src/HOL/Library/Saturated.thy

+(* Author: Brian Huffman *)
+(* Author: Peter Gammie *)
+(* Author: Florian Haftmann *)
+
+header {* Saturated arithmetic *}
+
+theory Saturated
+imports Main "~~/src/HOL/Library/Numeral_Type" "~~/src/HOL/Word/Type_Length"
+begin
+
+subsection {* The type of saturated naturals *}
+
+typedef (open) ('a::len) sat = "{.. len_of TYPE('a)}"
+  morphisms nat_of Abs_sat
+  by auto
+
+lemma sat_eqI:
+  "nat_of m = nat_of n \<Longrightarrow> m = n"
+  by (simp add: nat_of_inject)
+
+lemma sat_eq_iff:
+  "m = n \<longleftrightarrow> nat_of m = nat_of n"
+  by (simp add: nat_of_inject)
+
+lemma Abs_sa_nat_of [code abstype]:
+  "Abs_sat (nat_of n) = n"
+  by (fact nat_of_inverse)
+
+definition Sat :: "nat \<Rightarrow> 'a::len sat" where
+  "Sat n = Abs_sat (min (len_of TYPE('a)) n)"
+
+lemma nat_of_Sat [simp]:
+  "nat_of (Sat n :: ('a::len) sat) = min (len_of TYPE('a)) n"
+  unfolding Sat_def by (rule Abs_sat_inverse) simp
+
+lemma nat_of_le_len_of [simp]:
+  "nat_of (n :: ('a::len) sat) \<le> len_of TYPE('a)"
+  using nat_of [where x = n] by simp
+
+lemma min_len_of_nat_of [simp]:
+  "min (len_of TYPE('a)) (nat_of (n::('a::len) sat)) = nat_of n"
+  by (rule min_max.inf_absorb2 [OF nat_of_le_len_of])
+
+lemma min_nat_of_len_of [simp]:
+  "min (nat_of (n::('a::len) sat)) (len_of TYPE('a)) = nat_of n"
+  by (subst min_max.inf.commute) simp
+
+lemma Sat_nat_of [simp]:
+  "Sat (nat_of n) = n"
+  by (simp add: Sat_def nat_of_inverse)
+
+instantiation sat :: (len) linorder
+begin
+
+definition
+  less_eq_sat_def: "x \<le> y \<longleftrightarrow> nat_of x \<le> nat_of y"
+
+definition
+  less_sat_def: "x < y \<longleftrightarrow> nat_of x < nat_of y"
+
+instance
+by default (auto simp add: less_eq_sat_def less_sat_def not_le sat_eq_iff min_max.le_infI1 nat_mult_commute)
+
+end
+
+instantiation sat :: (len) "{minus, comm_semiring_0, comm_semiring_1}"
+begin
+
+definition
+  "0 = Sat 0"
+
+definition
+  "1 = Sat 1"
+
+lemma nat_of_zero_sat [simp, code abstract]:
+  "nat_of 0 = 0"
+  by (simp add: zero_sat_def)
+
+lemma nat_of_one_sat [simp, code abstract]:
+  "nat_of 1 = min 1 (len_of TYPE('a))"
+  by (simp add: one_sat_def)
+
+definition
+  "x + y = Sat (nat_of x + nat_of y)"
+
+lemma nat_of_plus_sat [simp, code abstract]:
+  "nat_of (x + y) = min (nat_of x + nat_of y) (len_of TYPE('a))"
+  by (simp add: plus_sat_def)
+
+definition
+  "x - y = Sat (nat_of x - nat_of y)"
+
+lemma nat_of_minus_sat [simp, code abstract]:
+  "nat_of (x - y) = nat_of x - nat_of y"
+proof -
+  from nat_of_le_len_of [of x] have "nat_of x - nat_of y \<le> len_of TYPE('a)" by arith
+  then show ?thesis by (simp add: minus_sat_def)
+qed
+
+definition
+  "x * y = Sat (nat_of x * nat_of y)"
+
+lemma nat_of_times_sat [simp, code abstract]:
+  "nat_of (x * y) = min (nat_of x * nat_of y) (len_of TYPE('a))"
+  by (simp add: times_sat_def)
+
+instance proof
+  fix a b c :: "('a::len) sat"
+  show "a * b * c = a * (b * c)"
+  proof(cases "a = 0")
+    case True thus ?thesis by (simp add: sat_eq_iff)
+  next
+    case False show ?thesis
+    proof(cases "c = 0")
+      case True thus ?thesis by (simp add: sat_eq_iff)
+    next
+      case False with `a \<noteq> 0` show ?thesis
+        by (simp add: sat_eq_iff nat_mult_min_left nat_mult_min_right mult_assoc min_max.inf_assoc min_max.inf_absorb2)
+    qed
+  qed
+next
+  fix a :: "('a::len) sat"
+  show "1 * a = a"
+    apply (simp add: sat_eq_iff)
+    apply (metis One_nat_def len_gt_0 less_Suc0 less_zeroE linorder_not_less min_max.le_iff_inf min_nat_of_len_of nat_mult_1_right nat_mult_commute)
+    done
+next
+  fix a b c :: "('a::len) sat"
+  show "(a + b) * c = a * c + b * c"
+  proof(cases "c = 0")
+    case True thus ?thesis by (simp add: sat_eq_iff)
+  next
+    case False thus ?thesis
+      by (simp add: sat_eq_iff nat_mult_min_left add_mult_distrib nat_add_min_left nat_add_min_right min_max.inf_assoc min_max.inf_absorb2)
+  qed
+qed (simp_all add: sat_eq_iff mult.commute)
+
+end
+
+instantiation sat :: (len) ordered_comm_semiring
+begin
+
+instance
+by default (auto simp add: less_eq_sat_def less_sat_def not_le sat_eq_iff min_max.le_infI1 nat_mult_commute)
+
+end
+
+instantiation sat :: (len) number
+begin
+
+definition
+  number_of_sat_def [code del]: "number_of = Sat \<circ> nat"
+
+instance ..
+
+end
+
+lemma [code abstract]:
+  "nat_of (number_of n :: ('a::len) sat) = min (nat n) (len_of TYPE('a))"
+  unfolding number_of_sat_def by simp
+
+instance sat :: (len) finite
+proof
+  show "finite (UNIV::'a sat set)"
+    unfolding type_definition.univ [OF type_definition_sat]
+    using finite by simp
+qed
+
+instantiation sat :: (len) equal
+begin
+
+definition
+  "HOL.equal A B \<longleftrightarrow> nat_of A = nat_of B"
+
+instance proof
+qed (simp add: equal_sat_def nat_of_inject)
+
+end
+
+instantiation sat :: (len) "{bounded_lattice, distrib_lattice}"
+begin
+
+definition
+  "(inf :: 'a sat \<Rightarrow> 'a sat \<Rightarrow> 'a sat) = min"
+
+definition
+  "(sup :: 'a sat \<Rightarrow> 'a sat \<Rightarrow> 'a sat) = max"
+
+definition
+  "bot = (0 :: 'a sat)"
+
+definition
+  "top = Sat (len_of TYPE('a))"
+
+instance proof
+qed (simp_all add: inf_sat_def sup_sat_def bot_sat_def top_sat_def min_max.sup_inf_distrib1,
+  simp_all add: less_eq_sat_def)
+
+end
+
+instantiation sat :: (len) complete_lattice
+begin
+
+definition
+  "Inf (A :: 'a sat set) = fold min top A"
+
+definition
+  "Sup (A :: 'a sat set) = fold max bot A"
+
+instance proof
+  fix x :: "'a sat"
+  fix A :: "'a sat set"
+  note finite
+  moreover assume "x \<in> A"
+  ultimately have "fold min top A \<le> min x top" by (rule min_max.fold_inf_le_inf)
+  then show "Inf A \<le> x" by (simp add: Inf_sat_def)
+next
+  fix z :: "'a sat"
+  fix A :: "'a sat set"
+  note finite
+  moreover assume z: "\<And>x. x \<in> A \<Longrightarrow> z \<le> x"
+  ultimately have "min z top \<le> fold min top A" by (blast intro: min_max.inf_le_fold_inf)
+  then show "z \<le> Inf A" by (simp add: Inf_sat_def min_def)
+next
+  fix x :: "'a sat"
+  fix A :: "'a sat set"
+  note finite
+  moreover assume "x \<in> A"
+  ultimately have "max x bot \<le> fold max bot A" by (rule min_max.sup_le_fold_sup)
+  then show "x \<le> Sup A" by (simp add: Sup_sat_def)
+next
+  fix z :: "'a sat"
+  fix A :: "'a sat set"
+  note finite
+  moreover assume z: "\<And>x. x \<in> A \<Longrightarrow> x \<le> z"
+  ultimately have "fold max bot A \<le> max z bot" by (blast intro: min_max.fold_sup_le_sup)
+  then show "Sup A \<le> z" by (simp add: Sup_sat_def max_def bot_unique)
+qed
+
+end
+
+end