(* Authors: Jeremy Avigad and Amine Chaieb *)

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

theory Nat_Transfer

imports Int

begin

subsection {* Generic transfer machinery *}

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

  where "transfer_morphism f A \<longleftrightarrow> True"

lemma transfer_morphismI[intro]: "transfer_morphism f A"

  by (simp add: transfer_morphism_def)

ML_file "Tools/legacy_transfer.ML"

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

text {* set up transfer direction *}

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

declare transfer_morphism_nat_int [transfer add

  mode: manual

  return: nat_0_le

  labels: nat_int

]

text {* basic functions and relations *}

lemma transfer_nat_int_numerals [transfer key: transfer_morphism_nat_int]:

    "(0::nat) = nat 0"

    "(1::nat) = nat 1"

    "(2::nat) = nat 2"

    "(3::nat) = nat 3"

  by auto

definition

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

where

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

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

  by (simp add: tsub_def)

lemma transfer_nat_int_functions [transfer key: transfer_morphism_nat_int]:

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

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

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

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

  by (auto simp add: eq_nat_nat_iff nat_mult_distrib

      nat_power_eq tsub_def)

lemma transfer_nat_int_function_closures [transfer key: transfer_morphism_nat_int]:

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

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

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

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

    "(0::int) >= 0"

    "(1::int) >= 0"

    "(2::int) >= 0"

    "(3::int) >= 0"

    "int z >= 0"

  by (auto simp add: zero_le_mult_iff tsub_def)

lemma transfer_nat_int_relations [transfer key: transfer_morphism_nat_int]:

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

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

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

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

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

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

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

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

  by (auto simp add: zdvd_int)

text {* first-order quantifiers *}

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

  by (simp split add: split_nat)

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

proof

  assume "\<exists>x. P x"

  then obtain x where "P x" ..

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

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

next

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

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

qed

lemma transfer_nat_int_quantifiers [transfer key: transfer_morphism_nat_int]:

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

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

  by (rule all_nat, rule ex_nat)

(* should we restrict these? *)

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

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

  by auto

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

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

  by auto

declare transfer_morphism_nat_int [transfer add

  cong: all_cong ex_cong]

text {* if *}

lemma nat_if_cong [transfer key: transfer_morphism_nat_int]:

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

  by auto

text {* operations with sets *}

definition

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

where

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

lemma transfer_nat_int_set_functions:

    "card A = card (int ` A)"

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

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

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

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

  apply (rule card_image [symmetric])

  apply (auto simp add: inj_on_def image_def)

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

  apply auto

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

  apply auto

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

  apply auto

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

  apply auto

done

lemma transfer_nat_int_set_function_closures:

    "nat_set {}"

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

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

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

    "nat_set (int ` C)"

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

  unfolding nat_set_def apply auto

done

lemma transfer_nat_int_set_relations:

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

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

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

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

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

  apply (rule iffI)

  apply (erule finite_imageI)

  apply (erule finite_imageD)

  apply (auto simp add: image_def set_eq_iff inj_on_def)

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

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

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

done

lemma transfer_nat_int_set_return_embed: "nat_set A \<Longrightarrow>

    (int ` nat ` A = A)"

  by (auto simp add: nat_set_def image_def)

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

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

  by auto

declare transfer_morphism_nat_int [transfer add

  return: transfer_nat_int_set_functions

    transfer_nat_int_set_function_closures

    transfer_nat_int_set_relations

    transfer_nat_int_set_return_embed

  cong: transfer_nat_int_set_cong

]

text {* setsum and setprod *}

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

lemma transfer_nat_int_sum_prod:

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

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

  apply (subst setsum.reindex)

  apply (unfold inj_on_def, auto)

  apply (subst setprod.reindex)

  apply (unfold inj_on_def o_def, auto)

done

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

lemma transfer_nat_int_sum_prod2:

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

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

  apply (subst int_setsum [symmetric])

  apply auto

  apply (subst int_setprod [symmetric])

  apply auto

done

lemma transfer_nat_int_sum_prod_closure:

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

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

  unfolding nat_set_def

  apply (rule setsum_nonneg)

  apply auto

  apply (rule setprod_nonneg)

  apply auto

done

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

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

  also: what does =simp=> do?

lemma transfer_nat_int_sum_prod_closure:

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

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

  unfolding nat_set_def simp_implies_def

  apply (rule setsum_nonneg)

  apply auto

  apply (rule setprod_nonneg)

  apply auto

done

*)

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

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

   with the previously mentioned rule turned on? *)

lemma transfer_nat_int_sum_prod_cong:

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

      setsum f A = setsum g B"

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

      setprod f A = setprod g B"

  unfolding nat_set_def

  apply (subst setsum.cong, assumption)

  apply auto [2]

  apply (subst setprod.cong, assumption, auto)

done

declare transfer_morphism_nat_int [transfer add

  return: transfer_nat_int_sum_prod transfer_nat_int_sum_prod2

    transfer_nat_int_sum_prod_closure

  cong: transfer_nat_int_sum_prod_cong]

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

text {* set up transfer direction *}

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

declare transfer_morphism_int_nat [transfer add

  mode: manual

  return: nat_int

  labels: int_nat

]

text {* basic functions and relations *}

definition

  is_nat :: "int \<Rightarrow> bool"

where

  "is_nat x = (x >= 0)"

lemma transfer_int_nat_numerals:

    "0 = int 0"

    "1 = int 1"

    "2 = int 2"

    "3 = int 3"

  by auto

lemma transfer_int_nat_functions:

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

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

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

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

  by (auto simp add: int_mult tsub_def int_power)

lemma transfer_int_nat_function_closures:

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

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

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

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

    "is_nat 0"

    "is_nat 1"

    "is_nat 2"

    "is_nat 3"

    "is_nat (int z)"

  by (simp_all only: is_nat_def transfer_nat_int_function_closures)

lemma transfer_int_nat_relations:

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

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

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

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

  by (auto simp add: zdvd_int)

declare transfer_morphism_int_nat [transfer add return:

  transfer_int_nat_numerals

  transfer_int_nat_functions

  transfer_int_nat_function_closures

  transfer_int_nat_relations

]

text {* first-order quantifiers *}

lemma transfer_int_nat_quantifiers:

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

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

  apply (subst all_nat)

  apply auto [1]

  apply (subst ex_nat)

  apply auto

done

declare transfer_morphism_int_nat [transfer add

  return: transfer_int_nat_quantifiers]

text {* if *}

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

    int (if P then x else y)"

  by auto

declare transfer_morphism_int_nat [transfer add return: int_if_cong]

text {* operations with sets *}

lemma transfer_int_nat_set_functions:

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

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

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

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

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

       (* need all variants of these! *)

  by (simp_all only: is_nat_def transfer_nat_int_set_functions

          transfer_nat_int_set_function_closures

          transfer_nat_int_set_return_embed nat_0_le

          cong: transfer_nat_int_set_cong)

lemma transfer_int_nat_set_function_closures:

    "nat_set {}"

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

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

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

    "nat_set (int ` C)"

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

  by (simp_all only: transfer_nat_int_set_function_closures is_nat_def)

lemma transfer_int_nat_set_relations:

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

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

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

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

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

  by (simp_all only: is_nat_def transfer_nat_int_set_relations

    transfer_nat_int_set_return_embed nat_0_le)

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

  by (simp only: transfer_nat_int_set_relations

    transfer_nat_int_set_function_closures

    transfer_nat_int_set_return_embed nat_0_le)

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

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

  by auto

declare transfer_morphism_int_nat [transfer add

  return: transfer_int_nat_set_functions

    transfer_int_nat_set_function_closures

    transfer_int_nat_set_relations

    transfer_int_nat_set_return_embed

  cong: transfer_int_nat_set_cong

]

text {* setsum and setprod *}

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

lemma transfer_int_nat_sum_prod:

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

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

  apply (subst setsum.reindex)

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

  apply (subst setprod.reindex)

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

            cong: setprod.cong)

done

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

lemma transfer_int_nat_sum_prod2:

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

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

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

  unfolding is_nat_def

  apply (subst int_setsum, auto)

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

done

declare transfer_morphism_int_nat [transfer add

  return: transfer_int_nat_sum_prod transfer_int_nat_sum_prod2

  cong: setsum.cong setprod.cong]

end