huffman@31708: 
haftmann@32554: (* Authors: Jeremy Avigad and Amine Chaieb *)
huffman@31708: 
haftmann@33318: header {* Generic transfer machinery;  specific transfer from nats to ints and back. *}
huffman@31708: 
haftmann@32558: theory Nat_Transfer
haftmann@33318: imports Nat_Numeral
haftmann@33318: uses ("Tools/transfer.ML")
huffman@31708: begin
huffman@31708: 
haftmann@33318: subsection {* Generic transfer machinery *}
haftmann@33318: 
haftmann@35821: definition transfer_morphism:: "('b \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> bool) \<Rightarrow> bool"
haftmann@35821:   where "transfer_morphism f A \<longleftrightarrow> (\<forall>P. (\<forall>x. P x) \<longrightarrow> (\<forall>y. A y \<longrightarrow> P (f y)))"
haftmann@35644: 
haftmann@35644: lemma transfer_morphismI:
haftmann@35821:   assumes "\<And>P y. (\<And>x. P x) \<Longrightarrow> A y \<Longrightarrow> P (f y)"
haftmann@35821:   shows "transfer_morphism f A"
haftmann@35821:   using assms by (auto simp add: transfer_morphism_def)
haftmann@33318: 
haftmann@33318: use "Tools/transfer.ML"
haftmann@33318: 
haftmann@33318: setup Transfer.setup
haftmann@33318: 
haftmann@33318: 
huffman@31708: subsection {* Set up transfer from nat to int *}
huffman@31708: 
haftmann@33318: text {* set up transfer direction *}
huffman@31708: 
haftmann@35644: lemma transfer_morphism_nat_int: "transfer_morphism nat (op <= (0::int))"
haftmann@35821:   by (rule transfer_morphismI) simp
huffman@31708: 
haftmann@35683: declare transfer_morphism_nat_int [transfer add
haftmann@35683:   mode: manual
huffman@31708:   return: nat_0_le
haftmann@35683:   labels: nat_int
huffman@31708: ]
huffman@31708: 
haftmann@33318: text {* basic functions and relations *}
huffman@31708: 
haftmann@35683: lemma transfer_nat_int_numerals [transfer key: transfer_morphism_nat_int]:
huffman@31708:     "(0::nat) = nat 0"
huffman@31708:     "(1::nat) = nat 1"
huffman@31708:     "(2::nat) = nat 2"
huffman@31708:     "(3::nat) = nat 3"
huffman@31708:   by auto
huffman@31708: 
huffman@31708: definition
huffman@31708:   tsub :: "int \<Rightarrow> int \<Rightarrow> int"
huffman@31708: where
huffman@31708:   "tsub x y = (if x >= y then x - y else 0)"
huffman@31708: 
huffman@31708: lemma tsub_eq: "x >= y \<Longrightarrow> tsub x y = x - y"
huffman@31708:   by (simp add: tsub_def)
huffman@31708: 
haftmann@35683: lemma transfer_nat_int_functions [transfer key: transfer_morphism_nat_int]:
huffman@31708:     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) + (nat y) = nat (x + y)"
huffman@31708:     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) * (nat y) = nat (x * y)"
huffman@31708:     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) - (nat y) = nat (tsub x y)"
huffman@31708:     "(x::int) >= 0 \<Longrightarrow> (nat x)^n = nat (x^n)"
huffman@31708:   by (auto simp add: eq_nat_nat_iff nat_mult_distrib
haftmann@33318:       nat_power_eq tsub_def)
huffman@31708: 
haftmann@35683: lemma transfer_nat_int_function_closures [transfer key: transfer_morphism_nat_int]:
huffman@31708:     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x + y >= 0"
huffman@31708:     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x * y >= 0"
huffman@31708:     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> tsub x y >= 0"
huffman@31708:     "(x::int) >= 0 \<Longrightarrow> x^n >= 0"
huffman@31708:     "(0::int) >= 0"
huffman@31708:     "(1::int) >= 0"
huffman@31708:     "(2::int) >= 0"
huffman@31708:     "(3::int) >= 0"
huffman@31708:     "int z >= 0"
haftmann@33340:   by (auto simp add: zero_le_mult_iff tsub_def)
huffman@31708: 
haftmann@35683: lemma transfer_nat_int_relations [transfer key: transfer_morphism_nat_int]:
huffman@31708:     "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow>
huffman@31708:       (nat (x::int) = nat y) = (x = y)"
huffman@31708:     "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow>
huffman@31708:       (nat (x::int) < nat y) = (x < y)"
huffman@31708:     "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow>
huffman@31708:       (nat (x::int) <= nat y) = (x <= y)"
huffman@31708:     "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow>
huffman@31708:       (nat (x::int) dvd nat y) = (x dvd y)"
haftmann@32558:   by (auto simp add: zdvd_int)
huffman@31708: 
huffman@31708: 
haftmann@33318: text {* first-order quantifiers *}
haftmann@33318: 
haftmann@33318: lemma all_nat: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x\<ge>0. P (nat x))"
haftmann@33318:   by (simp split add: split_nat)
haftmann@33318: 
haftmann@33318: lemma ex_nat: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. 0 \<le> x \<and> P (nat x))"
haftmann@33318: proof
haftmann@33318:   assume "\<exists>x. P x"
haftmann@33318:   then obtain x where "P x" ..
haftmann@33318:   then have "int x \<ge> 0 \<and> P (nat (int x))" by simp
haftmann@33318:   then show "\<exists>x\<ge>0. P (nat x)" ..
haftmann@33318: next
haftmann@33318:   assume "\<exists>x\<ge>0. P (nat x)"
haftmann@33318:   then show "\<exists>x. P x" by auto
haftmann@33318: qed
huffman@31708: 
haftmann@35683: lemma transfer_nat_int_quantifiers [transfer key: transfer_morphism_nat_int]:
huffman@31708:     "(ALL (x::nat). P x) = (ALL (x::int). x >= 0 \<longrightarrow> P (nat x))"
huffman@31708:     "(EX (x::nat). P x) = (EX (x::int). x >= 0 & P (nat x))"
huffman@31708:   by (rule all_nat, rule ex_nat)
huffman@31708: 
huffman@31708: (* should we restrict these? *)
huffman@31708: lemma all_cong: "(\<And>x. Q x \<Longrightarrow> P x = P' x) \<Longrightarrow>
huffman@31708:     (ALL x. Q x \<longrightarrow> P x) = (ALL x. Q x \<longrightarrow> P' x)"
huffman@31708:   by auto
huffman@31708: 
huffman@31708: lemma ex_cong: "(\<And>x. Q x \<Longrightarrow> P x = P' x) \<Longrightarrow>
huffman@31708:     (EX x. Q x \<and> P x) = (EX x. Q x \<and> P' x)"
huffman@31708:   by auto
huffman@31708: 
haftmann@35644: declare transfer_morphism_nat_int [transfer add
huffman@31708:   cong: all_cong ex_cong]
huffman@31708: 
huffman@31708: 
haftmann@33318: text {* if *}
huffman@31708: 
haftmann@35683: lemma nat_if_cong [transfer key: transfer_morphism_nat_int]:
haftmann@35683:   "(if P then (nat x) else (nat y)) = nat (if P then x else y)"
huffman@31708:   by auto
huffman@31708: 
huffman@31708: 
haftmann@33318: text {* operations with sets *}
huffman@31708: 
huffman@31708: definition
huffman@31708:   nat_set :: "int set \<Rightarrow> bool"
huffman@31708: where
huffman@31708:   "nat_set S = (ALL x:S. x >= 0)"
huffman@31708: 
huffman@31708: lemma transfer_nat_int_set_functions:
huffman@31708:     "card A = card (int ` A)"
huffman@31708:     "{} = nat ` ({}::int set)"
huffman@31708:     "A Un B = nat ` (int ` A Un int ` B)"
huffman@31708:     "A Int B = nat ` (int ` A Int int ` B)"
huffman@31708:     "{x. P x} = nat ` {x. x >= 0 & P(nat x)}"
huffman@31708:   apply (rule card_image [symmetric])
huffman@31708:   apply (auto simp add: inj_on_def image_def)
huffman@31708:   apply (rule_tac x = "int x" in bexI)
huffman@31708:   apply auto
huffman@31708:   apply (rule_tac x = "int x" in bexI)
huffman@31708:   apply auto
huffman@31708:   apply (rule_tac x = "int x" in bexI)
huffman@31708:   apply auto
huffman@31708:   apply (rule_tac x = "int x" in exI)
huffman@31708:   apply auto
huffman@31708: done
huffman@31708: 
huffman@31708: lemma transfer_nat_int_set_function_closures:
huffman@31708:     "nat_set {}"
huffman@31708:     "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> nat_set (A Un B)"
huffman@31708:     "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> nat_set (A Int B)"
huffman@31708:     "nat_set {x. x >= 0 & P x}"
huffman@31708:     "nat_set (int ` C)"
huffman@31708:     "nat_set A \<Longrightarrow> x : A \<Longrightarrow> x >= 0" (* does it hurt to turn this on? *)
huffman@31708:   unfolding nat_set_def apply auto
huffman@31708: done
huffman@31708: 
huffman@31708: lemma transfer_nat_int_set_relations:
huffman@31708:     "(finite A) = (finite (int ` A))"
huffman@31708:     "(x : A) = (int x : int ` A)"
huffman@31708:     "(A = B) = (int ` A = int ` B)"
huffman@31708:     "(A < B) = (int ` A < int ` B)"
huffman@31708:     "(A <= B) = (int ` A <= int ` B)"
huffman@31708:   apply (rule iffI)
huffman@31708:   apply (erule finite_imageI)
huffman@31708:   apply (erule finite_imageD)
huffman@31708:   apply (auto simp add: image_def expand_set_eq inj_on_def)
huffman@31708:   apply (drule_tac x = "int x" in spec, auto)
huffman@31708:   apply (drule_tac x = "int x" in spec, auto)
huffman@31708:   apply (drule_tac x = "int x" in spec, auto)
huffman@31708: done
huffman@31708: 
huffman@31708: lemma transfer_nat_int_set_return_embed: "nat_set A \<Longrightarrow>
huffman@31708:     (int ` nat ` A = A)"
huffman@31708:   by (auto simp add: nat_set_def image_def)
huffman@31708: 
huffman@31708: lemma transfer_nat_int_set_cong: "(!!x. x >= 0 \<Longrightarrow> P x = P' x) \<Longrightarrow>
huffman@31708:     {(x::int). x >= 0 & P x} = {x. x >= 0 & P' x}"
huffman@31708:   by auto
huffman@31708: 
haftmann@35644: declare transfer_morphism_nat_int [transfer add
huffman@31708:   return: transfer_nat_int_set_functions
huffman@31708:     transfer_nat_int_set_function_closures
huffman@31708:     transfer_nat_int_set_relations
huffman@31708:     transfer_nat_int_set_return_embed
huffman@31708:   cong: transfer_nat_int_set_cong
huffman@31708: ]
huffman@31708: 
huffman@31708: 
haftmann@33318: text {* setsum and setprod *}
huffman@31708: 
huffman@31708: (* this handles the case where the *domain* of f is nat *)
huffman@31708: lemma transfer_nat_int_sum_prod:
huffman@31708:     "setsum f A = setsum (%x. f (nat x)) (int ` A)"
huffman@31708:     "setprod f A = setprod (%x. f (nat x)) (int ` A)"
huffman@31708:   apply (subst setsum_reindex)
huffman@31708:   apply (unfold inj_on_def, auto)
huffman@31708:   apply (subst setprod_reindex)
huffman@31708:   apply (unfold inj_on_def o_def, auto)
huffman@31708: done
huffman@31708: 
huffman@31708: (* this handles the case where the *range* of f is nat *)
huffman@31708: lemma transfer_nat_int_sum_prod2:
huffman@31708:     "setsum f A = nat(setsum (%x. int (f x)) A)"
huffman@31708:     "setprod f A = nat(setprod (%x. int (f x)) A)"
huffman@31708:   apply (subst int_setsum [symmetric])
huffman@31708:   apply auto
huffman@31708:   apply (subst int_setprod [symmetric])
huffman@31708:   apply auto
huffman@31708: done
huffman@31708: 
huffman@31708: lemma transfer_nat_int_sum_prod_closure:
huffman@31708:     "nat_set A \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x >= (0::int)) \<Longrightarrow> setsum f A >= 0"
huffman@31708:     "nat_set A \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x >= (0::int)) \<Longrightarrow> setprod f A >= 0"
huffman@31708:   unfolding nat_set_def
huffman@31708:   apply (rule setsum_nonneg)
huffman@31708:   apply auto
huffman@31708:   apply (rule setprod_nonneg)
huffman@31708:   apply auto
huffman@31708: done
huffman@31708: 
huffman@31708: (* this version doesn't work, even with nat_set A \<Longrightarrow>
huffman@31708:       x : A \<Longrightarrow> x >= 0 turned on. Why not?
huffman@31708: 
huffman@31708:   also: what does =simp=> do?
huffman@31708: 
huffman@31708: lemma transfer_nat_int_sum_prod_closure:
huffman@31708:     "(!!x. x : A  ==> f x >= (0::int)) \<Longrightarrow> setsum f A >= 0"
huffman@31708:     "(!!x. x : A  ==> f x >= (0::int)) \<Longrightarrow> setprod f A >= 0"
huffman@31708:   unfolding nat_set_def simp_implies_def
huffman@31708:   apply (rule setsum_nonneg)
huffman@31708:   apply auto
huffman@31708:   apply (rule setprod_nonneg)
huffman@31708:   apply auto
huffman@31708: done
huffman@31708: *)
huffman@31708: 
huffman@31708: (* Making A = B in this lemma doesn't work. Why not?
huffman@31708:    Also, why aren't setsum_cong and setprod_cong enough,
huffman@31708:    with the previously mentioned rule turned on? *)
huffman@31708: 
huffman@31708: lemma transfer_nat_int_sum_prod_cong:
huffman@31708:     "A = B \<Longrightarrow> nat_set B \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x = g x) \<Longrightarrow>
huffman@31708:       setsum f A = setsum g B"
huffman@31708:     "A = B \<Longrightarrow> nat_set B \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x = g x) \<Longrightarrow>
huffman@31708:       setprod f A = setprod g B"
huffman@31708:   unfolding nat_set_def
huffman@31708:   apply (subst setsum_cong, assumption)
huffman@31708:   apply auto [2]
huffman@31708:   apply (subst setprod_cong, assumption, auto)
huffman@31708: done
huffman@31708: 
haftmann@35644: declare transfer_morphism_nat_int [transfer add
huffman@31708:   return: transfer_nat_int_sum_prod transfer_nat_int_sum_prod2
huffman@31708:     transfer_nat_int_sum_prod_closure
huffman@31708:   cong: transfer_nat_int_sum_prod_cong]
huffman@31708: 
huffman@31708: 
huffman@31708: subsection {* Set up transfer from int to nat *}
huffman@31708: 
haftmann@33318: text {* set up transfer direction *}
huffman@31708: 
haftmann@35683: lemma transfer_morphism_int_nat: "transfer_morphism int (\<lambda>n. True)"
haftmann@35821: by (rule transfer_morphismI) simp
huffman@31708: 
haftmann@35644: declare transfer_morphism_int_nat [transfer add
huffman@31708:   mode: manual
huffman@31708:   return: nat_int
haftmann@35683:   labels: int_nat
huffman@31708: ]
huffman@31708: 
huffman@31708: 
haftmann@33318: text {* basic functions and relations *}
haftmann@33318: 
huffman@31708: definition
huffman@31708:   is_nat :: "int \<Rightarrow> bool"
huffman@31708: where
huffman@31708:   "is_nat x = (x >= 0)"
huffman@31708: 
huffman@31708: lemma transfer_int_nat_numerals:
huffman@31708:     "0 = int 0"
huffman@31708:     "1 = int 1"
huffman@31708:     "2 = int 2"
huffman@31708:     "3 = int 3"
huffman@31708:   by auto
huffman@31708: 
huffman@31708: lemma transfer_int_nat_functions:
huffman@31708:     "(int x) + (int y) = int (x + y)"
huffman@31708:     "(int x) * (int y) = int (x * y)"
huffman@31708:     "tsub (int x) (int y) = int (x - y)"
huffman@31708:     "(int x)^n = int (x^n)"
haftmann@33318:   by (auto simp add: int_mult tsub_def int_power)
huffman@31708: 
huffman@31708: lemma transfer_int_nat_function_closures:
huffman@31708:     "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x + y)"
huffman@31708:     "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x * y)"
huffman@31708:     "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (tsub x y)"
huffman@31708:     "is_nat x \<Longrightarrow> is_nat (x^n)"
huffman@31708:     "is_nat 0"
huffman@31708:     "is_nat 1"
huffman@31708:     "is_nat 2"
huffman@31708:     "is_nat 3"
huffman@31708:     "is_nat (int z)"
huffman@31708:   by (simp_all only: is_nat_def transfer_nat_int_function_closures)
huffman@31708: 
huffman@31708: lemma transfer_int_nat_relations:
huffman@31708:     "(int x = int y) = (x = y)"
huffman@31708:     "(int x < int y) = (x < y)"
huffman@31708:     "(int x <= int y) = (x <= y)"
huffman@31708:     "(int x dvd int y) = (x dvd y)"
haftmann@33318:   by (auto simp add: zdvd_int)
haftmann@32121: 
haftmann@35644: declare transfer_morphism_int_nat [transfer add return:
huffman@31708:   transfer_int_nat_numerals
huffman@31708:   transfer_int_nat_functions
huffman@31708:   transfer_int_nat_function_closures
huffman@31708:   transfer_int_nat_relations
huffman@31708: ]
huffman@31708: 
huffman@31708: 
haftmann@33318: text {* first-order quantifiers *}
huffman@31708: 
huffman@31708: lemma transfer_int_nat_quantifiers:
huffman@31708:     "(ALL (x::int) >= 0. P x) = (ALL (x::nat). P (int x))"
huffman@31708:     "(EX (x::int) >= 0. P x) = (EX (x::nat). P (int x))"
huffman@31708:   apply (subst all_nat)
huffman@31708:   apply auto [1]
huffman@31708:   apply (subst ex_nat)
huffman@31708:   apply auto
huffman@31708: done
huffman@31708: 
haftmann@35644: declare transfer_morphism_int_nat [transfer add
huffman@31708:   return: transfer_int_nat_quantifiers]
huffman@31708: 
huffman@31708: 
haftmann@33318: text {* if *}
huffman@31708: 
huffman@31708: lemma int_if_cong: "(if P then (int x) else (int y)) =
huffman@31708:     int (if P then x else y)"
huffman@31708:   by auto
huffman@31708: 
haftmann@35644: declare transfer_morphism_int_nat [transfer add return: int_if_cong]
huffman@31708: 
huffman@31708: 
huffman@31708: 
haftmann@33318: text {* operations with sets *}
huffman@31708: 
huffman@31708: lemma transfer_int_nat_set_functions:
huffman@31708:     "nat_set A \<Longrightarrow> card A = card (nat ` A)"
huffman@31708:     "{} = int ` ({}::nat set)"
huffman@31708:     "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> A Un B = int ` (nat ` A Un nat ` B)"
huffman@31708:     "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> A Int B = int ` (nat ` A Int nat ` B)"
huffman@31708:     "{x. x >= 0 & P x} = int ` {x. P(int x)}"
huffman@31708:        (* need all variants of these! *)
huffman@31708:   by (simp_all only: is_nat_def transfer_nat_int_set_functions
huffman@31708:           transfer_nat_int_set_function_closures
huffman@31708:           transfer_nat_int_set_return_embed nat_0_le
huffman@31708:           cong: transfer_nat_int_set_cong)
huffman@31708: 
huffman@31708: lemma transfer_int_nat_set_function_closures:
huffman@31708:     "nat_set {}"
huffman@31708:     "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> nat_set (A Un B)"
huffman@31708:     "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> nat_set (A Int B)"
huffman@31708:     "nat_set {x. x >= 0 & P x}"
huffman@31708:     "nat_set (int ` C)"
huffman@31708:     "nat_set A \<Longrightarrow> x : A \<Longrightarrow> is_nat x"
huffman@31708:   by (simp_all only: transfer_nat_int_set_function_closures is_nat_def)
huffman@31708: 
huffman@31708: lemma transfer_int_nat_set_relations:
huffman@31708:     "nat_set A \<Longrightarrow> finite A = finite (nat ` A)"
huffman@31708:     "is_nat x \<Longrightarrow> nat_set A \<Longrightarrow> (x : A) = (nat x : nat ` A)"
huffman@31708:     "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> (A = B) = (nat ` A = nat ` B)"
huffman@31708:     "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> (A < B) = (nat ` A < nat ` B)"
huffman@31708:     "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> (A <= B) = (nat ` A <= nat ` B)"
huffman@31708:   by (simp_all only: is_nat_def transfer_nat_int_set_relations
huffman@31708:     transfer_nat_int_set_return_embed nat_0_le)
huffman@31708: 
huffman@31708: lemma transfer_int_nat_set_return_embed: "nat ` int ` A = A"
huffman@31708:   by (simp only: transfer_nat_int_set_relations
huffman@31708:     transfer_nat_int_set_function_closures
huffman@31708:     transfer_nat_int_set_return_embed nat_0_le)
huffman@31708: 
huffman@31708: lemma transfer_int_nat_set_cong: "(!!x. P x = P' x) \<Longrightarrow>
huffman@31708:     {(x::nat). P x} = {x. P' x}"
huffman@31708:   by auto
huffman@31708: 
haftmann@35644: declare transfer_morphism_int_nat [transfer add
huffman@31708:   return: transfer_int_nat_set_functions
huffman@31708:     transfer_int_nat_set_function_closures
huffman@31708:     transfer_int_nat_set_relations
huffman@31708:     transfer_int_nat_set_return_embed
huffman@31708:   cong: transfer_int_nat_set_cong
huffman@31708: ]
huffman@31708: 
huffman@31708: 
haftmann@33318: text {* setsum and setprod *}
huffman@31708: 
huffman@31708: (* this handles the case where the *domain* of f is int *)
huffman@31708: lemma transfer_int_nat_sum_prod:
huffman@31708:     "nat_set A \<Longrightarrow> setsum f A = setsum (%x. f (int x)) (nat ` A)"
huffman@31708:     "nat_set A \<Longrightarrow> setprod f A = setprod (%x. f (int x)) (nat ` A)"
huffman@31708:   apply (subst setsum_reindex)
huffman@31708:   apply (unfold inj_on_def nat_set_def, auto simp add: eq_nat_nat_iff)
huffman@31708:   apply (subst setprod_reindex)
huffman@31708:   apply (unfold inj_on_def nat_set_def o_def, auto simp add: eq_nat_nat_iff
huffman@31708:             cong: setprod_cong)
huffman@31708: done
huffman@31708: 
huffman@31708: (* this handles the case where the *range* of f is int *)
huffman@31708: lemma transfer_int_nat_sum_prod2:
huffman@31708:     "(!!x. x:A \<Longrightarrow> is_nat (f x)) \<Longrightarrow> setsum f A = int(setsum (%x. nat (f x)) A)"
huffman@31708:     "(!!x. x:A \<Longrightarrow> is_nat (f x)) \<Longrightarrow>
huffman@31708:       setprod f A = int(setprod (%x. nat (f x)) A)"
huffman@31708:   unfolding is_nat_def
huffman@31708:   apply (subst int_setsum, auto)
huffman@31708:   apply (subst int_setprod, auto simp add: cong: setprod_cong)
huffman@31708: done
huffman@31708: 
haftmann@35644: declare transfer_morphism_int_nat [transfer add
huffman@31708:   return: transfer_int_nat_sum_prod transfer_int_nat_sum_prod2
huffman@31708:   cong: setsum_cong setprod_cong]
huffman@31708: 
huffman@31708: end