# Theory Transfer_Int_Nat

theory Transfer_Int_Nat
imports Main
```(*  Title:      HOL/ex/Transfer_Int_Nat.thy
Author:     Brian Huffman, TU Muenchen
*)

section ‹Using the transfer method between nat and int›

theory Transfer_Int_Nat
imports Main
begin

subsection ‹Correspondence relation›

definition ZN :: "int ⇒ nat ⇒ bool"
where "ZN = (λz n. z = of_nat n)"

subsection ‹Transfer domain rules›

lemma Domainp_ZN [transfer_domain_rule]: "Domainp ZN = (λx. x ≥ 0)"
unfolding ZN_def Domainp_iff[abs_def] by (auto intro: zero_le_imp_eq_int)

subsection ‹Transfer rules›

context includes lifting_syntax
begin

lemma bi_unique_ZN [transfer_rule]: "bi_unique ZN"
unfolding ZN_def bi_unique_def by simp

lemma right_total_ZN [transfer_rule]: "right_total ZN"
unfolding ZN_def right_total_def by simp

lemma ZN_0 [transfer_rule]: "ZN 0 0"
unfolding ZN_def by simp

lemma ZN_1 [transfer_rule]: "ZN 1 1"
unfolding ZN_def by simp

lemma ZN_add [transfer_rule]: "(ZN ===> ZN ===> ZN) (+) (+)"
unfolding rel_fun_def ZN_def by simp

lemma ZN_mult [transfer_rule]: "(ZN ===> ZN ===> ZN) (( * )) (( * ))"
unfolding rel_fun_def ZN_def by simp

definition tsub :: "int ⇒ int ⇒ int"
where "tsub k l = max 0 (k - l)"

lemma ZN_diff [transfer_rule]: "(ZN ===> ZN ===> ZN) tsub (-)"
unfolding rel_fun_def ZN_def by (auto simp add: of_nat_diff tsub_def)

lemma ZN_power [transfer_rule]: "(ZN ===> (=) ===> ZN) (^) (^)"
unfolding rel_fun_def ZN_def by simp

lemma ZN_nat_id [transfer_rule]: "(ZN ===> (=)) nat id"
unfolding rel_fun_def ZN_def by simp

lemma ZN_id_int [transfer_rule]: "(ZN ===> (=)) id int"
unfolding rel_fun_def ZN_def by simp

lemma ZN_All [transfer_rule]:
"((ZN ===> (=)) ===> (=)) (Ball {0..}) All"
unfolding rel_fun_def ZN_def by (auto dest: zero_le_imp_eq_int)

lemma ZN_transfer_forall [transfer_rule]:
"((ZN ===> (=)) ===> (=)) (transfer_bforall (λx. 0 ≤ x)) transfer_forall"
unfolding transfer_forall_def transfer_bforall_def
unfolding rel_fun_def ZN_def by (auto dest: zero_le_imp_eq_int)

lemma ZN_Ex [transfer_rule]: "((ZN ===> (=)) ===> (=)) (Bex {0..}) Ex"
unfolding rel_fun_def ZN_def Bex_def atLeast_iff
by (metis zero_le_imp_eq_int of_nat_0_le_iff)

lemma ZN_le [transfer_rule]: "(ZN ===> ZN ===> (=)) (≤) (≤)"
unfolding rel_fun_def ZN_def by simp

lemma ZN_less [transfer_rule]: "(ZN ===> ZN ===> (=)) (<) (<)"
unfolding rel_fun_def ZN_def by simp

lemma ZN_eq [transfer_rule]: "(ZN ===> ZN ===> (=)) (=) (=)"
unfolding rel_fun_def ZN_def by simp

lemma ZN_Suc [transfer_rule]: "(ZN ===> ZN) (λx. x + 1) Suc"
unfolding rel_fun_def ZN_def by simp

lemma ZN_numeral [transfer_rule]:
"((=) ===> ZN) numeral numeral"
unfolding rel_fun_def ZN_def by simp

lemma ZN_dvd [transfer_rule]: "(ZN ===> ZN ===> (=)) (dvd) (dvd)"
unfolding rel_fun_def ZN_def by simp

lemma ZN_div [transfer_rule]: "(ZN ===> ZN ===> ZN) (div) (div)"
unfolding rel_fun_def ZN_def by (simp add: zdiv_int)

lemma ZN_mod [transfer_rule]: "(ZN ===> ZN ===> ZN) (mod) (mod)"
unfolding rel_fun_def ZN_def by (simp add: zmod_int)

lemma ZN_gcd [transfer_rule]: "(ZN ===> ZN ===> ZN) gcd gcd"
unfolding rel_fun_def ZN_def by (simp add: gcd_int_def)

lemma ZN_atMost [transfer_rule]:
"(ZN ===> rel_set ZN) (atLeastAtMost 0) atMost"
unfolding rel_fun_def ZN_def rel_set_def
by (clarsimp simp add: Bex_def, arith)

lemma ZN_atLeastAtMost [transfer_rule]:
"(ZN ===> ZN ===> rel_set ZN) atLeastAtMost atLeastAtMost"
unfolding rel_fun_def ZN_def rel_set_def
by (clarsimp simp add: Bex_def, arith)

lemma ZN_sum [transfer_rule]:
"bi_unique A ⟹ ((A ===> ZN) ===> rel_set A ===> ZN) sum sum"
apply (intro rel_funI)
apply (erule (1) bi_unique_rel_set_lemma)
apply (simp add: sum.reindex int_sum ZN_def rel_fun_def)
apply (rule sum.cong)
apply simp_all
done

text ‹For derived operations, we can use the ‹transfer_prover›
method to help generate transfer rules.›

lemma ZN_sum_list [transfer_rule]: "(list_all2 ZN ===> ZN) sum_list sum_list"
by transfer_prover

end

subsection ‹Transfer examples›

lemma
assumes "⋀i::int. 0 ≤ i ⟹ i + 0 = i"
shows "⋀i::nat. i + 0 = i"
apply transfer
apply fact
done

lemma
assumes "⋀i k::int. ⟦0 ≤ i; 0 ≤ k; i < k⟧ ⟹ ∃j∈{0..}. i + j = k"
shows "⋀i k::nat. i < k ⟹ ∃j. i + j = k"
apply transfer
apply fact
done

lemma
assumes "∀x∈{0::int..}. ∀y∈{0..}. x * y div y = x"
shows "∀x y :: nat. x * y div y = x"
apply transfer
apply fact
done

lemma
assumes "⋀m n::int. ⟦0 ≤ m; 0 ≤ n; m * n = 0⟧ ⟹ m = 0 ∨ n = 0"
shows "m * n = (0::nat) ⟹ m = 0 ∨ n = 0"
apply transfer
apply fact
done

lemma
assumes "∀x∈{0::int..}. ∃y∈{0..}. ∃z∈{0..}. x + 3 * y = 5 * z"
shows "∀x::nat. ∃y z. x + 3 * y = 5 * z"
apply transfer
apply fact
done

text ‹The ‹fixing› option prevents generalization over the free
variable ‹n›, allowing the local transfer rule to be used.›

lemma
assumes [transfer_rule]: "ZN x n"
assumes "∀i∈{0..}. i < x ⟶ 2 * i < 3 * x"
shows "∀i. i < n ⟶ 2 * i < 3 * n"
apply (transfer fixing: n)
apply fact
done

lemma
assumes "gcd (2^i) (3^j) = (1::int)"
shows "gcd (2^i) (3^j) = (1::nat)"
apply (transfer fixing: i j)
apply fact
done

lemma
assumes "⋀x y z::int. ⟦0 ≤ x; 0 ≤ y; 0 ≤ z⟧ ⟹
sum_list [x, y, z] = 0 ⟷ list_all (λx. x = 0) [x, y, z]"
shows "sum_list [x, y, z] = (0::nat) ⟷ list_all (λx. x = 0) [x, y, z]"
apply transfer
apply fact
done

text ‹Quantifiers over higher types (e.g. ‹nat list›) are
transferred to a readable formula thanks to the transfer domain rule @{thm Domainp_ZN}›

lemma
assumes "⋀xs::int list. list_all (λx. x ≥ 0) xs ⟹
(sum_list xs = 0) = list_all (λx. x = 0) xs"
shows "sum_list xs = (0::nat) ⟷ list_all (λx. x = 0) xs"
apply transfer
apply fact
done

text ‹Equality on a higher type can be transferred if the relations
involved are bi-unique.›

lemma
assumes "⋀xs::int list. ⟦list_all (λx. x ≥ 0) xs; xs ≠ []⟧ ⟹
sum_list xs < sum_list (map (λx. x + 1) xs)"
shows "xs ≠ [] ⟹ sum_list xs < sum_list (map Suc xs)"
apply transfer
apply fact
done

end
```