Theory Saturated

theory Saturated
imports Type_Length
(*  Title:      HOL/Library/Saturated.thy
Author: Brian Huffman
Author: Peter Gammie
Author: Florian Haftmann
*)


header {* Saturated arithmetic *}

theory Saturated
imports Numeral_Type "~~/src/HOL/Word/Type_Length"
begin

subsection {* The type of saturated naturals *}

typedef ('a::len) sat = "{.. len_of TYPE('a)}"
morphisms nat_of Abs_sat
by auto

lemma sat_eqI:
"nat_of m = nat_of n ==> m = n"
by (simp add: nat_of_inject)

lemma sat_eq_iff:
"m = n <-> nat_of m = nat_of n"
by (simp add: nat_of_inject)

lemma Abs_sat_nat_of [code abstype]:
"Abs_sat (nat_of n) = n"
by (fact nat_of_inverse)

definition Abs_sat' :: "nat => 'a::len sat" where
"Abs_sat' n = Abs_sat (min (len_of TYPE('a)) n)"

lemma nat_of_Abs_sat' [simp]:
"nat_of (Abs_sat' n :: ('a::len) sat) = min (len_of TYPE('a)) n"
unfolding Abs_sat'_def by (rule Abs_sat_inverse) simp

lemma nat_of_le_len_of [simp]:
"nat_of (n :: ('a::len) sat) ≤ 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.commute) simp

lemma Abs_sat'_nat_of [simp]:
"Abs_sat' (nat_of n) = n"
by (simp add: Abs_sat'_def nat_of_inverse)

instantiation sat :: (len) linorder
begin

definition
less_eq_sat_def: "x ≤ y <-> nat_of x ≤ nat_of y"

definition
less_sat_def: "x < y <-> 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_1}"
begin

definition
"0 = Abs_sat' 0"

definition
"1 = Abs_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 = Abs_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 = Abs_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 ≤ len_of TYPE('a)" by arith
then show ?thesis by (simp add: minus_sat_def)
qed

definition
"x * y = Abs_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 ≠ 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 min_add_distrib_left min_add_distrib_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

lemma Abs_sat'_eq_of_nat: "Abs_sat' n = of_nat n"
by (rule sat_eqI, induct n, simp_all)

abbreviation Sat :: "nat => 'a::len sat" where
"Sat ≡ of_nat"

lemma nat_of_Sat [simp]:
"nat_of (Sat n :: ('a::len) sat) = min (len_of TYPE('a)) n"
by (rule nat_of_Abs_sat' [unfolded Abs_sat'_eq_of_nat])

lemma [code_abbrev]:
"of_nat (numeral k) = (numeral k :: 'a::len sat)"
by simp

definition sat_of_nat :: "nat => ('a::len) sat"
where [code_abbrev]: "sat_of_nat = of_nat"

lemma [code abstract]:
"nat_of (sat_of_nat n :: ('a::len) sat) = min (len_of TYPE('a)) n"
by (simp add: sat_of_nat_def)

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 <-> 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 => 'a sat => 'a sat) = min"

definition
"(sup :: 'a sat => 'a sat => '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) "{Inf, Sup}"
begin

definition
"Inf = (semilattice_neutr_set.F min top :: 'a sat set => 'a sat)"

definition
"Sup = (semilattice_neutr_set.F max bot :: 'a sat set => 'a sat)"

instance ..

end

interpretation Inf_sat!: semilattice_neutr_set min "top :: 'a::len sat"
where
"semilattice_neutr_set.F min (top :: 'a sat) = Inf"
proof -
show "semilattice_neutr_set min (top :: 'a sat)" by default (simp add: min_def)
show "semilattice_neutr_set.F min (top :: 'a sat) = Inf" by (simp add: Inf_sat_def)
qed

interpretation Sup_sat!: semilattice_neutr_set max "bot :: 'a::len sat"
where
"semilattice_neutr_set.F max (bot :: 'a sat) = Sup"
proof -
show "semilattice_neutr_set max (bot :: 'a sat)" by default (simp add: max_def bot.extremum_unique)
show "semilattice_neutr_set.F max (bot :: 'a sat) = Sup" by (simp add: Sup_sat_def)
qed

instance sat :: (len) complete_lattice
proof
fix x :: "'a sat"
fix A :: "'a sat set"
note finite
moreover assume "x ∈ A"
ultimately show "Inf A ≤ x"
by (induct A) (auto intro: min_max.le_infI2)
next
fix z :: "'a sat"
fix A :: "'a sat set"
note finite
moreover assume z: "!!x. x ∈ A ==> z ≤ x"
ultimately show "z ≤ Inf A" by (induct A) simp_all
next
fix x :: "'a sat"
fix A :: "'a sat set"
note finite
moreover assume "x ∈ A"
ultimately show "x ≤ Sup A"
by (induct A) (auto intro: min_max.le_supI2)
next
fix z :: "'a sat"
fix A :: "'a sat set"
note finite
moreover assume z: "!!x. x ∈ A ==> x ≤ z"
ultimately show "Sup A ≤ z" by (induct A) auto
next
show "Inf {} = (top::'a sat)"
by (auto simp: top_sat_def)
next
show "Sup {} = (bot::'a sat)"
by (auto simp: bot_sat_def)
qed

hide_const (open) sat_of_nat

end