL/NatTransfer.thy
 author nipkow Fri Aug 28 18:52:41 2009 +0200 (2009-08-28) changeset 32436 10cd49e0c067 parent 32264 0be31453f698 child 32554 4ccd84fb19d3 permissions -rw-r--r--
Turned "x <= y ==> sup x y = y" (and relatives) into simp rules
```     1 (*  Title:      HOL/Library/NatTransfer.thy
```
```     2     Authors:    Jeremy Avigad and Amine Chaieb
```
```     3
```
```     4     Sets up transfer from nats to ints and
```
```     5     back.
```
```     6 *)
```
```     7
```
```     8
```
```     9 header {* NatTransfer *}
```
```    10
```
```    11 theory NatTransfer
```
```    12 imports Main Parity
```
```    13 uses ("Tools/transfer_data.ML")
```
```    14 begin
```
```    15
```
```    16 subsection {* A transfer Method between isomorphic domains*}
```
```    17
```
```    18 definition TransferMorphism:: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> bool"
```
```    19   where "TransferMorphism a B = True"
```
```    20
```
```    21 use "Tools/transfer_data.ML"
```
```    22
```
```    23 setup TransferData.setup
```
```    24
```
```    25
```
```    26 subsection {* Set up transfer from nat to int *}
```
```    27
```
```    28 (* set up transfer direction *)
```
```    29
```
```    30 lemma TransferMorphism_nat_int: "TransferMorphism nat (op <= (0::int))"
```
```    31   by (simp add: TransferMorphism_def)
```
```    32
```
```    33 declare TransferMorphism_nat_int[transfer
```
```    34   add mode: manual
```
```    35   return: nat_0_le
```
```    36   labels: natint
```
```    37 ]
```
```    38
```
```    39 (* basic functions and relations *)
```
```    40
```
```    41 lemma transfer_nat_int_numerals:
```
```    42     "(0::nat) = nat 0"
```
```    43     "(1::nat) = nat 1"
```
```    44     "(2::nat) = nat 2"
```
```    45     "(3::nat) = nat 3"
```
```    46   by auto
```
```    47
```
```    48 definition
```
```    49   tsub :: "int \<Rightarrow> int \<Rightarrow> int"
```
```    50 where
```
```    51   "tsub x y = (if x >= y then x - y else 0)"
```
```    52
```
```    53 lemma tsub_eq: "x >= y \<Longrightarrow> tsub x y = x - y"
```
```    54   by (simp add: tsub_def)
```
```    55
```
```    56
```
```    57 lemma transfer_nat_int_functions:
```
```    58     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) + (nat y) = nat (x + y)"
```
```    59     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) * (nat y) = nat (x * y)"
```
```    60     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) - (nat y) = nat (tsub x y)"
```
```    61     "(x::int) >= 0 \<Longrightarrow> (nat x)^n = nat (x^n)"
```
```    62     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) div (nat y) = nat (x div y)"
```
```    63     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) mod (nat y) = nat (x mod y)"
```
```    64   by (auto simp add: eq_nat_nat_iff nat_mult_distrib
```
```    65       nat_power_eq nat_div_distrib nat_mod_distrib tsub_def)
```
```    66
```
```    67 lemma transfer_nat_int_function_closures:
```
```    68     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x + y >= 0"
```
```    69     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x * y >= 0"
```
```    70     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> tsub x y >= 0"
```
```    71     "(x::int) >= 0 \<Longrightarrow> x^n >= 0"
```
```    72     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x div y >= 0"
```
```    73     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x mod y >= 0"
```
```    74     "(0::int) >= 0"
```
```    75     "(1::int) >= 0"
```
```    76     "(2::int) >= 0"
```
```    77     "(3::int) >= 0"
```
```    78     "int z >= 0"
```
```    79   apply (auto simp add: zero_le_mult_iff tsub_def)
```
```    80   apply (case_tac "y = 0")
```
```    81   apply auto
```
```    82   apply (subst pos_imp_zdiv_nonneg_iff, auto)
```
```    83   apply (case_tac "y = 0")
```
```    84   apply force
```
```    85   apply (rule pos_mod_sign)
```
```    86   apply arith
```
```    87 done
```
```    88
```
```    89 lemma transfer_nat_int_relations:
```
```    90     "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow>
```
```    91       (nat (x::int) = nat y) = (x = y)"
```
```    92     "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow>
```
```    93       (nat (x::int) < nat y) = (x < y)"
```
```    94     "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow>
```
```    95       (nat (x::int) <= nat y) = (x <= y)"
```
```    96     "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow>
```
```    97       (nat (x::int) dvd nat y) = (x dvd y)"
```
```    98   by (auto simp add: zdvd_int even_nat_def)
```
```    99
```
```   100 declare TransferMorphism_nat_int[transfer add return:
```
```   101   transfer_nat_int_numerals
```
```   102   transfer_nat_int_functions
```
```   103   transfer_nat_int_function_closures
```
```   104   transfer_nat_int_relations
```
```   105 ]
```
```   106
```
```   107
```
```   108 (* first-order quantifiers *)
```
```   109
```
```   110 lemma transfer_nat_int_quantifiers:
```
```   111     "(ALL (x::nat). P x) = (ALL (x::int). x >= 0 \<longrightarrow> P (nat x))"
```
```   112     "(EX (x::nat). P x) = (EX (x::int). x >= 0 & P (nat x))"
```
```   113   by (rule all_nat, rule ex_nat)
```
```   114
```
```   115 (* should we restrict these? *)
```
```   116 lemma all_cong: "(\<And>x. Q x \<Longrightarrow> P x = P' x) \<Longrightarrow>
```
```   117     (ALL x. Q x \<longrightarrow> P x) = (ALL x. Q x \<longrightarrow> P' x)"
```
```   118   by auto
```
```   119
```
```   120 lemma ex_cong: "(\<And>x. Q x \<Longrightarrow> P x = P' x) \<Longrightarrow>
```
```   121     (EX x. Q x \<and> P x) = (EX x. Q x \<and> P' x)"
```
```   122   by auto
```
```   123
```
```   124 declare TransferMorphism_nat_int[transfer add
```
```   125   return: transfer_nat_int_quantifiers
```
```   126   cong: all_cong ex_cong]
```
```   127
```
```   128
```
```   129 (* if *)
```
```   130
```
```   131 lemma nat_if_cong: "(if P then (nat x) else (nat y)) =
```
```   132     nat (if P then x else y)"
```
```   133   by auto
```
```   134
```
```   135 declare TransferMorphism_nat_int [transfer add return: nat_if_cong]
```
```   136
```
```   137
```
```   138 (* operations with sets *)
```
```   139
```
```   140 definition
```
```   141   nat_set :: "int set \<Rightarrow> bool"
```
```   142 where
```
```   143   "nat_set S = (ALL x:S. x >= 0)"
```
```   144
```
```   145 lemma transfer_nat_int_set_functions:
```
```   146     "card A = card (int ` A)"
```
```   147     "{} = nat ` ({}::int set)"
```
```   148     "A Un B = nat ` (int ` A Un int ` B)"
```
```   149     "A Int B = nat ` (int ` A Int int ` B)"
```
```   150     "{x. P x} = nat ` {x. x >= 0 & P(nat x)}"
```
```   151     "{..n} = nat ` {0..int n}"
```
```   152     "{m..n} = nat ` {int m..int n}"  (* need all variants of these! *)
```
```   153   apply (rule card_image [symmetric])
```
```   154   apply (auto simp add: inj_on_def image_def)
```
```   155   apply (rule_tac x = "int x" in bexI)
```
```   156   apply auto
```
```   157   apply (rule_tac x = "int x" in bexI)
```
```   158   apply auto
```
```   159   apply (rule_tac x = "int x" in bexI)
```
```   160   apply auto
```
```   161   apply (rule_tac x = "int x" in exI)
```
```   162   apply auto
```
```   163   apply (rule_tac x = "int x" in bexI)
```
```   164   apply auto
```
```   165   apply (rule_tac x = "int x" in bexI)
```
```   166   apply auto
```
```   167 done
```
```   168
```
```   169 lemma transfer_nat_int_set_function_closures:
```
```   170     "nat_set {}"
```
```   171     "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> nat_set (A Un B)"
```
```   172     "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> nat_set (A Int B)"
```
```   173     "x >= 0 \<Longrightarrow> nat_set {x..y}"
```
```   174     "nat_set {x. x >= 0 & P x}"
```
```   175     "nat_set (int ` C)"
```
```   176     "nat_set A \<Longrightarrow> x : A \<Longrightarrow> x >= 0" (* does it hurt to turn this on? *)
```
```   177   unfolding nat_set_def apply auto
```
```   178 done
```
```   179
```
```   180 lemma transfer_nat_int_set_relations:
```
```   181     "(finite A) = (finite (int ` A))"
```
```   182     "(x : A) = (int x : int ` A)"
```
```   183     "(A = B) = (int ` A = int ` B)"
```
```   184     "(A < B) = (int ` A < int ` B)"
```
```   185     "(A <= B) = (int ` A <= int ` B)"
```
```   186
```
```   187   apply (rule iffI)
```
```   188   apply (erule finite_imageI)
```
```   189   apply (erule finite_imageD)
```
```   190   apply (auto simp add: image_def expand_set_eq inj_on_def)
```
```   191   apply (drule_tac x = "int x" in spec, auto)
```
```   192   apply (drule_tac x = "int x" in spec, auto)
```
```   193   apply (drule_tac x = "int x" in spec, auto)
```
```   194 done
```
```   195
```
```   196 lemma transfer_nat_int_set_return_embed: "nat_set A \<Longrightarrow>
```
```   197     (int ` nat ` A = A)"
```
```   198   by (auto simp add: nat_set_def image_def)
```
```   199
```
```   200 lemma transfer_nat_int_set_cong: "(!!x. x >= 0 \<Longrightarrow> P x = P' x) \<Longrightarrow>
```
```   201     {(x::int). x >= 0 & P x} = {x. x >= 0 & P' x}"
```
```   202   by auto
```
```   203
```
```   204 declare TransferMorphism_nat_int[transfer add
```
```   205   return: transfer_nat_int_set_functions
```
```   206     transfer_nat_int_set_function_closures
```
```   207     transfer_nat_int_set_relations
```
```   208     transfer_nat_int_set_return_embed
```
```   209   cong: transfer_nat_int_set_cong
```
```   210 ]
```
```   211
```
```   212
```
```   213 (* setsum and setprod *)
```
```   214
```
```   215 (* this handles the case where the *domain* of f is nat *)
```
```   216 lemma transfer_nat_int_sum_prod:
```
```   217     "setsum f A = setsum (%x. f (nat x)) (int ` A)"
```
```   218     "setprod f A = setprod (%x. f (nat x)) (int ` A)"
```
```   219   apply (subst setsum_reindex)
```
```   220   apply (unfold inj_on_def, auto)
```
```   221   apply (subst setprod_reindex)
```
```   222   apply (unfold inj_on_def o_def, auto)
```
```   223 done
```
```   224
```
```   225 (* this handles the case where the *range* of f is nat *)
```
```   226 lemma transfer_nat_int_sum_prod2:
```
```   227     "setsum f A = nat(setsum (%x. int (f x)) A)"
```
```   228     "setprod f A = nat(setprod (%x. int (f x)) A)"
```
```   229   apply (subst int_setsum [symmetric])
```
```   230   apply auto
```
```   231   apply (subst int_setprod [symmetric])
```
```   232   apply auto
```
```   233 done
```
```   234
```
```   235 lemma transfer_nat_int_sum_prod_closure:
```
```   236     "nat_set A \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x >= (0::int)) \<Longrightarrow> setsum f A >= 0"
```
```   237     "nat_set A \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x >= (0::int)) \<Longrightarrow> setprod f A >= 0"
```
```   238   unfolding nat_set_def
```
```   239   apply (rule setsum_nonneg)
```
```   240   apply auto
```
```   241   apply (rule setprod_nonneg)
```
```   242   apply auto
```
```   243 done
```
```   244
```
```   245 (* this version doesn't work, even with nat_set A \<Longrightarrow>
```
```   246       x : A \<Longrightarrow> x >= 0 turned on. Why not?
```
```   247
```
```   248   also: what does =simp=> do?
```
```   249
```
```   250 lemma transfer_nat_int_sum_prod_closure:
```
```   251     "(!!x. x : A  ==> f x >= (0::int)) \<Longrightarrow> setsum f A >= 0"
```
```   252     "(!!x. x : A  ==> f x >= (0::int)) \<Longrightarrow> setprod f A >= 0"
```
```   253   unfolding nat_set_def simp_implies_def
```
```   254   apply (rule setsum_nonneg)
```
```   255   apply auto
```
```   256   apply (rule setprod_nonneg)
```
```   257   apply auto
```
```   258 done
```
```   259 *)
```
```   260
```
```   261 (* Making A = B in this lemma doesn't work. Why not?
```
```   262    Also, why aren't setsum_cong and setprod_cong enough,
```
```   263    with the previously mentioned rule turned on? *)
```
```   264
```
```   265 lemma transfer_nat_int_sum_prod_cong:
```
```   266     "A = B \<Longrightarrow> nat_set B \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x = g x) \<Longrightarrow>
```
```   267       setsum f A = setsum g B"
```
```   268     "A = B \<Longrightarrow> nat_set B \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x = g x) \<Longrightarrow>
```
```   269       setprod f A = setprod g B"
```
```   270   unfolding nat_set_def
```
```   271   apply (subst setsum_cong, assumption)
```
```   272   apply auto [2]
```
```   273   apply (subst setprod_cong, assumption, auto)
```
```   274 done
```
```   275
```
```   276 declare TransferMorphism_nat_int[transfer add
```
```   277   return: transfer_nat_int_sum_prod transfer_nat_int_sum_prod2
```
```   278     transfer_nat_int_sum_prod_closure
```
```   279   cong: transfer_nat_int_sum_prod_cong]
```
```   280
```
```   281 (* lists *)
```
```   282
```
```   283 definition
```
```   284   embed_list :: "nat list \<Rightarrow> int list"
```
```   285 where
```
```   286   "embed_list l = map int l";
```
```   287
```
```   288 definition
```
```   289   nat_list :: "int list \<Rightarrow> bool"
```
```   290 where
```
```   291   "nat_list l = nat_set (set l)";
```
```   292
```
```   293 definition
```
```   294   return_list :: "int list \<Rightarrow> nat list"
```
```   295 where
```
```   296   "return_list l = map nat l";
```
```   297
```
```   298 thm nat_0_le;
```
```   299
```
```   300 lemma transfer_nat_int_list_return_embed: "nat_list l \<longrightarrow>
```
```   301     embed_list (return_list l) = l";
```
```   302   unfolding embed_list_def return_list_def nat_list_def nat_set_def
```
```   303   apply (induct l);
```
```   304   apply auto;
```
```   305 done;
```
```   306
```
```   307 lemma transfer_nat_int_list_functions:
```
```   308   "l @ m = return_list (embed_list l @ embed_list m)"
```
```   309   "[] = return_list []";
```
```   310   unfolding return_list_def embed_list_def;
```
```   311   apply auto;
```
```   312   apply (induct l, auto);
```
```   313   apply (induct m, auto);
```
```   314 done;
```
```   315
```
```   316 (*
```
```   317 lemma transfer_nat_int_fold1: "fold f l x =
```
```   318     fold (%x. f (nat x)) (embed_list l) x";
```
```   319 *)
```
```   320
```
```   321
```
```   322
```
```   323
```
```   324 subsection {* Set up transfer from int to nat *}
```
```   325
```
```   326 (* set up transfer direction *)
```
```   327
```
```   328 lemma TransferMorphism_int_nat: "TransferMorphism int (UNIV :: nat set)"
```
```   329   by (simp add: TransferMorphism_def)
```
```   330
```
```   331 declare TransferMorphism_int_nat[transfer add
```
```   332   mode: manual
```
```   333 (*  labels: int-nat *)
```
```   334   return: nat_int
```
```   335 ]
```
```   336
```
```   337
```
```   338 (* basic functions and relations *)
```
```   339
```
```   340 definition
```
```   341   is_nat :: "int \<Rightarrow> bool"
```
```   342 where
```
```   343   "is_nat x = (x >= 0)"
```
```   344
```
```   345 lemma transfer_int_nat_numerals:
```
```   346     "0 = int 0"
```
```   347     "1 = int 1"
```
```   348     "2 = int 2"
```
```   349     "3 = int 3"
```
```   350   by auto
```
```   351
```
```   352 lemma transfer_int_nat_functions:
```
```   353     "(int x) + (int y) = int (x + y)"
```
```   354     "(int x) * (int y) = int (x * y)"
```
```   355     "tsub (int x) (int y) = int (x - y)"
```
```   356     "(int x)^n = int (x^n)"
```
```   357     "(int x) div (int y) = int (x div y)"
```
```   358     "(int x) mod (int y) = int (x mod y)"
```
```   359   by (auto simp add: int_mult tsub_def int_power zdiv_int zmod_int)
```
```   360
```
```   361 lemma transfer_int_nat_function_closures:
```
```   362     "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x + y)"
```
```   363     "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x * y)"
```
```   364     "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (tsub x y)"
```
```   365     "is_nat x \<Longrightarrow> is_nat (x^n)"
```
```   366     "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x div y)"
```
```   367     "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x mod y)"
```
```   368     "is_nat 0"
```
```   369     "is_nat 1"
```
```   370     "is_nat 2"
```
```   371     "is_nat 3"
```
```   372     "is_nat (int z)"
```
```   373   by (simp_all only: is_nat_def transfer_nat_int_function_closures)
```
```   374
```
```   375 lemma transfer_int_nat_relations:
```
```   376     "(int x = int y) = (x = y)"
```
```   377     "(int x < int y) = (x < y)"
```
```   378     "(int x <= int y) = (x <= y)"
```
```   379     "(int x dvd int y) = (x dvd y)"
```
```   380     "(even (int x)) = (even x)"
```
```   381   by (auto simp add: zdvd_int even_nat_def)
```
```   382
```
```   383 lemma UNIV_apply:
```
```   384   "UNIV x = True"
```
```   385   by (simp add: top_fun_eq top_bool_eq)
```
```   386
```
```   387 declare TransferMorphism_int_nat[transfer add return:
```
```   388   transfer_int_nat_numerals
```
```   389   transfer_int_nat_functions
```
```   390   transfer_int_nat_function_closures
```
```   391   transfer_int_nat_relations
```
```   392   UNIV_apply
```
```   393 ]
```
```   394
```
```   395
```
```   396 (* first-order quantifiers *)
```
```   397
```
```   398 lemma transfer_int_nat_quantifiers:
```
```   399     "(ALL (x::int) >= 0. P x) = (ALL (x::nat). P (int x))"
```
```   400     "(EX (x::int) >= 0. P x) = (EX (x::nat). P (int x))"
```
```   401   apply (subst all_nat)
```
```   402   apply auto [1]
```
```   403   apply (subst ex_nat)
```
```   404   apply auto
```
```   405 done
```
```   406
```
```   407 declare TransferMorphism_int_nat[transfer add
```
```   408   return: transfer_int_nat_quantifiers]
```
```   409
```
```   410
```
```   411 (* if *)
```
```   412
```
```   413 lemma int_if_cong: "(if P then (int x) else (int y)) =
```
```   414     int (if P then x else y)"
```
```   415   by auto
```
```   416
```
```   417 declare TransferMorphism_int_nat [transfer add return: int_if_cong]
```
```   418
```
```   419
```
```   420
```
```   421 (* operations with sets *)
```
```   422
```
```   423 lemma transfer_int_nat_set_functions:
```
```   424     "nat_set A \<Longrightarrow> card A = card (nat ` A)"
```
```   425     "{} = int ` ({}::nat set)"
```
```   426     "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> A Un B = int ` (nat ` A Un nat ` B)"
```
```   427     "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> A Int B = int ` (nat ` A Int nat ` B)"
```
```   428     "{x. x >= 0 & P x} = int ` {x. P(int x)}"
```
```   429     "is_nat m \<Longrightarrow> is_nat n \<Longrightarrow> {m..n} = int ` {nat m..nat n}"
```
```   430        (* need all variants of these! *)
```
```   431   by (simp_all only: is_nat_def transfer_nat_int_set_functions
```
```   432           transfer_nat_int_set_function_closures
```
```   433           transfer_nat_int_set_return_embed nat_0_le
```
```   434           cong: transfer_nat_int_set_cong)
```
```   435
```
```   436 lemma transfer_int_nat_set_function_closures:
```
```   437     "nat_set {}"
```
```   438     "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> nat_set (A Un B)"
```
```   439     "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> nat_set (A Int B)"
```
```   440     "is_nat x \<Longrightarrow> nat_set {x..y}"
```
```   441     "nat_set {x. x >= 0 & P x}"
```
```   442     "nat_set (int ` C)"
```
```   443     "nat_set A \<Longrightarrow> x : A \<Longrightarrow> is_nat x"
```
```   444   by (simp_all only: transfer_nat_int_set_function_closures is_nat_def)
```
```   445
```
```   446 lemma transfer_int_nat_set_relations:
```
```   447     "nat_set A \<Longrightarrow> finite A = finite (nat ` A)"
```
```   448     "is_nat x \<Longrightarrow> nat_set A \<Longrightarrow> (x : A) = (nat x : nat ` A)"
```
```   449     "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> (A = B) = (nat ` A = nat ` B)"
```
```   450     "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> (A < B) = (nat ` A < nat ` B)"
```
```   451     "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> (A <= B) = (nat ` A <= nat ` B)"
```
```   452   by (simp_all only: is_nat_def transfer_nat_int_set_relations
```
```   453     transfer_nat_int_set_return_embed nat_0_le)
```
```   454
```
```   455 lemma transfer_int_nat_set_return_embed: "nat ` int ` A = A"
```
```   456   by (simp only: transfer_nat_int_set_relations
```
```   457     transfer_nat_int_set_function_closures
```
```   458     transfer_nat_int_set_return_embed nat_0_le)
```
```   459
```
```   460 lemma transfer_int_nat_set_cong: "(!!x. P x = P' x) \<Longrightarrow>
```
```   461     {(x::nat). P x} = {x. P' x}"
```
```   462   by auto
```
```   463
```
```   464 declare TransferMorphism_int_nat[transfer add
```
```   465   return: transfer_int_nat_set_functions
```
```   466     transfer_int_nat_set_function_closures
```
```   467     transfer_int_nat_set_relations
```
```   468     transfer_int_nat_set_return_embed
```
```   469   cong: transfer_int_nat_set_cong
```
```   470 ]
```
```   471
```
```   472
```
```   473 (* setsum and setprod *)
```
```   474
```
```   475 (* this handles the case where the *domain* of f is int *)
```
```   476 lemma transfer_int_nat_sum_prod:
```
```   477     "nat_set A \<Longrightarrow> setsum f A = setsum (%x. f (int x)) (nat ` A)"
```
```   478     "nat_set A \<Longrightarrow> setprod f A = setprod (%x. f (int x)) (nat ` A)"
```
```   479   apply (subst setsum_reindex)
```
```   480   apply (unfold inj_on_def nat_set_def, auto simp add: eq_nat_nat_iff)
```
```   481   apply (subst setprod_reindex)
```
```   482   apply (unfold inj_on_def nat_set_def o_def, auto simp add: eq_nat_nat_iff
```
```   483             cong: setprod_cong)
```
```   484 done
```
```   485
```
```   486 (* this handles the case where the *range* of f is int *)
```
```   487 lemma transfer_int_nat_sum_prod2:
```
```   488     "(!!x. x:A \<Longrightarrow> is_nat (f x)) \<Longrightarrow> setsum f A = int(setsum (%x. nat (f x)) A)"
```
```   489     "(!!x. x:A \<Longrightarrow> is_nat (f x)) \<Longrightarrow>
```
```   490       setprod f A = int(setprod (%x. nat (f x)) A)"
```
```   491   unfolding is_nat_def
```
```   492   apply (subst int_setsum, auto)
```
```   493   apply (subst int_setprod, auto simp add: cong: setprod_cong)
```
```   494 done
```
```   495
```
```   496 declare TransferMorphism_int_nat[transfer add
```
```   497   return: transfer_int_nat_sum_prod transfer_int_nat_sum_prod2
```
```   498   cong: setsum_cong setprod_cong]
```
```   499
```
```   500
```
```   501 subsection {* Test it out *}
```
```   502
```
```   503 (* nat to int *)
```
```   504
```
```   505 lemma ex1: "(x::nat) + y = y + x"
```
```   506   by auto
```
```   507
```
```   508 thm ex1 [transferred]
```
```   509
```
```   510 lemma ex2: "(a::nat) div b * b + a mod b = a"
```
```   511   by (rule mod_div_equality)
```
```   512
```
```   513 thm ex2 [transferred]
```
```   514
```
```   515 lemma ex3: "ALL (x::nat). ALL y. EX z. z >= x + y"
```
```   516   by auto
```
```   517
```
```   518 thm ex3 [transferred natint]
```
```   519
```
```   520 lemma ex4: "(x::nat) >= y \<Longrightarrow> (x - y) + y = x"
```
```   521   by auto
```
```   522
```
```   523 thm ex4 [transferred]
```
```   524
```
```   525 lemma ex5: "(2::nat) * (SUM i <= n. i) = n * (n + 1)"
```
```   526   by (induct n rule: nat_induct, auto)
```
```   527
```
```   528 thm ex5 [transferred]
```
```   529
```
```   530 theorem ex6: "0 <= (n::int) \<Longrightarrow> 2 * \<Sum>{0..n} = n * (n + 1)"
```
```   531   by (rule ex5 [transferred])
```
```   532
```
```   533 thm ex6 [transferred]
```
```   534
```
```   535 thm ex5 [transferred, transferred]
```
```   536
```
```   537 end
```