# Theory Int

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theory Int
imports Quotient
(*  Title:      HOL/Int.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Author: Tobias Nipkow, Florian Haftmann, TU Muenchen
*)

header {* The Integers as Equivalence Classes over Pairs of Natural Numbers *}

theory Int
imports Equiv_Relations Wellfounded Quotient
begin

subsection {* Definition of integers as a quotient type *}

definition intrel :: "(nat × nat) => (nat × nat) => bool" where
"intrel = (λ(x, y) (u, v). x + v = u + y)"

lemma intrel_iff [simp]: "intrel (x, y) (u, v) <-> x + v = u + y"

quotient_type int = "nat × nat" / "intrel"
morphisms Rep_Integ Abs_Integ
proof (rule equivpI)
show "reflp intrel"
unfolding reflp_def by auto
show "symp intrel"
unfolding symp_def by auto
show "transp intrel"
unfolding transp_def by auto
qed

lemma eq_Abs_Integ [case_names Abs_Integ, cases type: int]:
"(!!x y. z = Abs_Integ (x, y) ==> P) ==> P"
by (induct z) auto

subsection {* Integers form a commutative ring *}

instantiation int :: comm_ring_1
begin

lift_definition zero_int :: "int" is "(0, 0)"
by simp

lift_definition one_int :: "int" is "(1, 0)"
by simp

lift_definition plus_int :: "int => int => int"
is "λ(x, y) (u, v). (x + u, y + v)"
by clarsimp

lift_definition uminus_int :: "int => int"
is "λ(x, y). (y, x)"
by clarsimp

lift_definition minus_int :: "int => int => int"
is "λ(x, y) (u, v). (x + v, y + u)"
by clarsimp

lift_definition times_int :: "int => int => int"
is "λ(x, y) (u, v). (x*u + y*v, x*v + y*u)"
proof (clarsimp)
fix s t u v w x y z :: nat
assume "s + v = u + t" and "w + z = y + x"
hence "(s + v) * w + (u + t) * x + u * (w + z) + v * (y + x)
= (u + t) * w + (s + v) * x + u * (y + x) + v * (w + z)"

by simp
thus "(s * w + t * x) + (u * z + v * y) = (u * y + v * z) + (s * x + t * w)"
qed

instance
by default (transfer, clarsimp simp: algebra_simps)+

end

abbreviation int :: "nat => int" where
"int ≡ of_nat"

lemma int_def: "int n = Abs_Integ (n, 0)"
by (induct n, simp add: zero_int.abs_eq,

lemma int_transfer [transfer_rule]:
"(fun_rel (op =) cr_int) (λn. (n, 0)) int"
unfolding fun_rel_def cr_int_def int_def by simp

lemma int_diff_cases:
obtains (diff) m n where "z = int m - int n"
by transfer clarsimp

subsection {* Integers are totally ordered *}

instantiation int :: linorder
begin

lift_definition less_eq_int :: "int => int => bool"
is "λ(x, y) (u, v). x + v ≤ u + y"
by auto

lift_definition less_int :: "int => int => bool"
is "λ(x, y) (u, v). x + v < u + y"
by auto

instance
by default (transfer, force)+

end

instantiation int :: distrib_lattice
begin

definition
"(inf :: int => int => int) = min"

definition
"(sup :: int => int => int) = max"

instance
by intro_classes
(auto simp add: inf_int_def sup_int_def min_max.sup_inf_distrib1)

end

subsection {* Ordering properties of arithmetic operations *}

proof
fix i j k :: int
show "i ≤ j ==> k + i ≤ k + j"
by transfer clarsimp
qed

text{*Strict Monotonicity of Multiplication*}

text{*strict, in 1st argument; proof is by induction on k>0*}
lemma zmult_zless_mono2_lemma:
"(i::int)<j ==> 0<k ==> int k * i < int k * j"
apply (induct k)
apply simp
apply (case_tac "k=0")
done

lemma zero_le_imp_eq_int: "(0::int) ≤ k ==> ∃n. k = int n"
apply transfer
apply clarsimp
apply (rule_tac x="a - b" in exI, simp)
done

lemma zero_less_imp_eq_int: "(0::int) < k ==> ∃n>0. k = int n"
apply transfer
apply clarsimp
apply (rule_tac x="a - b" in exI, simp)
done

lemma zmult_zless_mono2: "[| i<j; (0::int) < k |] ==> k*i < k*j"
apply (drule zero_less_imp_eq_int)
done

text{*The integers form an ordered integral domain*}
instantiation int :: linordered_idom
begin

definition
zabs_def: "¦i::int¦ = (if i < 0 then - i else i)"

definition
zsgn_def: "sgn (i::int) = (if i=0 then 0 else if 0<i then 1 else - 1)"

instance proof
fix i j k :: int
show "i < j ==> 0 < k ==> k * i < k * j"
by (rule zmult_zless_mono2)
show "¦i¦ = (if i < 0 then -i else i)"
by (simp only: zabs_def)
show "sgn (i::int) = (if i=0 then 0 else if 0<i then 1 else - 1)"
by (simp only: zsgn_def)
qed

end

lemma zless_imp_add1_zle: "w < z ==> w + (1::int) ≤ z"
by transfer clarsimp

"(w :: int) < z <-> (∃n. z = w + int (Suc n))"
apply transfer
apply auto
apply (rename_tac a b c d)
apply (rule_tac x="c+b - Suc(a+d)" in exI)
apply arith
done

lemmas int_distrib =
distrib_right [of z1 z2 w]
distrib_left [of w z1 z2]
left_diff_distrib [of z1 z2 w]
right_diff_distrib [of w z1 z2]
for z1 z2 w :: int

subsection {* Embedding of the Integers into any @{text ring_1}: @{text of_int}*}

context ring_1
begin

lift_definition of_int :: "int => 'a" is "λ(i, j). of_nat i - of_nat j"

lemma of_int_0 [simp]: "of_int 0 = 0"
by transfer simp

lemma of_int_1 [simp]: "of_int 1 = 1"
by transfer simp

lemma of_int_add [simp]: "of_int (w+z) = of_int w + of_int z"
by transfer (clarsimp simp add: algebra_simps)

lemma of_int_minus [simp]: "of_int (-z) = - (of_int z)"
by (transfer fixing: uminus) clarsimp

lemma of_int_diff [simp]: "of_int (w - z) = of_int w - of_int z"

lemma of_int_mult [simp]: "of_int (w*z) = of_int w * of_int z"
by (transfer fixing: times) (clarsimp simp add: algebra_simps of_nat_mult)

text{*Collapse nested embeddings*}
lemma of_int_of_nat_eq [simp]: "of_int (int n) = of_nat n"
by (induct n) auto

lemma of_int_numeral [simp, code_post]: "of_int (numeral k) = numeral k"
by (simp add: of_nat_numeral [symmetric] of_int_of_nat_eq [symmetric])

lemma of_int_neg_numeral [simp, code_post]: "of_int (neg_numeral k) = neg_numeral k"
unfolding neg_numeral_def neg_numeral_class.neg_numeral_def
by (simp only: of_int_minus of_int_numeral)

lemma of_int_power:
"of_int (z ^ n) = of_int z ^ n"
by (induct n) simp_all

end

context ring_char_0
begin

lemma of_int_eq_iff [simp]:
"of_int w = of_int z <-> w = z"
by transfer (clarsimp simp add: algebra_simps

text{*Special cases where either operand is zero*}
lemma of_int_eq_0_iff [simp]:
"of_int z = 0 <-> z = 0"
using of_int_eq_iff [of z 0] by simp

lemma of_int_0_eq_iff [simp]:
"0 = of_int z <-> z = 0"
using of_int_eq_iff [of 0 z] by simp

end

context linordered_idom
begin

text{*Every @{text linordered_idom} has characteristic zero.*}
subclass ring_char_0 ..

lemma of_int_le_iff [simp]:
"of_int w ≤ of_int z <-> w ≤ z"
by (transfer fixing: less_eq) (clarsimp simp add: algebra_simps

lemma of_int_less_iff [simp]:
"of_int w < of_int z <-> w < z"

lemma of_int_0_le_iff [simp]:
"0 ≤ of_int z <-> 0 ≤ z"
using of_int_le_iff [of 0 z] by simp

lemma of_int_le_0_iff [simp]:
"of_int z ≤ 0 <-> z ≤ 0"
using of_int_le_iff [of z 0] by simp

lemma of_int_0_less_iff [simp]:
"0 < of_int z <-> 0 < z"
using of_int_less_iff [of 0 z] by simp

lemma of_int_less_0_iff [simp]:
"of_int z < 0 <-> z < 0"
using of_int_less_iff [of z 0] by simp

end

lemma of_int_eq_id [simp]: "of_int = id"
proof
fix z show "of_int z = id z"
by (cases z rule: int_diff_cases, simp)
qed

subsection {* Magnitude of an Integer, as a Natural Number: @{text nat} *}

lift_definition nat :: "int => nat" is "λ(x, y). x - y"
by auto

lemma nat_int [simp]: "nat (int n) = n"
by transfer simp

lemma int_nat_eq [simp]: "int (nat z) = (if 0 ≤ z then z else 0)"
by transfer clarsimp

corollary nat_0_le: "0 ≤ z ==> int (nat z) = z"
by simp

lemma nat_le_0 [simp]: "z ≤ 0 ==> nat z = 0"
by transfer clarsimp

lemma nat_le_eq_zle: "0 < w | 0 ≤ z ==> (nat w ≤ nat z) = (w≤z)"
by transfer (clarsimp, arith)

text{*An alternative condition is @{term "0 ≤ w"} *}
corollary nat_mono_iff: "0 < z ==> (nat w < nat z) = (w < z)"
by (simp add: nat_le_eq_zle linorder_not_le [symmetric])

corollary nat_less_eq_zless: "0 ≤ w ==> (nat w < nat z) = (w<z)"
by (simp add: nat_le_eq_zle linorder_not_le [symmetric])

lemma zless_nat_conj [simp]: "(nat w < nat z) = (0 < z & w < z)"
by transfer (clarsimp, arith)

lemma nonneg_eq_int:
fixes z :: int
assumes "0 ≤ z" and "!!m. z = int m ==> P"
shows P
using assms by (blast dest: nat_0_le sym)

lemma nat_eq_iff: "(nat w = m) = (if 0 ≤ w then w = int m else m=0)"

corollary nat_eq_iff2: "(m = nat w) = (if 0 ≤ w then w = int m else m=0)"
by (simp only: eq_commute [of m] nat_eq_iff)

lemma nat_less_iff: "0 ≤ w ==> (nat w < m) = (w < of_nat m)"
by transfer (clarsimp, arith)

lemma nat_le_iff: "nat x ≤ n <-> x ≤ int n"
by transfer (clarsimp simp add: le_diff_conv)

lemma nat_mono: "x ≤ y ==> nat x ≤ nat y"
by transfer auto

lemma nat_0_iff[simp]: "nat(i::int) = 0 <-> i≤0"
by transfer clarsimp

lemma int_eq_iff: "(of_nat m = z) = (m = nat z & 0 ≤ z)"

lemma zero_less_nat_eq [simp]: "(0 < nat z) = (0 < z)"
by (insert zless_nat_conj [of 0], auto)

"[| (0::int) ≤ z; 0 ≤ z' |] ==> nat (z+z') = nat z + nat z'"
by transfer clarsimp

lemma nat_diff_distrib:
"[| (0::int) ≤ z'; z' ≤ z |] ==> nat (z-z') = nat z - nat z'"
by transfer clarsimp

lemma nat_zminus_int [simp]: "nat (- int n) = 0"
by transfer simp

lemma zless_nat_eq_int_zless: "(m < nat z) = (int m < z)"
by transfer (clarsimp simp add: less_diff_conv)

context ring_1
begin

lemma of_nat_nat: "0 ≤ z ==> of_nat (nat z) = of_int z"
by transfer (clarsimp simp add: of_nat_diff)

end

text {* For termination proofs: *}
lemma measure_function_int[measure_function]: "is_measure (nat o abs)" ..

subsection{*Lemmas about the Function @{term of_nat} and Orderings*}

lemma negative_zless_0: "- (int (Suc n)) < (0 :: int)"
by (simp add: order_less_le del: of_nat_Suc)

lemma negative_zless [iff]: "- (int (Suc n)) < int m"
by (rule negative_zless_0 [THEN order_less_le_trans], simp)

lemma negative_zle_0: "- int n ≤ 0"

lemma negative_zle [iff]: "- int n ≤ int m"
by (rule order_trans [OF negative_zle_0 of_nat_0_le_iff])

lemma not_zle_0_negative [simp]: "~ (0 ≤ - (int (Suc n)))"
by (subst le_minus_iff, simp del: of_nat_Suc)

lemma int_zle_neg: "(int n ≤ - int m) = (n = 0 & m = 0)"
by transfer simp

lemma not_int_zless_negative [simp]: "~ (int n < - int m)"

lemma negative_eq_positive [simp]: "(- int n = of_nat m) = (n = 0 & m = 0)"
by (force simp add: order_eq_iff [of "- of_nat n"] int_zle_neg)

lemma zle_iff_zadd: "w ≤ z <-> (∃n. z = w + int n)"
proof -
have "(w ≤ z) = (0 ≤ z - w)"
also have "… = (∃n. z - w = of_nat n)"
by (auto elim: zero_le_imp_eq_int)
also have "… = (∃n. z = w + of_nat n)"
by (simp only: algebra_simps)
finally show ?thesis .
qed

lemma zadd_int_left: "int m + (int n + z) = int (m + n) + z"
by simp

lemma int_Suc0_eq_1: "int (Suc 0) = 1"
by simp

text{*This version is proved for all ordered rings, not just integers!
It is proved here because attribute @{text arith_split} is not available
in theory @{text Rings}.
But is it really better than just rewriting with @{text abs_if}?*}

lemma abs_split [arith_split,no_atp]:
"P(abs(a::'a::linordered_idom)) = ((0 ≤ a --> P a) & (a < 0 --> P(-a)))"
by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)

lemma negD: "x < 0 ==> ∃n. x = - (int (Suc n))"
apply transfer
apply clarsimp
apply (rule_tac x="b - Suc a" in exI, arith)
done

subsection {* Cases and induction *}

text{*Now we replace the case analysis rule by a more conventional one:
whether an integer is negative or not.*}

theorem int_cases [case_names nonneg neg, cases type: int]:
"[|!! n. z = int n ==> P; !! n. z = - (int (Suc n)) ==> P |] ==> P"
apply (cases "z < 0")
apply (blast dest!: negD)
apply (simp add: linorder_not_less del: of_nat_Suc)
apply auto
apply (blast dest: nat_0_le [THEN sym])
done

theorem int_of_nat_induct [case_names nonneg neg, induct type: int]:
"[|!! n. P (int n); !!n. P (- (int (Suc n))) |] ==> P z"
by (cases z) auto

lemma nonneg_int_cases:
assumes "0 ≤ k" obtains n where "k = int n"
using assms by (cases k, simp, simp del: of_nat_Suc)

lemma Let_numeral [simp]: "Let (numeral v) f = f (numeral v)"
-- {* Unfold all @{text let}s involving constants *}
unfolding Let_def ..

lemma Let_neg_numeral [simp]: "Let (neg_numeral v) f = f (neg_numeral v)"
-- {* Unfold all @{text let}s involving constants *}
unfolding Let_def ..

text {* Unfold @{text min} and @{text max} on numerals. *}

lemmas max_number_of [simp] =
max_def [of "numeral u" "numeral v"]
max_def [of "numeral u" "neg_numeral v"]
max_def [of "neg_numeral u" "numeral v"]
max_def [of "neg_numeral u" "neg_numeral v"] for u v

lemmas min_number_of [simp] =
min_def [of "numeral u" "numeral v"]
min_def [of "numeral u" "neg_numeral v"]
min_def [of "neg_numeral u" "numeral v"]
min_def [of "neg_numeral u" "neg_numeral v"] for u v

subsubsection {* Binary comparisons *}

text {* Preliminaries *}

lemma even_less_0_iff:
"a + a < 0 <-> a < (0::'a::linordered_idom)"
proof -
have "a + a < 0 <-> (1+1)*a < 0" by (simp add: distrib_right del: one_add_one)
also have "(1+1)*a < 0 <-> a < 0"
order_less_not_sym [OF zero_less_two])
finally show ?thesis .
qed

lemma le_imp_0_less:
assumes le: "0 ≤ z"
shows "(0::int) < 1 + z"
proof -
have "0 ≤ z" by fact
also have "... < z + 1" by (rule less_add_one)
finally show "0 < 1 + z" .
qed

lemma odd_less_0_iff:
"(1 + z + z < 0) = (z < (0::int))"
proof (cases z)
case (nonneg n)
le_imp_0_less [THEN order_less_imp_le])
next
case (neg n)
thus ?thesis by (simp del: of_nat_Suc of_nat_add of_nat_1
qed

subsubsection {* Comparisons, for Ordered Rings *}

lemmas double_eq_0_iff = double_zero

lemma odd_nonzero:
"1 + z + z ≠ (0::int)"
proof (cases z)
case (nonneg n)
thus ?thesis using le_imp_0_less [OF le]
next
case (neg n)
show ?thesis
proof
assume eq: "1 + z + z = 0"
have "(0::int) < 1 + (int n + int n)"
also have "... = - (1 + z + z)"
also have "... = 0" by (simp add: eq)
finally have "0<0" ..
thus False by blast
qed
qed

subsection {* The Set of Integers *}

context ring_1
begin

definition Ints :: "'a set" where
"Ints = range of_int"

notation (xsymbols)
Ints ("\<int>")

lemma Ints_of_int [simp]: "of_int z ∈ \<int>"

lemma Ints_of_nat [simp]: "of_nat n ∈ \<int>"
using Ints_of_int [of "of_nat n"] by simp

lemma Ints_0 [simp]: "0 ∈ \<int>"
using Ints_of_int [of "0"] by simp

lemma Ints_1 [simp]: "1 ∈ \<int>"
using Ints_of_int [of "1"] by simp

lemma Ints_add [simp]: "a ∈ \<int> ==> b ∈ \<int> ==> a + b ∈ \<int>"
apply (rule range_eqI)
done

lemma Ints_minus [simp]: "a ∈ \<int> ==> -a ∈ \<int>"
apply (rule range_eqI)
apply (rule of_int_minus [symmetric])
done

lemma Ints_diff [simp]: "a ∈ \<int> ==> b ∈ \<int> ==> a - b ∈ \<int>"
apply (rule range_eqI)
apply (rule of_int_diff [symmetric])
done

lemma Ints_mult [simp]: "a ∈ \<int> ==> b ∈ \<int> ==> a * b ∈ \<int>"
apply (rule range_eqI)
apply (rule of_int_mult [symmetric])
done

lemma Ints_power [simp]: "a ∈ \<int> ==> a ^ n ∈ \<int>"
by (induct n) simp_all

lemma Ints_cases [cases set: Ints]:
assumes "q ∈ \<int>"
obtains (of_int) z where "q = of_int z"
unfolding Ints_def
proof -
from `q ∈ \<int>` have "q ∈ range of_int" unfolding Ints_def .
then obtain z where "q = of_int z" ..
then show thesis ..
qed

lemma Ints_induct [case_names of_int, induct set: Ints]:
"q ∈ \<int> ==> (!!z. P (of_int z)) ==> P q"
by (rule Ints_cases) auto

end

text {* The premise involving @{term Ints} prevents @{term "a = 1/2"}. *}

lemma Ints_double_eq_0_iff:
assumes in_Ints: "a ∈ Ints"
shows "(a + a = 0) = (a = (0::'a::ring_char_0))"
proof -
from in_Ints have "a ∈ range of_int" unfolding Ints_def [symmetric] .
then obtain z where a: "a = of_int z" ..
show ?thesis
proof
assume "a = 0"
thus "a + a = 0" by simp
next
assume eq: "a + a = 0"
hence "of_int (z + z) = (of_int 0 :: 'a)" by (simp add: a)
hence "z + z = 0" by (simp only: of_int_eq_iff)
hence "z = 0" by (simp only: double_eq_0_iff)
thus "a = 0" by (simp add: a)
qed
qed

lemma Ints_odd_nonzero:
assumes in_Ints: "a ∈ Ints"
shows "1 + a + a ≠ (0::'a::ring_char_0)"
proof -
from in_Ints have "a ∈ range of_int" unfolding Ints_def [symmetric] .
then obtain z where a: "a = of_int z" ..
show ?thesis
proof
assume eq: "1 + a + a = 0"
hence "of_int (1 + z + z) = (of_int 0 :: 'a)" by (simp add: a)
hence "1 + z + z = 0" by (simp only: of_int_eq_iff)
with odd_nonzero show False by blast
qed
qed

lemma Nats_numeral [simp]: "numeral w ∈ Nats"
using of_nat_in_Nats [of "numeral w"] by simp

lemma Ints_odd_less_0:
assumes in_Ints: "a ∈ Ints"
shows "(1 + a + a < 0) = (a < (0::'a::linordered_idom))"
proof -
from in_Ints have "a ∈ range of_int" unfolding Ints_def [symmetric] .
then obtain z where a: "a = of_int z" ..
hence "((1::'a) + a + a < 0) = (of_int (1 + z + z) < (of_int 0 :: 'a))"
also have "... = (z < 0)" by (simp only: of_int_less_iff odd_less_0_iff)
also have "... = (a < 0)" by (simp add: a)
finally show ?thesis .
qed

subsection {* @{term setsum} and @{term setprod} *}

lemma of_nat_setsum: "of_nat (setsum f A) = (∑x∈A. of_nat(f x))"
apply (cases "finite A")
apply (erule finite_induct, auto)
done

lemma of_int_setsum: "of_int (setsum f A) = (∑x∈A. of_int(f x))"
apply (cases "finite A")
apply (erule finite_induct, auto)
done

lemma of_nat_setprod: "of_nat (setprod f A) = (∏x∈A. of_nat(f x))"
apply (cases "finite A")
apply (erule finite_induct, auto simp add: of_nat_mult)
done

lemma of_int_setprod: "of_int (setprod f A) = (∏x∈A. of_int(f x))"
apply (cases "finite A")
apply (erule finite_induct, auto)
done

lemmas int_setsum = of_nat_setsum [where 'a=int]
lemmas int_setprod = of_nat_setprod [where 'a=int]

text {* Legacy theorems *}

lemmas zle_int = of_nat_le_iff [where 'a=int]
lemmas int_int_eq = of_nat_eq_iff [where 'a=int]
lemmas numeral_1_eq_1 = numeral_One

subsection {* Setting up simplification procedures *}

lemmas int_arith_rules =
neg_le_iff_le numeral_One
minus_zero diff_minus left_minus right_minus
mult_zero_left mult_zero_right mult_1_left mult_1_right
mult_minus_left mult_minus_right

ML_file "Tools/int_arith.ML"
declaration {* K Int_Arith.setup *}

simproc_setup fast_arith ("(m::'a::linordered_idom) < n" |
"(m::'a::linordered_idom) <= n" |
"(m::'a::linordered_idom) = n") =
{* fn _ => fn ss => fn ct => Lin_Arith.simproc ss (term_of ct) *}

lemma abs_power_minus_one [simp]:
"abs(-1 ^ n) = (1::'a::linordered_idom)"

subsection{*More Inequality Reasoning*}

lemma zless_add1_eq: "(w < z + (1::int)) = (w<z | w=z)"
by arith

lemma add1_zle_eq: "(w + (1::int) ≤ z) = (w<z)"
by arith

lemma zle_diff1_eq [simp]: "(w ≤ z - (1::int)) = (w<z)"
by arith

lemma zle_add1_eq_le [simp]: "(w < z + (1::int)) = (w≤z)"
by arith

lemma int_one_le_iff_zero_less: "((1::int) ≤ z) = (0 < z)"
by arith

subsection{*The functions @{term nat} and @{term int}*}

text{*Simplify the term @{term "w + - z"}*}
lemmas diff_int_def_symmetric = diff_def [where 'a=int, symmetric, simp]

lemma nat_0 [simp]: "nat 0 = 0"

lemma nat_1 [simp]: "nat 1 = Suc 0"
by (subst nat_eq_iff, simp)

lemma nat_2: "nat 2 = Suc (Suc 0)"
by (subst nat_eq_iff, simp)

lemma one_less_nat_eq [simp]: "(Suc 0 < nat z) = (1 < z)"
apply (insert zless_nat_conj [of 1 z])
apply auto
done

text{*This simplifies expressions of the form @{term "int n = z"} where
z is an integer literal.*}

lemmas int_eq_iff_numeral [simp] = int_eq_iff [of _ "numeral v"] for v

lemma split_nat [arith_split]:
"P(nat(i::int)) = ((∀n. i = int n --> P n) & (i < 0 --> P 0))"
(is "?P = (?L & ?R)")
proof (cases "i < 0")
case True thus ?thesis by auto
next
case False
have "?P = ?L"
proof
assume ?P thus ?L using False by clarsimp
next
assume ?L thus ?P using False by simp
qed
with False show ?thesis by simp
qed

context ring_1
begin

lemma of_int_of_nat [nitpick_simp]:
"of_int k = (if k < 0 then - of_nat (nat (- k)) else of_nat (nat k))"
proof (cases "k < 0")
case True then have "0 ≤ - k" by simp
then have "of_nat (nat (- k)) = of_int (- k)" by (rule of_nat_nat)
with True show ?thesis by simp
next
case False then show ?thesis by (simp add: not_less of_nat_nat)
qed

end

lemma nat_mult_distrib:
fixes z z' :: int
assumes "0 ≤ z"
shows "nat (z * z') = nat z * nat z'"
proof (cases "0 ≤ z'")
case False with assms have "z * z' ≤ 0"
then have "nat (z * z') = 0" by simp
moreover from False have "nat z' = 0" by simp
ultimately show ?thesis by simp
next
case True with assms have ge_0: "z * z' ≥ 0" by (simp add: zero_le_mult_iff)
show ?thesis
by (rule injD [of "of_nat :: nat => int", OF inj_of_nat])
(simp only: of_nat_mult of_nat_nat [OF True]
of_nat_nat [OF assms] of_nat_nat [OF ge_0], simp)
qed

lemma nat_mult_distrib_neg: "z ≤ (0::int) ==> nat(z*z') = nat(-z) * nat(-z')"
apply (rule trans)
apply (rule_tac [2] nat_mult_distrib, auto)
done

lemma nat_abs_mult_distrib: "nat (abs (w * z)) = nat (abs w) * nat (abs z)"
apply (cases "z=0 | w=0")
apply (auto simp add: abs_if nat_mult_distrib [symmetric]
nat_mult_distrib_neg [symmetric] mult_less_0_iff)
done

lemma Suc_nat_eq_nat_zadd1: "(0::int) <= z ==> Suc (nat z) = nat (1 + z)"
apply (rule sym)
done

lemma diff_nat_eq_if:
"nat z - nat z' =
(if z' < 0 then nat z
else let d = z-z' in
if d < 0 then 0 else nat d)"

by (simp add: Let_def nat_diff_distrib [symmetric])

(* nat_diff_distrib has too-strong premises *)
lemma nat_diff_distrib': "[|0 ≤ x; 0 ≤ y|] ==> nat (x - y) = nat x - nat y"
apply (rule int_int_eq [THEN iffD1], clarsimp)
apply (subst of_nat_diff)
apply (rule nat_mono, simp_all)
done

lemma nat_numeral [simp, code_abbrev]:
"nat (numeral k) = numeral k"

lemma nat_neg_numeral [simp]:
"nat (neg_numeral k) = 0"
by simp

lemma diff_nat_numeral [simp]:
"(numeral v :: nat) - numeral v' = nat (numeral v - numeral v')"
by (simp only: nat_diff_distrib' zero_le_numeral nat_numeral)

lemma nat_numeral_diff_1 [simp]:
"numeral v - (1::nat) = nat (numeral v - 1)"
using diff_nat_numeral [of v Num.One] by simp

lemmas nat_arith = diff_nat_numeral

subsection "Induction principles for int"

text{*Well-founded segments of the integers*}

definition
int_ge_less_than :: "int => (int * int) set"
where
"int_ge_less_than d = {(z',z). d ≤ z' & z' < z}"

theorem wf_int_ge_less_than: "wf (int_ge_less_than d)"
proof -
have "int_ge_less_than d ⊆ measure (%z. nat (z-d))"
thus ?thesis
by (rule wf_subset [OF wf_measure])
qed

text{*This variant looks odd, but is typical of the relations suggested
by RankFinder.*}

definition
int_ge_less_than2 :: "int => (int * int) set"
where
"int_ge_less_than2 d = {(z',z). d ≤ z & z' < z}"

theorem wf_int_ge_less_than2: "wf (int_ge_less_than2 d)"
proof -
have "int_ge_less_than2 d ⊆ measure (%z. nat (1+z-d))"
thus ?thesis
by (rule wf_subset [OF wf_measure])
qed

(* `set:int': dummy construction *)
theorem int_ge_induct [case_names base step, induct set: int]:
fixes i :: int
assumes ge: "k ≤ i" and
base: "P k" and
step: "!!i. k ≤ i ==> P i ==> P (i + 1)"
shows "P i"
proof -
{ fix n
have "!!i::int. n = nat (i - k) ==> k ≤ i ==> P i"
proof (induct n)
case 0
hence "i = k" by arith
thus "P i" using base by simp
next
case (Suc n)
then have "n = nat((i - 1) - k)" by arith
moreover
have ki1: "k ≤ i - 1" using Suc.prems by arith
ultimately
have "P (i - 1)" by (rule Suc.hyps)
from step [OF ki1 this] show ?case by simp
qed
}
with ge show ?thesis by fast
qed

(* `set:int': dummy construction *)
theorem int_gr_induct [case_names base step, induct set: int]:
assumes gr: "k < (i::int)" and
base: "P(k+1)" and
step: "!!i. [|k < i; P i|] ==> P(i+1)"
shows "P i"
apply(rule int_ge_induct[of "k + 1"])
using gr apply arith
apply(rule base)
apply (rule step, simp+)
done

theorem int_le_induct [consumes 1, case_names base step]:
assumes le: "i ≤ (k::int)" and
base: "P(k)" and
step: "!!i. [|i ≤ k; P i|] ==> P(i - 1)"
shows "P i"
proof -
{ fix n
have "!!i::int. n = nat(k-i) ==> i ≤ k ==> P i"
proof (induct n)
case 0
hence "i = k" by arith
thus "P i" using base by simp
next
case (Suc n)
hence "n = nat (k - (i + 1))" by arith
moreover
have ki1: "i + 1 ≤ k" using Suc.prems by arith
ultimately
have "P (i + 1)" by(rule Suc.hyps)
from step[OF ki1 this] show ?case by simp
qed
}
with le show ?thesis by fast
qed

theorem int_less_induct [consumes 1, case_names base step]:
assumes less: "(i::int) < k" and
base: "P(k - 1)" and
step: "!!i. [|i < k; P i|] ==> P(i - 1)"
shows "P i"
apply(rule int_le_induct[of _ "k - 1"])
using less apply arith
apply(rule base)
apply (rule step, simp+)
done

theorem int_induct [case_names base step1 step2]:
fixes k :: int
assumes base: "P k"
and step1: "!!i. k ≤ i ==> P i ==> P (i + 1)"
and step2: "!!i. k ≥ i ==> P i ==> P (i - 1)"
shows "P i"
proof -
have "i ≤ k ∨ i ≥ k" by arith
then show ?thesis
proof
assume "i ≥ k"
then show ?thesis using base
by (rule int_ge_induct) (fact step1)
next
assume "i ≤ k"
then show ?thesis using base
by (rule int_le_induct) (fact step2)
qed
qed

subsection{*Intermediate value theorems*}

lemma int_val_lemma:
"(∀i<n::nat. abs(f(i+1) - f i) ≤ 1) -->
f 0 ≤ k --> k ≤ f n --> (∃i ≤ n. f i = (k::int))"

unfolding One_nat_def
apply (induct n)
apply simp
apply (intro strip)
apply (erule impE, simp)
apply (erule_tac x = n in allE, simp)
apply (case_tac "k = f (Suc n)")
apply force
apply (erule impE)
apply (blast intro: le_SucI)
done

lemmas nat0_intermed_int_val = int_val_lemma [rule_format (no_asm)]

lemma nat_intermed_int_val:
"[| ∀i. m ≤ i & i < n --> abs(f(i + 1::nat) - f i) ≤ 1; m < n;
f m ≤ k; k ≤ f n |] ==> ? i. m ≤ i & i ≤ n & f i = (k::int)"

apply (cut_tac n = "n-m" and f = "%i. f (i+m) " and k = k
in int_val_lemma)
unfolding One_nat_def
apply simp
apply (erule exE)
apply (rule_tac x = "i+m" in exI, arith)
done

subsection{*Products and 1, by T. M. Rasmussen*}

lemma zabs_less_one_iff [simp]: "(¦z¦ < 1) = (z = (0::int))"
by arith

lemma abs_zmult_eq_1:
assumes mn: "¦m * n¦ = 1"
shows "¦m¦ = (1::int)"
proof -
have 0: "m ≠ 0 & n ≠ 0" using mn
by auto
have "~ (2 ≤ ¦m¦)"
proof
assume "2 ≤ ¦m¦"
hence "2*¦n¦ ≤ ¦m¦*¦n¦"
also have "... = ¦m*n¦"
also have "... = 1"
finally have "2*¦n¦ ≤ 1" .
thus "False" using 0
by arith
qed
thus ?thesis using 0
by auto
qed

ML_val {* @{const_name neg_numeral} *}

lemma pos_zmult_eq_1_iff_lemma: "(m * n = 1) ==> m = (1::int) | m = -1"
by (insert abs_zmult_eq_1 [of m n], arith)

lemma pos_zmult_eq_1_iff:
assumes "0 < (m::int)" shows "(m * n = 1) = (m = 1 & n = 1)"
proof -
from assms have "m * n = 1 ==> m = 1" by (auto dest: pos_zmult_eq_1_iff_lemma)
thus ?thesis by (auto dest: pos_zmult_eq_1_iff_lemma)
qed

lemma zmult_eq_1_iff: "(m*n = (1::int)) = ((m = 1 & n = 1) | (m = -1 & n = -1))"
apply (rule iffI)
apply (frule pos_zmult_eq_1_iff_lemma)
apply (simp add: mult_commute [of m])
apply (frule pos_zmult_eq_1_iff_lemma, auto)
done

lemma infinite_UNIV_int: "¬ finite (UNIV::int set)"
proof
assume "finite (UNIV::int set)"
moreover have "inj (λi::int. 2 * i)"
by (rule injI) simp
ultimately have "surj (λi::int. 2 * i)"
by (rule finite_UNIV_inj_surj)
then obtain i :: int where "1 = 2 * i" by (rule surjE)
then show False by (simp add: pos_zmult_eq_1_iff)
qed

subsection {* Further theorems on numerals *}

subsubsection{*Special Simplification for Constants*}

text{*These distributive laws move literals inside sums and differences.*}

lemmas distrib_right_numeral [simp] = distrib_right [of _ _ "numeral v"] for v
lemmas distrib_left_numeral [simp] = distrib_left [of "numeral v"] for v
lemmas left_diff_distrib_numeral [simp] = left_diff_distrib [of _ _ "numeral v"] for v
lemmas right_diff_distrib_numeral [simp] = right_diff_distrib [of "numeral v"] for v

text{*These are actually for fields, like real: but where else to put them?*}

lemmas zero_less_divide_iff_numeral [simp, no_atp] = zero_less_divide_iff [of "numeral w"] for w
lemmas divide_less_0_iff_numeral [simp, no_atp] = divide_less_0_iff [of "numeral w"] for w
lemmas zero_le_divide_iff_numeral [simp, no_atp] = zero_le_divide_iff [of "numeral w"] for w
lemmas divide_le_0_iff_numeral [simp, no_atp] = divide_le_0_iff [of "numeral w"] for w

text {*Replaces @{text "inverse #nn"} by @{text "1/#nn"}. It looks
strange, but then other simprocs simplify the quotient.*}

lemmas inverse_eq_divide_numeral [simp] =
inverse_eq_divide [of "numeral w"] for w

lemmas inverse_eq_divide_neg_numeral [simp] =
inverse_eq_divide [of "neg_numeral w"] for w

text {*These laws simplify inequalities, moving unary minus from a term
into the literal.*}

lemmas le_minus_iff_numeral [simp, no_atp] =
le_minus_iff [of "numeral v"]
le_minus_iff [of "neg_numeral v"] for v

lemmas equation_minus_iff_numeral [simp, no_atp] =
equation_minus_iff [of "numeral v"]
equation_minus_iff [of "neg_numeral v"] for v

lemmas minus_less_iff_numeral [simp, no_atp] =
minus_less_iff [of _ "numeral v"]
minus_less_iff [of _ "neg_numeral v"] for v

lemmas minus_le_iff_numeral [simp, no_atp] =
minus_le_iff [of _ "numeral v"]
minus_le_iff [of _ "neg_numeral v"] for v

lemmas minus_equation_iff_numeral [simp, no_atp] =
minus_equation_iff [of _ "numeral v"]
minus_equation_iff [of _ "neg_numeral v"] for v

text{*To Simplify Inequalities Where One Side is the Constant 1*}

lemma less_minus_iff_1 [simp,no_atp]:
fixes b::"'b::linordered_idom"
shows "(1 < - b) = (b < -1)"
by auto

lemma le_minus_iff_1 [simp,no_atp]:
fixes b::"'b::linordered_idom"
shows "(1 ≤ - b) = (b ≤ -1)"
by auto

lemma equation_minus_iff_1 [simp,no_atp]:
fixes b::"'b::ring_1"
shows "(1 = - b) = (b = -1)"
by (subst equation_minus_iff, auto)

lemma minus_less_iff_1 [simp,no_atp]:
fixes a::"'b::linordered_idom"
shows "(- a < 1) = (-1 < a)"
by auto

lemma minus_le_iff_1 [simp,no_atp]:
fixes a::"'b::linordered_idom"
shows "(- a ≤ 1) = (-1 ≤ a)"
by auto

lemma minus_equation_iff_1 [simp,no_atp]:
fixes a::"'b::ring_1"
shows "(- a = 1) = (a = -1)"
by (subst minus_equation_iff, auto)

text {*Cancellation of constant factors in comparisons (@{text "<"} and @{text "≤"}) *}

lemmas mult_less_cancel_left_numeral [simp, no_atp] = mult_less_cancel_left [of "numeral v"] for v
lemmas mult_less_cancel_right_numeral [simp, no_atp] = mult_less_cancel_right [of _ "numeral v"] for v
lemmas mult_le_cancel_left_numeral [simp, no_atp] = mult_le_cancel_left [of "numeral v"] for v
lemmas mult_le_cancel_right_numeral [simp, no_atp] = mult_le_cancel_right [of _ "numeral v"] for v

text {*Multiplying out constant divisors in comparisons (@{text "<"}, @{text "≤"} and @{text "="}) *}

lemmas le_divide_eq_numeral1 [simp] =
pos_le_divide_eq [of "numeral w", OF zero_less_numeral]
neg_le_divide_eq [of "neg_numeral w", OF neg_numeral_less_zero] for w

lemmas divide_le_eq_numeral1 [simp] =
pos_divide_le_eq [of "numeral w", OF zero_less_numeral]
neg_divide_le_eq [of "neg_numeral w", OF neg_numeral_less_zero] for w

lemmas less_divide_eq_numeral1 [simp] =
pos_less_divide_eq [of "numeral w", OF zero_less_numeral]
neg_less_divide_eq [of "neg_numeral w", OF neg_numeral_less_zero] for w

lemmas divide_less_eq_numeral1 [simp] =
pos_divide_less_eq [of "numeral w", OF zero_less_numeral]
neg_divide_less_eq [of "neg_numeral w", OF neg_numeral_less_zero] for w

lemmas eq_divide_eq_numeral1 [simp] =
eq_divide_eq [of _ _ "numeral w"]
eq_divide_eq [of _ _ "neg_numeral w"] for w

lemmas divide_eq_eq_numeral1 [simp] =
divide_eq_eq [of _ "numeral w"]
divide_eq_eq [of _ "neg_numeral w"] for w

subsubsection{*Optional Simplification Rules Involving Constants*}

text{*Simplify quotients that are compared with a literal constant.*}

lemmas le_divide_eq_numeral =
le_divide_eq [of "numeral w"]
le_divide_eq [of "neg_numeral w"] for w

lemmas divide_le_eq_numeral =
divide_le_eq [of _ _ "numeral w"]
divide_le_eq [of _ _ "neg_numeral w"] for w

lemmas less_divide_eq_numeral =
less_divide_eq [of "numeral w"]
less_divide_eq [of "neg_numeral w"] for w

lemmas divide_less_eq_numeral =
divide_less_eq [of _ _ "numeral w"]
divide_less_eq [of _ _ "neg_numeral w"] for w

lemmas eq_divide_eq_numeral =
eq_divide_eq [of "numeral w"]
eq_divide_eq [of "neg_numeral w"] for w

lemmas divide_eq_eq_numeral =
divide_eq_eq [of _ _ "numeral w"]
divide_eq_eq [of _ _ "neg_numeral w"] for w

text{*Not good as automatic simprules because they cause case splits.*}
lemmas divide_const_simps =
le_divide_eq_numeral divide_le_eq_numeral less_divide_eq_numeral
divide_less_eq_numeral eq_divide_eq_numeral divide_eq_eq_numeral
le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1

text{*Division By @{text "-1"}*}

lemma divide_minus1 [simp]: "(x::'a::field) / -1 = - x"
unfolding minus_one [symmetric]
unfolding nonzero_minus_divide_right [OF one_neq_zero, symmetric]
by simp

lemma minus1_divide [simp]: "-1 / (x::'a::field) = - (1 / x)"
unfolding minus_one [symmetric] by (rule divide_minus_left)

lemma half_gt_zero_iff:
"(0 < r/2) = (0 < (r::'a::linordered_field_inverse_zero))"
by auto

lemmas half_gt_zero [simp] = half_gt_zero_iff [THEN iffD2]

lemma divide_Numeral1: "(x::'a::field) / Numeral1 = x"
by simp

subsection {* The divides relation *}

lemma zdvd_antisym_nonneg:
"0 <= m ==> 0 <= n ==> m dvd n ==> n dvd m ==> m = (n::int)"
apply (auto simp add: mult_assoc zero_le_mult_iff zmult_eq_1_iff)
done

lemma zdvd_antisym_abs: assumes "(a::int) dvd b" and "b dvd a"
shows "¦a¦ = ¦b¦"
proof cases
assume "a = 0" with assms show ?thesis by simp
next
assume "a ≠ 0"
from `a dvd b` obtain k where k:"b = a*k" unfolding dvd_def by blast
from `b dvd a` obtain k' where k':"a = b*k'" unfolding dvd_def by blast
from k k' have "a = a*k*k'" by simp
with mult_cancel_left1[where c="a" and b="k*k'"]
have kk':"k*k' = 1" using `a≠0` by (simp add: mult_assoc)
hence "k = 1 ∧ k' = 1 ∨ k = -1 ∧ k' = -1" by (simp add: zmult_eq_1_iff)
thus ?thesis using k k' by auto
qed

lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"
apply (subgoal_tac "m = n + (m - n)")
apply (erule ssubst)
done

lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"
apply (rule iffI)
apply (subgoal_tac "n = (n + k * m) - k * m")
apply (erule ssubst)
apply (erule dvd_diff)
apply(simp_all)
done

lemma dvd_imp_le_int:
fixes d i :: int
assumes "i ≠ 0" and "d dvd i"
shows "¦d¦ ≤ ¦i¦"
proof -
from `d dvd i` obtain k where "i = d * k" ..
with `i ≠ 0` have "k ≠ 0" by auto
then have "1 ≤ ¦k¦" and "0 ≤ ¦d¦" by auto
then have "¦d¦ * 1 ≤ ¦d¦ * ¦k¦" by (rule mult_left_mono)
with `i = d * k` show ?thesis by (simp add: abs_mult)
qed

lemma zdvd_not_zless:
fixes m n :: int
assumes "0 < m" and "m < n"
shows "¬ n dvd m"
proof
from assms have "0 < n" by auto
assume "n dvd m" then obtain k where k: "m = n * k" ..
with `0 < m` have "0 < n * k" by auto
with `0 < n` have "0 < k" by (simp add: zero_less_mult_iff)
with k `0 < n` `m < n` have "n * k < n * 1" by simp
with `0 < n` `0 < k` show False unfolding mult_less_cancel_left by auto
qed

lemma zdvd_mult_cancel: assumes d:"k * m dvd k * n" and kz:"k ≠ (0::int)"
shows "m dvd n"
proof-
from d obtain h where h: "k*n = k*m * h" unfolding dvd_def by blast
{assume "n ≠ m*h" hence "k* n ≠ k* (m*h)" using kz by simp
with h have False by (simp add: mult_assoc)}
hence "n = m * h" by blast
thus ?thesis by simp
qed

theorem zdvd_int: "(x dvd y) = (int x dvd int y)"
proof -
have "!!k. int y = int x * k ==> x dvd y"
proof -
fix k
assume A: "int y = int x * k"
then show "x dvd y"
proof (cases k)
case (nonneg n)
with A have "y = x * n" by (simp add: of_nat_mult [symmetric])
then show ?thesis ..
next
case (neg n)
with A have "int y = int x * (- int (Suc n))" by simp
also have "… = - (int x * int (Suc n))" by (simp only: mult_minus_right)
also have "… = - int (x * Suc n)" by (simp only: of_nat_mult [symmetric])
finally have "- int (x * Suc n) = int y" ..
then show ?thesis by (simp only: negative_eq_positive) auto
qed
qed
then show ?thesis by (auto elim!: dvdE simp only: dvd_triv_left of_nat_mult)
qed

lemma zdvd1_eq[simp]: "(x::int) dvd 1 = (¦x¦ = 1)"
proof
assume d: "x dvd 1" hence "int (nat ¦x¦) dvd int (nat 1)" by simp
hence "nat ¦x¦ dvd 1" by (simp add: zdvd_int)
hence "nat ¦x¦ = 1" by simp
thus "¦x¦ = 1" by (cases "x < 0") auto
next
assume "¦x¦=1"
then have "x = 1 ∨ x = -1" by auto
then show "x dvd 1" by (auto intro: dvdI)
qed

lemma zdvd_mult_cancel1:
assumes mp:"m ≠(0::int)" shows "(m * n dvd m) = (¦n¦ = 1)"
proof
assume n1: "¦n¦ = 1" thus "m * n dvd m"
by (cases "n >0") (auto simp add: minus_equation_iff)
next
assume H: "m * n dvd m" hence H2: "m * n dvd m * 1" by simp
from zdvd_mult_cancel[OF H2 mp] show "¦n¦ = 1" by (simp only: zdvd1_eq)
qed

lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))"
unfolding zdvd_int by (cases "z ≥ 0") simp_all

lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)"
unfolding zdvd_int by (cases "z ≥ 0") simp_all

lemma nat_dvd_iff: "(nat z dvd m) = (if 0 ≤ z then (z dvd int m) else m = 0)"

lemma eq_nat_nat_iff:
"0 ≤ z ==> 0 ≤ z' ==> nat z = nat z' <-> z = z'"
by (auto elim!: nonneg_eq_int)

lemma nat_power_eq:
"0 ≤ z ==> nat (z ^ n) = nat z ^ n"
by (induct n) (simp_all add: nat_mult_distrib)

lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z ≤ (n::int)"
apply (cases n)
apply (cases z)
done

lemma zdvd_period:
fixes a d :: int
assumes "a dvd d"
shows "a dvd (x + t) <-> a dvd ((x + c * d) + t)"
proof -
from assms obtain k where "d = a * k" by (rule dvdE)
show ?thesis
proof
assume "a dvd (x + t)"
then obtain l where "x + t = a * l" by (rule dvdE)
then have "x = a * l - t" by simp
with `d = a * k` show "a dvd x + c * d + t" by simp
next
assume "a dvd x + c * d + t"
then obtain l where "x + c * d + t = a * l" by (rule dvdE)
then have "x = a * l - c * d - t" by simp
with `d = a * k` show "a dvd (x + t)" by simp
qed
qed

subsection {* Finiteness of intervals *}

lemma finite_interval_int1 [iff]: "finite {i :: int. a <= i & i <= b}"
proof (cases "a <= b")
case True
from this show ?thesis
proof (induct b rule: int_ge_induct)
case base
have "{i. a <= i & i <= a} = {a}" by auto
from this show ?case by simp
next
case (step b)
from this have "{i. a <= i & i <= b + 1} = {i. a <= i & i <= b} ∪ {b + 1}" by auto
from this step show ?case by simp
qed
next
case False from this show ?thesis
by (metis (lifting, no_types) Collect_empty_eq finite.emptyI order_trans)
qed

lemma finite_interval_int2 [iff]: "finite {i :: int. a <= i & i < b}"
by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto

lemma finite_interval_int3 [iff]: "finite {i :: int. a < i & i <= b}"
by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto

lemma finite_interval_int4 [iff]: "finite {i :: int. a < i & i < b}"
by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto

subsection {* Configuration of the code generator *}

text {* Constructors *}

definition Pos :: "num => int" where
[simp, code_abbrev]: "Pos = numeral"

definition Neg :: "num => int" where
[simp, code_abbrev]: "Neg = neg_numeral"

code_datatype "0::int" Pos Neg

text {* Auxiliary operations *}

definition dup :: "int => int" where
[simp]: "dup k = k + k"

lemma dup_code [code]:
"dup 0 = 0"
"dup (Pos n) = Pos (Num.Bit0 n)"
"dup (Neg n) = Neg (Num.Bit0 n)"
unfolding Pos_def Neg_def neg_numeral_def

definition sub :: "num => num => int" where
[simp]: "sub m n = numeral m - numeral n"

lemma sub_code [code]:
"sub Num.One Num.One = 0"
"sub (Num.Bit0 m) Num.One = Pos (Num.BitM m)"
"sub (Num.Bit1 m) Num.One = Pos (Num.Bit0 m)"
"sub Num.One (Num.Bit0 n) = Neg (Num.BitM n)"
"sub Num.One (Num.Bit1 n) = Neg (Num.Bit0 n)"
"sub (Num.Bit0 m) (Num.Bit0 n) = dup (sub m n)"
"sub (Num.Bit1 m) (Num.Bit1 n) = dup (sub m n)"
"sub (Num.Bit1 m) (Num.Bit0 n) = dup (sub m n) + 1"
"sub (Num.Bit0 m) (Num.Bit1 n) = dup (sub m n) - 1"
unfolding sub_def dup_def numeral.simps Pos_def Neg_def
neg_numeral_def numeral_BitM
by (simp_all only: algebra_simps)

text {* Implementations *}

lemma one_int_code [code, code_unfold]:
"1 = Pos Num.One"
by simp

lemma plus_int_code [code]:
"k + 0 = (k::int)"
"0 + l = (l::int)"
"Pos m + Pos n = Pos (m + n)"
"Pos m + Neg n = sub m n"
"Neg m + Pos n = sub n m"
"Neg m + Neg n = Neg (m + n)"
by simp_all

lemma uminus_int_code [code]:
"uminus 0 = (0::int)"
"uminus (Pos m) = Neg m"
"uminus (Neg m) = Pos m"
by simp_all

lemma minus_int_code [code]:
"k - 0 = (k::int)"
"0 - l = uminus (l::int)"
"Pos m - Pos n = sub m n"
"Pos m - Neg n = Pos (m + n)"
"Neg m - Pos n = Neg (m + n)"
"Neg m - Neg n = sub n m"
by simp_all

lemma times_int_code [code]:
"k * 0 = (0::int)"
"0 * l = (0::int)"
"Pos m * Pos n = Pos (m * n)"
"Pos m * Neg n = Neg (m * n)"
"Neg m * Pos n = Neg (m * n)"
"Neg m * Neg n = Pos (m * n)"
by simp_all

instantiation int :: equal
begin

definition
"HOL.equal k l <-> k = (l::int)"

instance by default (rule equal_int_def)

end

lemma equal_int_code [code]:
"HOL.equal 0 (0::int) <-> True"
"HOL.equal 0 (Pos l) <-> False"
"HOL.equal 0 (Neg l) <-> False"
"HOL.equal (Pos k) 0 <-> False"
"HOL.equal (Pos k) (Pos l) <-> HOL.equal k l"
"HOL.equal (Pos k) (Neg l) <-> False"
"HOL.equal (Neg k) 0 <-> False"
"HOL.equal (Neg k) (Pos l) <-> False"
"HOL.equal (Neg k) (Neg l) <-> HOL.equal k l"

lemma equal_int_refl [code nbe]:
"HOL.equal (k::int) k <-> True"
by (fact equal_refl)

lemma less_eq_int_code [code]:
"0 ≤ (0::int) <-> True"
"0 ≤ Pos l <-> True"
"0 ≤ Neg l <-> False"
"Pos k ≤ 0 <-> False"
"Pos k ≤ Pos l <-> k ≤ l"
"Pos k ≤ Neg l <-> False"
"Neg k ≤ 0 <-> True"
"Neg k ≤ Pos l <-> True"
"Neg k ≤ Neg l <-> l ≤ k"
by simp_all

lemma less_int_code [code]:
"0 < (0::int) <-> False"
"0 < Pos l <-> True"
"0 < Neg l <-> False"
"Pos k < 0 <-> False"
"Pos k < Pos l <-> k < l"
"Pos k < Neg l <-> False"
"Neg k < 0 <-> True"
"Neg k < Pos l <-> True"
"Neg k < Neg l <-> l < k"
by simp_all

lemma nat_code [code]:
"nat (Int.Neg k) = 0"
"nat 0 = 0"
"nat (Int.Pos k) = nat_of_num k"

lemma (in ring_1) of_int_code [code]:
"of_int (Int.Neg k) = neg_numeral k"
"of_int 0 = 0"
"of_int (Int.Pos k) = numeral k"
by simp_all

text {* Serializer setup *}

code_modulename SML
Int Arith

code_modulename OCaml
Int Arith

Int Arith

quickcheck_params [default_type = int]

hide_const (open) Pos Neg sub dup

subsection {* Legacy theorems *}

lemmas inj_int = inj_of_nat [where 'a=int]
lemmas int_mult = of_nat_mult [where 'a=int]
lemmas zmult_int = of_nat_mult [where 'a=int, symmetric]
lemmas int_eq_0_conv = of_nat_eq_0_iff [where 'a=int and m="n"] for n
lemmas zless_int = of_nat_less_iff [where 'a=int]
lemmas int_less_0_conv = of_nat_less_0_iff [where 'a=int and m="k"] for k
lemmas zero_less_int_conv = of_nat_0_less_iff [where 'a=int]
lemmas zero_zle_int = of_nat_0_le_iff [where 'a=int]
lemmas int_le_0_conv = of_nat_le_0_iff [where 'a=int and m="n"] for n
lemmas int_0 = of_nat_0 [where 'a=int]
lemmas int_1 = of_nat_1 [where 'a=int]
lemmas int_Suc = of_nat_Suc [where 'a=int]
lemmas int_numeral = of_nat_numeral [where 'a=int]
lemmas abs_int_eq = abs_of_nat [where 'a=int and n="m"] for m
lemmas of_int_int_eq = of_int_of_nat_eq [where 'a=int]
lemmas zdiff_int = of_nat_diff [where 'a=int, symmetric]
lemmas zpower_numeral_even = power_numeral_even [where 'a=int]
lemmas zpower_numeral_odd = power_numeral_odd [where 'a=int]

lemma zpower_zpower:
"(x ^ y) ^ z = (x ^ (y * z)::int)"
by (rule power_mult [symmetric])

lemma int_power:
"int (m ^ n) = int m ^ n"
by (rule of_nat_power)

lemmas zpower_int = int_power [symmetric]

text {* De-register @{text "int"} as a quotient type: *}

lemmas [transfer_rule del] =
int.id_abs_transfer int.rel_eq_transfer zero_int.transfer one_int.transfer
plus_int.transfer uminus_int.transfer minus_int.transfer times_int.transfer
int_transfer less_eq_int.transfer less_int.transfer of_int.transfer
nat.transfer

declare Quotient_int [quot_del]

end