(* Authors: Jeremy Avigad and Amine Chaieb *)
header {* Sets up transfer from nats to ints and back. *}
theory Nat_Transfer
imports Main Parity
begin
subsection {* Set up transfer from nat to int *}
(* set up transfer direction *)
lemma TransferMorphism_nat_int: "TransferMorphism nat (op <= (0::int))"
by (simp add: TransferMorphism_def)
declare TransferMorphism_nat_int[transfer
add mode: manual
return: nat_0_le
labels: natint
]
(* basic functions and relations *)
lemma transfer_nat_int_numerals:
"(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:
"(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)"
"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) div (nat y) = nat (x div y)"
"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) mod (nat y) = nat (x mod y)"
by (auto simp add: eq_nat_nat_iff nat_mult_distrib
nat_power_eq nat_div_distrib nat_mod_distrib tsub_def)
lemma transfer_nat_int_function_closures:
"(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"
"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x div y >= 0"
"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x mod y >= 0"
"(0::int) >= 0"
"(1::int) >= 0"
"(2::int) >= 0"
"(3::int) >= 0"
"int z >= 0"
apply (auto simp add: zero_le_mult_iff tsub_def)
apply (case_tac "y = 0")
apply auto
apply (subst pos_imp_zdiv_nonneg_iff, auto)
apply (case_tac "y = 0")
apply force
apply (rule pos_mod_sign)
apply arith
done
lemma transfer_nat_int_relations:
"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)
declare TransferMorphism_nat_int[transfer add return:
transfer_nat_int_numerals
transfer_nat_int_functions
transfer_nat_int_function_closures
transfer_nat_int_relations
]
(* first-order quantifiers *)
lemma transfer_nat_int_quantifiers:
"(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 TransferMorphism_nat_int[transfer add
return: transfer_nat_int_quantifiers
cong: all_cong ex_cong]
(* if *)
lemma nat_if_cong: "(if P then (nat x) else (nat y)) =
nat (if P then x else y)"
by auto
declare TransferMorphism_nat_int [transfer add return: nat_if_cong]
(* 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)}"
"{..n} = nat ` {0..int n}"
"{m..n} = nat ` {int m..int n}" (* need all variants of these! *)
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
apply (rule_tac x = "int x" in bexI)
apply auto
apply (rule_tac x = "int x" in bexI)
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)"
"x >= 0 \<Longrightarrow> nat_set {x..y}"
"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 expand_set_eq 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 TransferMorphism_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
]
(* 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 TransferMorphism_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]
(* lists *)
definition
embed_list :: "nat list \<Rightarrow> int list"
where
"embed_list l = map int l";
definition
nat_list :: "int list \<Rightarrow> bool"
where
"nat_list l = nat_set (set l)";
definition
return_list :: "int list \<Rightarrow> nat list"
where
"return_list l = map nat l";
thm nat_0_le;
lemma transfer_nat_int_list_return_embed: "nat_list l \<longrightarrow>
embed_list (return_list l) = l";
unfolding embed_list_def return_list_def nat_list_def nat_set_def
apply (induct l);
apply auto;
done;
lemma transfer_nat_int_list_functions:
"l @ m = return_list (embed_list l @ embed_list m)"
"[] = return_list []";
unfolding return_list_def embed_list_def;
apply auto;
apply (induct l, auto);
apply (induct m, auto);
done;
(*
lemma transfer_nat_int_fold1: "fold f l x =
fold (%x. f (nat x)) (embed_list l) x";
*)
subsection {* Set up transfer from int to nat *}
(* set up transfer direction *)
lemma TransferMorphism_int_nat: "TransferMorphism int (UNIV :: nat set)"
by (simp add: TransferMorphism_def)
declare TransferMorphism_int_nat[transfer add
mode: manual
(* labels: int-nat *)
return: nat_int
]
(* 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)"
"(int x) div (int y) = int (x div y)"
"(int x) mod (int y) = int (x mod y)"
by (auto simp add: int_mult tsub_def int_power zdiv_int zmod_int)
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 x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x div y)"
"is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x mod y)"
"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)"
"(even (int x)) = (even x)"
by (auto simp add: zdvd_int even_nat_def)
lemma UNIV_apply:
"UNIV x = True"
by (simp add: top_fun_eq top_bool_eq)
declare TransferMorphism_int_nat[transfer add return:
transfer_int_nat_numerals
transfer_int_nat_functions
transfer_int_nat_function_closures
transfer_int_nat_relations
UNIV_apply
]
(* 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 TransferMorphism_int_nat[transfer add
return: transfer_int_nat_quantifiers]
(* if *)
lemma int_if_cong: "(if P then (int x) else (int y)) =
int (if P then x else y)"
by auto
declare TransferMorphism_int_nat [transfer add return: int_if_cong]
(* 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)}"
"is_nat m \<Longrightarrow> is_nat n \<Longrightarrow> {m..n} = int ` {nat m..nat n}"
(* 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)"
"is_nat x \<Longrightarrow> nat_set {x..y}"
"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 TransferMorphism_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
]
(* 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 TransferMorphism_int_nat[transfer add
return: transfer_int_nat_sum_prod transfer_int_nat_sum_prod2
cong: setsum_cong setprod_cong]
end