isabelle: src/HOL/Nat_Transfer.thy@f16e4fb20652

     2 (* Authors: Jeremy Avigad and Amine Chaieb *)

     4 section {* Generic transfer machinery;  specific transfer from nats to ints and back. *}

     6 theory Nat_Transfer

     7 imports Int

     8 begin

    10 subsection {* Generic transfer machinery *}

    12 definition transfer_morphism:: "('b \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> bool) \<Rightarrow> bool"

    13   where "transfer_morphism f A \<longleftrightarrow> True"

    15 lemma transfer_morphismI[intro]: "transfer_morphism f A"

    16   by (simp add: transfer_morphism_def)

    18 ML_file "Tools/legacy_transfer.ML"

    21 subsection {* Set up transfer from nat to int *}

    23 text {* set up transfer direction *}

    25 lemma transfer_morphism_nat_int: "transfer_morphism nat (op <= (0::int))" ..

    27 declare transfer_morphism_nat_int [transfer add

    28   mode: manual

    29   return: nat_0_le

    30   labels: nat_int

    31 ]

    33 text {* basic functions and relations *}

    35 lemma transfer_nat_int_numerals [transfer key: transfer_morphism_nat_int]:

    36     "(0::nat) = nat 0"

    37     "(1::nat) = nat 1"

    38     "(2::nat) = nat 2"

    39     "(3::nat) = nat 3"

    40   by auto

    42 definition

    43   tsub :: "int \<Rightarrow> int \<Rightarrow> int"

    44 where

    45   "tsub x y = (if x >= y then x - y else 0)"

    47 lemma tsub_eq: "x >= y \<Longrightarrow> tsub x y = x - y"

    48   by (simp add: tsub_def)

    50 lemma transfer_nat_int_functions [transfer key: transfer_morphism_nat_int]:

    51     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) + (nat y) = nat (x + y)"

    52     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) * (nat y) = nat (x * y)"

    53     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) - (nat y) = nat (tsub x y)"

    54     "(x::int) >= 0 \<Longrightarrow> (nat x)^n = nat (x^n)"

    55   by (auto simp add: eq_nat_nat_iff nat_mult_distrib

    56       nat_power_eq tsub_def)

    58 lemma transfer_nat_int_function_closures [transfer key: transfer_morphism_nat_int]:

    59     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x + y >= 0"

    60     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x * y >= 0"

    61     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> tsub x y >= 0"

    62     "(x::int) >= 0 \<Longrightarrow> x^n >= 0"

    63     "(0::int) >= 0"

    64     "(1::int) >= 0"

    65     "(2::int) >= 0"

    66     "(3::int) >= 0"

    67     "int z >= 0"

    68   by (auto simp add: zero_le_mult_iff tsub_def)

    70 lemma transfer_nat_int_relations [transfer key: transfer_morphism_nat_int]:

    71     "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow>

    72       (nat (x::int) = nat y) = (x = y)"

    73     "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow>

    74       (nat (x::int) < nat y) = (x < y)"

    75     "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow>

    76       (nat (x::int) <= nat y) = (x <= y)"

    77     "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow>

    78       (nat (x::int) dvd nat y) = (x dvd y)"

    79   by (auto simp add: zdvd_int)

    82 text {* first-order quantifiers *}

    84 lemma all_nat: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x\<ge>0. P (nat x))"

    85   by (simp split add: split_nat)

    87 lemma ex_nat: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. 0 \<le> x \<and> P (nat x))"

    88 proof

    89   assume "\<exists>x. P x"

    90   then obtain x where "P x" ..

    91   then have "int x \<ge> 0 \<and> P (nat (int x))" by simp

    92   then show "\<exists>x\<ge>0. P (nat x)" ..

    93 next

    94   assume "\<exists>x\<ge>0. P (nat x)"

    95   then show "\<exists>x. P x" by auto

    96 qed

    98 lemma transfer_nat_int_quantifiers [transfer key: transfer_morphism_nat_int]:

    99     "(ALL (x::nat). P x) = (ALL (x::int). x >= 0 \<longrightarrow> P (nat x))"

   100     "(EX (x::nat). P x) = (EX (x::int). x >= 0 & P (nat x))"

   101   by (rule all_nat, rule ex_nat)

   103 (* should we restrict these? *)

   104 lemma all_cong: "(\<And>x. Q x \<Longrightarrow> P x = P' x) \<Longrightarrow>

   105     (ALL x. Q x \<longrightarrow> P x) = (ALL x. Q x \<longrightarrow> P' x)"

   106   by auto

   108 lemma ex_cong: "(\<And>x. Q x \<Longrightarrow> P x = P' x) \<Longrightarrow>

   109     (EX x. Q x \<and> P x) = (EX x. Q x \<and> P' x)"

   110   by auto

   112 declare transfer_morphism_nat_int [transfer add

   113   cong: all_cong ex_cong]

   116 text {* if *}

   118 lemma nat_if_cong [transfer key: transfer_morphism_nat_int]:

   119   "(if P then (nat x) else (nat y)) = nat (if P then x else y)"

   120   by auto

   123 text {* operations with sets *}

   125 definition

   126   nat_set :: "int set \<Rightarrow> bool"

   127 where

   128   "nat_set S = (ALL x:S. x >= 0)"

   130 lemma transfer_nat_int_set_functions:

   131     "card A = card (int ` A)"

   132     "{} = nat ` ({}::int set)"

   133     "A Un B = nat ` (int ` A Un int ` B)"

   134     "A Int B = nat ` (int ` A Int int ` B)"

   135     "{x. P x} = nat ` {x. x >= 0 & P(nat x)}"

   136   apply (rule card_image [symmetric])

   137   apply (auto simp add: inj_on_def image_def)

   138   apply (rule_tac x = "int x" in bexI)

   139   apply auto

   140   apply (rule_tac x = "int x" in bexI)

   141   apply auto

   142   apply (rule_tac x = "int x" in bexI)

   143   apply auto

   144   apply (rule_tac x = "int x" in exI)

   145   apply auto

   146 done

   148 lemma transfer_nat_int_set_function_closures:

   149     "nat_set {}"

   150     "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> nat_set (A Un B)"

   151     "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> nat_set (A Int B)"

   152     "nat_set {x. x >= 0 & P x}"

   153     "nat_set (int ` C)"

   154     "nat_set A \<Longrightarrow> x : A \<Longrightarrow> x >= 0" (* does it hurt to turn this on? *)

   155   unfolding nat_set_def apply auto

   156 done

   158 lemma transfer_nat_int_set_relations:

   159     "(finite A) = (finite (int ` A))"

   160     "(x : A) = (int x : int ` A)"

   161     "(A = B) = (int ` A = int ` B)"

   162     "(A < B) = (int ` A < int ` B)"

   163     "(A <= B) = (int ` A <= int ` B)"

   164   apply (rule iffI)

   165   apply (erule finite_imageI)

   166   apply (erule finite_imageD)

   167   apply (auto simp add: image_def set_eq_iff inj_on_def)

   168   apply (drule_tac x = "int x" in spec, auto)

   169   apply (drule_tac x = "int x" in spec, auto)

   170   apply (drule_tac x = "int x" in spec, auto)

   171 done

   173 lemma transfer_nat_int_set_return_embed: "nat_set A \<Longrightarrow>

   174     (int ` nat ` A = A)"

   175   by (auto simp add: nat_set_def image_def)

   177 lemma transfer_nat_int_set_cong: "(!!x. x >= 0 \<Longrightarrow> P x = P' x) \<Longrightarrow>

   178     {(x::int). x >= 0 & P x} = {x. x >= 0 & P' x}"

   179   by auto

   181 declare transfer_morphism_nat_int [transfer add

   182   return: transfer_nat_int_set_functions

   183     transfer_nat_int_set_function_closures

   184     transfer_nat_int_set_relations

   185     transfer_nat_int_set_return_embed

   186   cong: transfer_nat_int_set_cong

   187 ]

   190 text {* setsum and setprod *}

   192 (* this handles the case where the *domain* of f is nat *)

   193 lemma transfer_nat_int_sum_prod:

   194     "setsum f A = setsum (%x. f (nat x)) (int ` A)"

   195     "setprod f A = setprod (%x. f (nat x)) (int ` A)"

   196   apply (subst setsum.reindex)

   197   apply (unfold inj_on_def, auto)

   198   apply (subst setprod.reindex)

   199   apply (unfold inj_on_def o_def, auto)

   200 done

   202 (* this handles the case where the *range* of f is nat *)

   203 lemma transfer_nat_int_sum_prod2:

   204     "setsum f A = nat(setsum (%x. int (f x)) A)"

   205     "setprod f A = nat(setprod (%x. int (f x)) A)"

   206   apply (subst int_setsum [symmetric])

   207   apply auto

   208   apply (subst int_setprod [symmetric])

   209   apply auto

   210 done

   212 lemma transfer_nat_int_sum_prod_closure:

   213     "nat_set A \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x >= (0::int)) \<Longrightarrow> setsum f A >= 0"

   214     "nat_set A \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x >= (0::int)) \<Longrightarrow> setprod f A >= 0"

   215   unfolding nat_set_def

   216   apply (rule setsum_nonneg)

   217   apply auto

   218   apply (rule setprod_nonneg)

   219   apply auto

   220 done

   222 (* this version doesn't work, even with nat_set A \<Longrightarrow>

   223       x : A \<Longrightarrow> x >= 0 turned on. Why not?

   225   also: what does =simp=> do?

   227 lemma transfer_nat_int_sum_prod_closure:

   228     "(!!x. x : A  ==> f x >= (0::int)) \<Longrightarrow> setsum f A >= 0"

   229     "(!!x. x : A  ==> f x >= (0::int)) \<Longrightarrow> setprod f A >= 0"

   230   unfolding nat_set_def simp_implies_def

   231   apply (rule setsum_nonneg)

   232   apply auto

   233   apply (rule setprod_nonneg)

   234   apply auto

   235 done

   236 *)

   238 (* Making A = B in this lemma doesn't work. Why not?

   239    Also, why aren't setsum.cong and setprod.cong enough,

   240    with the previously mentioned rule turned on? *)

   242 lemma transfer_nat_int_sum_prod_cong:

   243     "A = B \<Longrightarrow> nat_set B \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x = g x) \<Longrightarrow>

   244       setsum f A = setsum g B"

   245     "A = B \<Longrightarrow> nat_set B \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x = g x) \<Longrightarrow>

   246       setprod f A = setprod g B"

   247   unfolding nat_set_def

   248   apply (subst setsum.cong, assumption)

   249   apply auto [2]

   250   apply (subst setprod.cong, assumption, auto)

   251 done

   253 declare transfer_morphism_nat_int [transfer add

   254   return: transfer_nat_int_sum_prod transfer_nat_int_sum_prod2

   255     transfer_nat_int_sum_prod_closure

   256   cong: transfer_nat_int_sum_prod_cong]

   259 subsection {* Set up transfer from int to nat *}

   261 text {* set up transfer direction *}

   263 lemma transfer_morphism_int_nat: "transfer_morphism int (\<lambda>n. True)" ..

   265 declare transfer_morphism_int_nat [transfer add

   266   mode: manual

   267   return: nat_int

   268   labels: int_nat

   269 ]

   272 text {* basic functions and relations *}

   274 definition

   275   is_nat :: "int \<Rightarrow> bool"

   276 where

   277   "is_nat x = (x >= 0)"

   279 lemma transfer_int_nat_numerals:

   280     "0 = int 0"

   281     "1 = int 1"

   282     "2 = int 2"

   283     "3 = int 3"

   284   by auto

   286 lemma transfer_int_nat_functions:

   287     "(int x) + (int y) = int (x + y)"

   288     "(int x) * (int y) = int (x * y)"

   289     "tsub (int x) (int y) = int (x - y)"

   290     "(int x)^n = int (x^n)"

   291   by (auto simp add: int_mult tsub_def int_power)

   293 lemma transfer_int_nat_function_closures:

   294     "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x + y)"

   295     "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x * y)"

   296     "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (tsub x y)"

   297     "is_nat x \<Longrightarrow> is_nat (x^n)"

   298     "is_nat 0"

   299     "is_nat 1"

   300     "is_nat 2"

   301     "is_nat 3"

   302     "is_nat (int z)"

   303   by (simp_all only: is_nat_def transfer_nat_int_function_closures)

   305 lemma transfer_int_nat_relations:

   306     "(int x = int y) = (x = y)"

   307     "(int x < int y) = (x < y)"

   308     "(int x <= int y) = (x <= y)"

   309     "(int x dvd int y) = (x dvd y)"

   310   by (auto simp add: zdvd_int)

   312 declare transfer_morphism_int_nat [transfer add return:

   313   transfer_int_nat_numerals

   314   transfer_int_nat_functions

   315   transfer_int_nat_function_closures

   316   transfer_int_nat_relations

   317 ]

   320 text {* first-order quantifiers *}

   322 lemma transfer_int_nat_quantifiers:

   323     "(ALL (x::int) >= 0. P x) = (ALL (x::nat). P (int x))"

   324     "(EX (x::int) >= 0. P x) = (EX (x::nat). P (int x))"

   325   apply (subst all_nat)

   326   apply auto [1]

   327   apply (subst ex_nat)

   328   apply auto

   329 done

   331 declare transfer_morphism_int_nat [transfer add

   332   return: transfer_int_nat_quantifiers]

   335 text {* if *}

   337 lemma int_if_cong: "(if P then (int x) else (int y)) =

   338     int (if P then x else y)"

   339   by auto

   341 declare transfer_morphism_int_nat [transfer add return: int_if_cong]

   345 text {* operations with sets *}

   347 lemma transfer_int_nat_set_functions:

   348     "nat_set A \<Longrightarrow> card A = card (nat ` A)"

   349     "{} = int ` ({}::nat set)"

   350     "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> A Un B = int ` (nat ` A Un nat ` B)"

   351     "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> A Int B = int ` (nat ` A Int nat ` B)"

   352     "{x. x >= 0 & P x} = int ` {x. P(int x)}"

   353        (* need all variants of these! *)

   354   by (simp_all only: is_nat_def transfer_nat_int_set_functions

   355           transfer_nat_int_set_function_closures

   356           transfer_nat_int_set_return_embed nat_0_le

   357           cong: transfer_nat_int_set_cong)

   359 lemma transfer_int_nat_set_function_closures:

   360     "nat_set {}"

   361     "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> nat_set (A Un B)"

   362     "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> nat_set (A Int B)"

   363     "nat_set {x. x >= 0 & P x}"

   364     "nat_set (int ` C)"

   365     "nat_set A \<Longrightarrow> x : A \<Longrightarrow> is_nat x"

   366   by (simp_all only: transfer_nat_int_set_function_closures is_nat_def)

   368 lemma transfer_int_nat_set_relations:

   369     "nat_set A \<Longrightarrow> finite A = finite (nat ` A)"

   370     "is_nat x \<Longrightarrow> nat_set A \<Longrightarrow> (x : A) = (nat x : nat ` A)"

   371     "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> (A = B) = (nat ` A = nat ` B)"

   372     "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> (A < B) = (nat ` A < nat ` B)"

   373     "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> (A <= B) = (nat ` A <= nat ` B)"

   374   by (simp_all only: is_nat_def transfer_nat_int_set_relations

   375     transfer_nat_int_set_return_embed nat_0_le)

   377 lemma transfer_int_nat_set_return_embed: "nat ` int ` A = A"

   378   by (simp only: transfer_nat_int_set_relations

   379     transfer_nat_int_set_function_closures

   380     transfer_nat_int_set_return_embed nat_0_le)

   382 lemma transfer_int_nat_set_cong: "(!!x. P x = P' x) \<Longrightarrow>

   383     {(x::nat). P x} = {x. P' x}"

   384   by auto

   386 declare transfer_morphism_int_nat [transfer add

   387   return: transfer_int_nat_set_functions

   388     transfer_int_nat_set_function_closures

   389     transfer_int_nat_set_relations

   390     transfer_int_nat_set_return_embed

   391   cong: transfer_int_nat_set_cong

   392 ]

   395 text {* setsum and setprod *}

   397 (* this handles the case where the *domain* of f is int *)

   398 lemma transfer_int_nat_sum_prod:

   399     "nat_set A \<Longrightarrow> setsum f A = setsum (%x. f (int x)) (nat ` A)"

   400     "nat_set A \<Longrightarrow> setprod f A = setprod (%x. f (int x)) (nat ` A)"

   401   apply (subst setsum.reindex)

   402   apply (unfold inj_on_def nat_set_def, auto simp add: eq_nat_nat_iff)

   403   apply (subst setprod.reindex)

   404   apply (unfold inj_on_def nat_set_def o_def, auto simp add: eq_nat_nat_iff

   405             cong: setprod.cong)

   406 done

   408 (* this handles the case where the *range* of f is int *)

   409 lemma transfer_int_nat_sum_prod2:

   410     "(!!x. x:A \<Longrightarrow> is_nat (f x)) \<Longrightarrow> setsum f A = int(setsum (%x. nat (f x)) A)"

   411     "(!!x. x:A \<Longrightarrow> is_nat (f x)) \<Longrightarrow>

   412       setprod f A = int(setprod (%x. nat (f x)) A)"

   413   unfolding is_nat_def

   414   apply (subst int_setsum, auto)

   415   apply (subst int_setprod, auto simp add: cong: setprod.cong)

   416 done

   418 declare transfer_morphism_int_nat [transfer add

   419   return: transfer_int_nat_sum_prod transfer_int_nat_sum_prod2

   420   cong: setsum.cong setprod.cong]

   422 end

author	haftmann
	Fri Jun 19 07:53:35 2015 +0200 (2015-06-19)
changeset 60517	f16e4fb20652
parent 58889	5b7a9633cfa8
child 60758	d8d85a8172b5
permissions	-rw-r--r--