(* Title : HOL/Hyperreal/StarClasses.thy
ID : $Id$
Author : Brian Huffman
*)
header {* Class Instances *}
theory StarClasses
imports StarDef
begin
subsection {* Syntactic classes *}
instance star :: (zero) zero
star_zero_def: "0 \<equiv> star_of 0" ..
instance star :: (one) one
star_one_def: "1 \<equiv> star_of 1" ..
instance star :: (plus) plus
star_add_def: "(op +) \<equiv> *f2* (op +)" ..
instance star :: (times) times
star_mult_def: "(op *) \<equiv> *f2* (op *)" ..
instance star :: (minus) minus
star_minus_def: "uminus \<equiv> *f* uminus"
star_diff_def: "(op -) \<equiv> *f2* (op -)"
star_abs_def: "abs \<equiv> *f* abs" ..
instance star :: (inverse) inverse
star_divide_def: "(op /) \<equiv> *f2* (op /)"
star_inverse_def: "inverse \<equiv> *f* inverse" ..
instance star :: (number) number
star_number_def: "number_of b \<equiv> star_of (number_of b)" ..
instance star :: ("Divides.div") "Divides.div"
star_div_def: "(op div) \<equiv> *f2* (op div)"
star_mod_def: "(op mod) \<equiv> *f2* (op mod)" ..
instance star :: (power) power
star_power_def: "(op ^) \<equiv> \<lambda>x n. ( *f* (\<lambda>x. x ^ n)) x" ..
instance star :: (ord) ord
star_le_def: "(op \<le>) \<equiv> *p2* (op \<le>)"
star_less_def: "(op <) \<equiv> *p2* (op <)" ..
lemmas star_class_defs [transfer_unfold] =
star_zero_def star_one_def star_number_def
star_add_def star_diff_def star_minus_def
star_mult_def star_divide_def star_inverse_def
star_le_def star_less_def star_abs_def
star_div_def star_mod_def star_power_def
text {* Class operations preserve standard elements *}
lemma Standard_zero: "0 \<in> Standard"
by (simp add: star_zero_def)
lemma Standard_one: "1 \<in> Standard"
by (simp add: star_one_def)
lemma Standard_number_of: "number_of b \<in> Standard"
by (simp add: star_number_def)
lemma Standard_add: "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> x + y \<in> Standard"
by (simp add: star_add_def)
lemma Standard_diff: "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> x - y \<in> Standard"
by (simp add: star_diff_def)
lemma Standard_minus: "x \<in> Standard \<Longrightarrow> - x \<in> Standard"
by (simp add: star_minus_def)
lemma Standard_mult: "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> x * y \<in> Standard"
by (simp add: star_mult_def)
lemma Standard_divide: "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> x / y \<in> Standard"
by (simp add: star_divide_def)
lemma Standard_inverse: "x \<in> Standard \<Longrightarrow> inverse x \<in> Standard"
by (simp add: star_inverse_def)
lemma Standard_abs: "x \<in> Standard \<Longrightarrow> abs x \<in> Standard"
by (simp add: star_abs_def)
lemma Standard_div: "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> x div y \<in> Standard"
by (simp add: star_div_def)
lemma Standard_mod: "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> x mod y \<in> Standard"
by (simp add: star_mod_def)
lemma Standard_power: "x \<in> Standard \<Longrightarrow> x ^ n \<in> Standard"
by (simp add: star_power_def)
lemmas Standard_simps [simp] =
Standard_zero Standard_one Standard_number_of
Standard_add Standard_diff Standard_minus
Standard_mult Standard_divide Standard_inverse
Standard_abs Standard_div Standard_mod
Standard_power
text {* @{term star_of} preserves class operations *}
lemma star_of_add: "star_of (x + y) = star_of x + star_of y"
by transfer (rule refl)
lemma star_of_diff: "star_of (x - y) = star_of x - star_of y"
by transfer (rule refl)
lemma star_of_minus: "star_of (-x) = - star_of x"
by transfer (rule refl)
lemma star_of_mult: "star_of (x * y) = star_of x * star_of y"
by transfer (rule refl)
lemma star_of_divide: "star_of (x / y) = star_of x / star_of y"
by transfer (rule refl)
lemma star_of_inverse: "star_of (inverse x) = inverse (star_of x)"
by transfer (rule refl)
lemma star_of_div: "star_of (x div y) = star_of x div star_of y"
by transfer (rule refl)
lemma star_of_mod: "star_of (x mod y) = star_of x mod star_of y"
by transfer (rule refl)
lemma star_of_power: "star_of (x ^ n) = star_of x ^ n"
by transfer (rule refl)
lemma star_of_abs: "star_of (abs x) = abs (star_of x)"
by transfer (rule refl)
text {* @{term star_of} preserves numerals *}
lemma star_of_zero: "star_of 0 = 0"
by transfer (rule refl)
lemma star_of_one: "star_of 1 = 1"
by transfer (rule refl)
lemma star_of_number_of: "star_of (number_of x) = number_of x"
by transfer (rule refl)
text {* @{term star_of} preserves orderings *}
lemma star_of_less: "(star_of x < star_of y) = (x < y)"
by transfer (rule refl)
lemma star_of_le: "(star_of x \<le> star_of y) = (x \<le> y)"
by transfer (rule refl)
lemma star_of_eq: "(star_of x = star_of y) = (x = y)"
by transfer (rule refl)
text{*As above, for 0*}
lemmas star_of_0_less = star_of_less [of 0, simplified star_of_zero]
lemmas star_of_0_le = star_of_le [of 0, simplified star_of_zero]
lemmas star_of_0_eq = star_of_eq [of 0, simplified star_of_zero]
lemmas star_of_less_0 = star_of_less [of _ 0, simplified star_of_zero]
lemmas star_of_le_0 = star_of_le [of _ 0, simplified star_of_zero]
lemmas star_of_eq_0 = star_of_eq [of _ 0, simplified star_of_zero]
text{*As above, for 1*}
lemmas star_of_1_less = star_of_less [of 1, simplified star_of_one]
lemmas star_of_1_le = star_of_le [of 1, simplified star_of_one]
lemmas star_of_1_eq = star_of_eq [of 1, simplified star_of_one]
lemmas star_of_less_1 = star_of_less [of _ 1, simplified star_of_one]
lemmas star_of_le_1 = star_of_le [of _ 1, simplified star_of_one]
lemmas star_of_eq_1 = star_of_eq [of _ 1, simplified star_of_one]
text{*As above, for numerals*}
lemmas star_of_number_less =
star_of_less [of "number_of w", standard, simplified star_of_number_of]
lemmas star_of_number_le =
star_of_le [of "number_of w", standard, simplified star_of_number_of]
lemmas star_of_number_eq =
star_of_eq [of "number_of w", standard, simplified star_of_number_of]
lemmas star_of_less_number =
star_of_less [of _ "number_of w", standard, simplified star_of_number_of]
lemmas star_of_le_number =
star_of_le [of _ "number_of w", standard, simplified star_of_number_of]
lemmas star_of_eq_number =
star_of_eq [of _ "number_of w", standard, simplified star_of_number_of]
lemmas star_of_simps [simp] =
star_of_add star_of_diff star_of_minus
star_of_mult star_of_divide star_of_inverse
star_of_div star_of_mod
star_of_power star_of_abs
star_of_zero star_of_one star_of_number_of
star_of_less star_of_le star_of_eq
star_of_0_less star_of_0_le star_of_0_eq
star_of_less_0 star_of_le_0 star_of_eq_0
star_of_1_less star_of_1_le star_of_1_eq
star_of_less_1 star_of_le_1 star_of_eq_1
star_of_number_less star_of_number_le star_of_number_eq
star_of_less_number star_of_le_number star_of_eq_number
subsection {* Ordering and lattice classes *}
instance star :: (order) order
apply (intro_classes)
apply (transfer, rule order_less_le)
apply (transfer, rule order_refl)
apply (transfer, erule (1) order_trans)
apply (transfer, erule (1) order_antisym)
done
instance star :: (lower_semilattice) lower_semilattice
star_inf_def [transfer_unfold]: "inf \<equiv> *f2* inf"
by default (transfer star_inf_def, auto)+
instance star :: (upper_semilattice) upper_semilattice
star_sup_def [transfer_unfold]: "sup \<equiv> *f2* sup"
by default (transfer star_sup_def, auto)+
instance star :: (lattice) lattice ..
instance star :: (distrib_lattice) distrib_lattice
by default (transfer, auto simp add: sup_inf_distrib1)
lemma Standard_inf [simp]:
"\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> inf x y \<in> Standard"
by (simp add: star_inf_def)
lemma Standard_sup [simp]:
"\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> sup x y \<in> Standard"
by (simp add: star_sup_def)
lemma star_of_inf [simp]: "star_of (inf x y) = inf (star_of x) (star_of y)"
by transfer (rule refl)
lemma star_of_sup [simp]: "star_of (sup x y) = sup (star_of x) (star_of y)"
by transfer (rule refl)
instance star :: (linorder) linorder
by (intro_classes, transfer, rule linorder_linear)
lemma star_max_def [transfer_unfold]: "max = *f2* max"
apply (rule ext, rule ext)
apply (unfold max_def, transfer, fold max_def)
apply (rule refl)
done
lemma star_min_def [transfer_unfold]: "min = *f2* min"
apply (rule ext, rule ext)
apply (unfold min_def, transfer, fold min_def)
apply (rule refl)
done
lemma Standard_max [simp]:
"\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> max x y \<in> Standard"
by (simp add: star_max_def)
lemma Standard_min [simp]:
"\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> min x y \<in> Standard"
by (simp add: star_min_def)
lemma star_of_max [simp]: "star_of (max x y) = max (star_of x) (star_of y)"
by transfer (rule refl)
lemma star_of_min [simp]: "star_of (min x y) = min (star_of x) (star_of y)"
by transfer (rule refl)
subsection {* Ordered group classes *}
instance star :: (semigroup_add) semigroup_add
by (intro_classes, transfer, rule add_assoc)
instance star :: (ab_semigroup_add) ab_semigroup_add
by (intro_classes, transfer, rule add_commute)
instance star :: (semigroup_mult) semigroup_mult
by (intro_classes, transfer, rule mult_assoc)
instance star :: (ab_semigroup_mult) ab_semigroup_mult
by (intro_classes, transfer, rule mult_commute)
instance star :: (comm_monoid_add) comm_monoid_add
by (intro_classes, transfer, rule comm_monoid_add_class.zero_plus.add_0)
instance star :: (monoid_mult) monoid_mult
apply (intro_classes)
apply (transfer, rule mult_1_left)
apply (transfer, rule mult_1_right)
done
instance star :: (comm_monoid_mult) comm_monoid_mult
by (intro_classes, transfer, rule mult_1)
instance star :: (cancel_semigroup_add) cancel_semigroup_add
apply (intro_classes)
apply (transfer, erule add_left_imp_eq)
apply (transfer, erule add_right_imp_eq)
done
instance star :: (cancel_ab_semigroup_add) cancel_ab_semigroup_add
by (intro_classes, transfer, rule add_imp_eq)
instance star :: (ab_group_add) ab_group_add
apply (intro_classes)
apply (transfer, rule left_minus)
apply (transfer, rule diff_minus)
done
instance star :: (pordered_ab_semigroup_add) pordered_ab_semigroup_add
by (intro_classes, transfer, rule add_left_mono)
instance star :: (pordered_cancel_ab_semigroup_add) pordered_cancel_ab_semigroup_add ..
instance star :: (pordered_ab_semigroup_add_imp_le) pordered_ab_semigroup_add_imp_le
by (intro_classes, transfer, rule add_le_imp_le_left)
instance star :: (pordered_ab_group_add) pordered_ab_group_add ..
instance star :: (ordered_cancel_ab_semigroup_add) ordered_cancel_ab_semigroup_add ..
instance star :: (lordered_ab_group_meet) lordered_ab_group_meet ..
instance star :: (lordered_ab_group_meet) lordered_ab_group_meet ..
instance star :: (lordered_ab_group) lordered_ab_group ..
instance star :: (lordered_ab_group_abs) lordered_ab_group_abs
by (intro_classes, transfer, rule abs_lattice)
subsection {* Ring and field classes *}
instance star :: (semiring) semiring
apply (intro_classes)
apply (transfer, rule left_distrib)
apply (transfer, rule right_distrib)
done
instance star :: (semiring_0) semiring_0
by intro_classes (transfer, simp)+
instance star :: (semiring_0_cancel) semiring_0_cancel ..
instance star :: (comm_semiring) comm_semiring
by (intro_classes, transfer, rule distrib)
instance star :: (comm_semiring_0) comm_semiring_0 ..
instance star :: (comm_semiring_0_cancel) comm_semiring_0_cancel ..
instance star :: (zero_neq_one) zero_neq_one
by (intro_classes, transfer, rule zero_neq_one)
instance star :: (semiring_1) semiring_1 ..
instance star :: (comm_semiring_1) comm_semiring_1 ..
instance star :: (no_zero_divisors) no_zero_divisors
by (intro_classes, transfer, rule no_zero_divisors)
instance star :: (semiring_1_cancel) semiring_1_cancel ..
instance star :: (comm_semiring_1_cancel) comm_semiring_1_cancel ..
instance star :: (ring) ring ..
instance star :: (comm_ring) comm_ring ..
instance star :: (ring_1) ring_1 ..
instance star :: (comm_ring_1) comm_ring_1 ..
instance star :: (idom) idom ..
instance star :: (division_ring) division_ring
apply (intro_classes)
apply (transfer, erule left_inverse)
apply (transfer, erule right_inverse)
done
instance star :: (field) field
apply (intro_classes)
apply (transfer, erule left_inverse)
apply (transfer, rule divide_inverse)
done
instance star :: (division_by_zero) division_by_zero
by (intro_classes, transfer, rule inverse_zero)
instance star :: (pordered_semiring) pordered_semiring
apply (intro_classes)
apply (transfer, erule (1) mult_left_mono)
apply (transfer, erule (1) mult_right_mono)
done
instance star :: (pordered_cancel_semiring) pordered_cancel_semiring ..
instance star :: (ordered_semiring_strict) ordered_semiring_strict
apply (intro_classes)
apply (transfer, erule (1) mult_strict_left_mono)
apply (transfer, erule (1) mult_strict_right_mono)
done
instance star :: (pordered_comm_semiring) pordered_comm_semiring
by (intro_classes, transfer, rule mult_mono1_class.times_zero_less_eq_less.mult_mono)
instance star :: (pordered_cancel_comm_semiring) pordered_cancel_comm_semiring ..
instance star :: (ordered_comm_semiring_strict) ordered_comm_semiring_strict
by (intro_classes, transfer, rule ordered_comm_semiring_strict_class.plus_less_eq_less_zero_times.mult_strict_mono)
instance star :: (pordered_ring) pordered_ring ..
instance star :: (lordered_ring) lordered_ring ..
instance star :: (abs_if) abs_if
by (intro_classes, transfer, rule abs_if)
instance star :: (ordered_ring_strict) ordered_ring_strict ..
instance star :: (pordered_comm_ring) pordered_comm_ring ..
instance star :: (ordered_semidom) ordered_semidom
by (intro_classes, transfer, rule zero_less_one)
instance star :: (ordered_idom) ordered_idom ..
instance star :: (ordered_field) ordered_field ..
subsection {* Power classes *}
text {*
Proving the class axiom @{thm [source] power_Suc} for type
@{typ "'a star"} is a little tricky, because it quantifies
over values of type @{typ nat}. The transfer principle does
not handle quantification over non-star types in general,
but we can work around this by fixing an arbitrary @{typ nat}
value, and then applying the transfer principle.
*}
instance star :: (recpower) recpower
proof
show "\<And>a::'a star. a ^ 0 = 1"
by transfer (rule power_0)
next
fix n show "\<And>a::'a star. a ^ Suc n = a * a ^ n"
by transfer (rule power_Suc)
qed
subsection {* Number classes *}
lemma star_of_nat_def [transfer_unfold]: "of_nat n = star_of (of_nat n)"
by (induct_tac n, simp_all)
lemma Standard_of_nat [simp]: "of_nat n \<in> Standard"
by (simp add: star_of_nat_def)
lemma star_of_of_nat [simp]: "star_of (of_nat n) = of_nat n"
by transfer (rule refl)
lemma star_of_int_def [transfer_unfold]: "of_int z = star_of (of_int z)"
by (rule_tac z=z in int_diff_cases, simp)
lemma Standard_of_int [simp]: "of_int z \<in> Standard"
by (simp add: star_of_int_def)
lemma star_of_of_int [simp]: "star_of (of_int z) = of_int z"
by transfer (rule refl)
instance star :: (number_ring) number_ring
by (intro_classes, simp only: star_number_def star_of_int_def number_of_eq)
subsection {* Finite class *}
lemma starset_finite: "finite A \<Longrightarrow> *s* A = star_of ` A"
by (erule finite_induct, simp_all)
instance star :: (finite) finite
apply (intro_classes)
apply (subst starset_UNIV [symmetric])
apply (subst starset_finite [OF finite])
apply (rule finite_imageI [OF finite])
done
end