# Theory Nat_Transfer

theory Nat_Transfer
imports Int
```(* 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 ⇒ 'a) ⇒ ('b ⇒ bool) ⇒ bool"
where "transfer_morphism f A ⟷ True"

lemma transfer_morphismI[intro]: "transfer_morphism f A"

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))" ..

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

lemma tsub_eq: "x >= y ⟹ tsub x y = x - y"

lemma transfer_nat_int_functions [transfer key: transfer_morphism_nat_int]:
"(x::int) >= 0 ⟹ y >= 0 ⟹ (nat x) + (nat y) = nat (x + y)"
"(x::int) >= 0 ⟹ y >= 0 ⟹ (nat x) * (nat y) = nat (x * y)"
"(x::int) >= 0 ⟹ y >= 0 ⟹ (nat x) - (nat y) = nat (tsub x y)"
"(x::int) >= 0 ⟹ (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 ⟹ y >= 0 ⟹ x + y >= 0"
"(x::int) >= 0 ⟹ y >= 0 ⟹ x * y >= 0"
"(x::int) >= 0 ⟹ y >= 0 ⟹ tsub x y >= 0"
"(x::int) >= 0 ⟹ 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 ⟹ y >= 0 ⟹
(nat (x::int) = nat y) = (x = y)"
"x >= 0 ⟹ y >= 0 ⟹
(nat (x::int) < nat y) = (x < y)"
"x >= 0 ⟹ y >= 0 ⟹
(nat (x::int) <= nat y) = (x <= y)"
"x >= 0 ⟹ y >= 0 ⟹
(nat (x::int) dvd nat y) = (x dvd y)"

text ‹first-order quantifiers›

lemma all_nat: "(∀x. P x) ⟷ (∀x≥0. P (nat x))"
by (simp split: split_nat)

lemma ex_nat: "(∃x. P x) ⟷ (∃x. 0 ≤ x ∧ P (nat x))"
proof
assume "∃x. P x"
then obtain x where "P x" ..
then have "int x ≥ 0 ∧ P (nat (int x))" by simp
then show "∃x≥0. P (nat x)" ..
next
assume "∃x≥0. P (nat x)"
then show "∃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 ⟶ 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: "(⋀x. Q x ⟹ P x = P' x) ⟹
(ALL x. Q x ⟶ P x) = (ALL x. Q x ⟶ P' x)"
by auto

lemma ex_cong: "(⋀x. Q x ⟹ P x = P' x) ⟹
(EX x. Q x ∧ P x) = (EX x. Q x ∧ P' x)"
by auto

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 ⇒ 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 ⟹ nat_set B ⟹ nat_set (A Un B)"
"nat_set A ⟹ nat_set B ⟹ nat_set (A Int B)"
"nat_set {x. x >= 0 & P x}"
"nat_set (int ` C)"
"nat_set A ⟹ x : A ⟹ 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 ⟹
(int ` nat ` A = A)"
by (auto simp add: nat_set_def image_def)

lemma transfer_nat_int_set_cong: "(!!x. x >= 0 ⟹ P x = P' x) ⟹
{(x::int). x >= 0 & P x} = {x. x >= 0 & P' x}"
by auto

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 ‹sum and prod›

(* this handles the case where the *domain* of f is nat *)
lemma transfer_nat_int_sum_prod:
"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:
"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:
"nat_set A ⟹ (!!x. x >= 0 ⟹ f x >= (0::int)) ⟹ sum f A >= 0"
"nat_set A ⟹ (!!x. x >= 0 ⟹ f x >= (0::int)) ⟹ 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 ⟹
x : A ⟹ 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)) ⟹ sum f A >= 0"
"(!!x. x : A  ==> f x >= (0::int)) ⟹ 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:
"A = B ⟹ nat_set B ⟹ (!!x. x >= 0 ⟹ f x = g x) ⟹
sum f A = sum g B"
"A = B ⟹ nat_set B ⟹ (!!x. x >= 0 ⟹ f x = g x) ⟹
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

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 (λn. True)" ..

mode: manual
return: nat_int
labels: int_nat
]

text ‹basic functions and relations›

definition
is_nat :: "int ⇒ 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: of_nat_mult tsub_def of_nat_power)

lemma transfer_int_nat_function_closures:
"is_nat x ⟹ is_nat y ⟹ is_nat (x + y)"
"is_nat x ⟹ is_nat y ⟹ is_nat (x * y)"
"is_nat x ⟹ is_nat y ⟹ is_nat (tsub x y)"
"is_nat x ⟹ 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)"

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

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 ⟹ card A = card (nat ` A)"
"{} = int ` ({}::nat set)"
"nat_set A ⟹ nat_set B ⟹ A Un B = int ` (nat ` A Un nat ` B)"
"nat_set A ⟹ nat_set B ⟹ 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 ⟹ nat_set B ⟹ nat_set (A Un B)"
"nat_set A ⟹ nat_set B ⟹ nat_set (A Int B)"
"nat_set {x. x >= 0 & P x}"
"nat_set (int ` C)"
"nat_set A ⟹ x : A ⟹ 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 ⟹ finite A = finite (nat ` A)"
"is_nat x ⟹ nat_set A ⟹ (x : A) = (nat x : nat ` A)"
"nat_set A ⟹ nat_set B ⟹ (A = B) = (nat ` A = nat ` B)"
"nat_set A ⟹ nat_set B ⟹ (A < B) = (nat ` A < nat ` B)"
"nat_set A ⟹ nat_set B ⟹ (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) ⟹
{(x::nat). P x} = {x. P' x}"
by auto

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 ‹sum and prod›

(* this handles the case where the *domain* of f is int *)
lemma transfer_int_nat_sum_prod:
"nat_set A ⟹ sum f A = sum (%x. f (int x)) (nat ` A)"
"nat_set A ⟹ 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:
"(!!x. x:A ⟹ is_nat (f x)) ⟹ sum f A = int(sum (%x. nat (f x)) A)"
"(!!x. x:A ⟹ is_nat (f x)) ⟹
prod f A = int(prod (%x. nat (f x)) A)"
unfolding is_nat_def
by (subst int_sum) auto