(* Authors: Jeremy Avigad and Amine Chaieb *)
section \<open>Generic transfer machinery; specific transfer from nats to ints and back.\<close>
theory Nat_Transfer
imports List GCD
begin
subsection \<open>Generic transfer machinery\<close>
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 \<open>Set up transfer from nat to int\<close>
text \<open>set up transfer direction\<close>
lemma transfer_morphism_nat_int [no_atp]:
"transfer_morphism nat (op <= (0::int))" ..
declare transfer_morphism_nat_int [transfer add
mode: manual
return: nat_0_le
labels: nat_int
]
text \<open>basic functions and relations\<close>
lemma transfer_nat_int_numerals [no_atp, 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 [no_atp, 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)"
"(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 tsub_def nat_div_distrib nat_mod_distrib)
lemma transfer_nat_int_function_closures [no_atp, 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"
"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x div y >= 0"
"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x mod y >= 0"
apply (auto simp add: zero_le_mult_iff tsub_def pos_imp_zdiv_nonneg_iff)
apply (cases "y = 0")
apply (auto simp add: pos_imp_zdiv_nonneg_iff)
apply (cases "y = 0")
apply auto
done
lemma transfer_nat_int_relations [no_atp, 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 \<open>first-order quantifiers\<close>
lemma transfer_nat_int_quantifiers [no_atp, 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)
declare transfer_morphism_nat_int [transfer add
cong: all_cong ex_cong]
text \<open>if\<close>
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 \<open>operations with sets\<close>
definition
nat_set :: "int set \<Rightarrow> bool"
where
"nat_set S = (ALL x:S. x >= 0)"
lemma transfer_nat_int_set_functions [no_atp]:
"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! *)
by (rule card_image [symmetric])
(auto simp add: inj_on_def image_def intro: bexI [of _ "int x" for x] exI [of _ "int x" for x])
lemma transfer_nat_int_set_function_closures [no_atp]:
"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? *)
"x >= 0 \<Longrightarrow> nat_set {x..y}"
unfolding nat_set_def apply auto
done
lemma transfer_nat_int_set_relations [no_atp]:
"(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 [no_atp]: "nat_set A \<Longrightarrow>
(int ` nat ` A = A)"
by (auto simp add: nat_set_def image_def)
lemma transfer_nat_int_set_cong [no_atp]: "(!!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 \<open>sum and prod\<close>
(* this handles the case where the *domain* of f is nat *)
lemma transfer_nat_int_sum_prod [no_atp]:
"sum f A = sum (%x. f (nat x)) (int ` A)"
"prod f A = prod (%x. f (nat x)) (int ` A)"
apply (subst sum.reindex)
apply (unfold inj_on_def, auto)
apply (subst prod.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 [no_atp]:
"sum f A = nat(sum (%x. int (f x)) A)"
"prod f A = nat(prod (%x. int (f x)) A)"
apply (simp only: int_sum [symmetric] nat_int)
apply (simp only: int_prod [symmetric] nat_int)
done
lemma transfer_nat_int_sum_prod_closure [no_atp]:
"nat_set A \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x >= (0::int)) \<Longrightarrow> sum f A >= 0"
"nat_set A \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x >= (0::int)) \<Longrightarrow> prod f A >= 0"
unfolding nat_set_def
apply (rule sum_nonneg)
apply auto
apply (rule prod_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> sum f A >= 0"
"(!!x. x : A ==> f x >= (0::int)) \<Longrightarrow> prod f A >= 0"
unfolding nat_set_def simp_implies_def
apply (rule sum_nonneg)
apply auto
apply (rule prod_nonneg)
apply auto
done
*)
(* Making A = B in this lemma doesn't work. Why not?
Also, why aren't sum.cong and prod.cong enough,
with the previously mentioned rule turned on? *)
lemma transfer_nat_int_sum_prod_cong [no_atp]:
"A = B \<Longrightarrow> nat_set B \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x = g x) \<Longrightarrow>
sum f A = sum g B"
"A = B \<Longrightarrow> nat_set B \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x = g x) \<Longrightarrow>
prod f A = prod g B"
unfolding nat_set_def
apply (subst sum.cong, assumption)
apply auto [2]
apply (subst prod.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 \<open>Set up transfer from int to nat\<close>
text \<open>set up transfer direction\<close>
lemma transfer_morphism_int_nat [no_atp]: "transfer_morphism int (\<lambda>n. True)" ..
declare transfer_morphism_int_nat [transfer add
mode: manual
return: nat_int
labels: int_nat
]
text \<open>basic functions and relations\<close>
definition
is_nat :: "int \<Rightarrow> bool"
where
"is_nat x = (x >= 0)"
lemma transfer_int_nat_numerals [no_atp]:
"0 = int 0"
"1 = int 1"
"2 = int 2"
"3 = int 3"
by auto
lemma transfer_int_nat_functions [no_atp]:
"(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: zdiv_int zmod_int tsub_def)
lemma transfer_int_nat_function_closures [no_atp]:
"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)"
"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)"
by (simp_all only: is_nat_def transfer_nat_int_function_closures)
lemma transfer_int_nat_relations [no_atp]:
"(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 \<open>first-order quantifiers\<close>
lemma transfer_int_nat_quantifiers [no_atp]:
"(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 \<open>if\<close>
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 \<open>operations with sets\<close>
lemma transfer_int_nat_set_functions [no_atp]:
"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 [no_atp]:
"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"
"is_nat x \<Longrightarrow> nat_set {x..y}"
by (simp_all only: transfer_nat_int_set_function_closures is_nat_def)
lemma transfer_int_nat_set_relations [no_atp]:
"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 [no_atp]: "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 [no_atp]: "(!!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 \<open>sum and prod\<close>
(* this handles the case where the *domain* of f is int *)
lemma transfer_int_nat_sum_prod [no_atp]:
"nat_set A \<Longrightarrow> sum f A = sum (%x. f (int x)) (nat ` A)"
"nat_set A \<Longrightarrow> prod f A = prod (%x. f (int x)) (nat ` A)"
apply (subst sum.reindex)
apply (unfold inj_on_def nat_set_def, auto simp add: eq_nat_nat_iff)
apply (subst prod.reindex)
apply (unfold inj_on_def nat_set_def o_def, auto simp add: eq_nat_nat_iff
cong: prod.cong)
done
(* this handles the case where the *range* of f is int *)
lemma transfer_int_nat_sum_prod2 [no_atp]:
"(!!x. x:A \<Longrightarrow> is_nat (f x)) \<Longrightarrow> sum f A = int(sum (%x. nat (f x)) A)"
"(!!x. x:A \<Longrightarrow> is_nat (f x)) \<Longrightarrow>
prod f A = int(prod (%x. nat (f x)) A)"
unfolding is_nat_def
by (subst int_sum) auto
declare transfer_morphism_int_nat [transfer add
return: transfer_int_nat_sum_prod transfer_int_nat_sum_prod2
cong: sum.cong prod.cong]
declare transfer_morphism_int_nat [transfer add return: even_int_iff]
lemma transfer_nat_int_gcd [no_atp]:
"x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> gcd (nat x) (nat y) = nat (gcd x y)"
"x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> lcm (nat x) (nat y) = nat (lcm x y)"
for x y :: int
unfolding gcd_int_def lcm_int_def by auto
lemma transfer_nat_int_gcd_closures [no_atp]:
"x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> gcd x y \<ge> 0"
"x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> lcm x y \<ge> 0"
for x y :: int
by (auto simp add: gcd_int_def lcm_int_def)
declare transfer_morphism_nat_int
[transfer add return: transfer_nat_int_gcd transfer_nat_int_gcd_closures]
lemma transfer_int_nat_gcd [no_atp]:
"gcd (int x) (int y) = int (gcd x y)"
"lcm (int x) (int y) = int (lcm x y)"
by (auto simp: gcd_int_def lcm_int_def)
lemma transfer_int_nat_gcd_closures [no_atp]:
"is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> gcd x y >= 0"
"is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> lcm x y >= 0"
by (auto simp: gcd_int_def lcm_int_def)
declare transfer_morphism_int_nat
[transfer add return: transfer_int_nat_gcd transfer_int_nat_gcd_closures]
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"
lemma transfer_nat_int_list_return_embed [no_atp]:
"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 [no_atp]:
"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";
*)
end