(* Title : HOL/Hyperreal/StarClasses.thy ID : $Id: StarClasses.thy,v 1.3 2005/09/15 21:46:22 huffman Exp $ Author : Brian Huffman *) header {* Class Instances *} theory StarClasses imports StarDef begin subsection {* Syntactic classes *} instance star :: (ord) ord .. instance star :: (zero) zero .. instance star :: (one) one .. instance star :: (plus) plus .. instance star :: (times) times .. instance star :: (minus) minus .. instance star :: (inverse) inverse .. instance star :: (number) number .. instance star :: ("Divides.div") "Divides.div" .. instance star :: (power) power .. defs (overloaded) star_zero_def: "0 ≡ star_of 0" star_one_def: "1 ≡ star_of 1" star_number_def: "number_of b ≡ star_of (number_of b)" star_add_def: "(op +) ≡ *f2* (op +)" star_diff_def: "(op -) ≡ *f2* (op -)" star_minus_def: "uminus ≡ *f* uminus" star_mult_def: "(op *) ≡ *f2* (op *)" star_divide_def: "(op /) ≡ *f2* (op /)" star_inverse_def: "inverse ≡ *f* inverse" star_le_def: "(op ≤) ≡ *p2* (op ≤)" star_less_def: "(op <) ≡ *p2* (op <)" star_abs_def: "abs ≡ *f* abs" star_div_def: "(op div) ≡ *f2* (op div)" star_mod_def: "(op mod) ≡ *f2* (op mod)" star_power_def: "(op ^) ≡ λx n. ( *f* (λx. x ^ n)) x" 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 {* @{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 ≤ star_of y) = (x ≤ 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 classes *} instance star :: (order) order apply (intro_classes) apply (transfer, rule order_refl) apply (transfer, erule (1) order_trans) apply (transfer, erule (1) order_antisym) apply (transfer, rule order_less_le) done instance star :: (linorder) linorder by (intro_classes, transfer, rule linorder_linear) subsection {* Lattice ordering classes *} text {* Some extra trouble is necessary because the class axioms for @{term meet} and @{term join} use quantification over function spaces. *} lemma ex_star_fun: "∃f::('a => 'b) star. P (λx. f ∗ x) ==> ∃f::'a star => 'b star. P f" by (erule exE, erule exI) lemma ex_star_fun2: "∃f::('a => 'b => 'c) star. P (λx y. f ∗ x ∗ y) ==> ∃f::'a star => 'b star => 'c star. P f" by (erule exE, erule exI) instance star :: (join_semilorder) join_semilorder apply (intro_classes) apply (rule ex_star_fun2) apply (transfer is_join_def) apply (rule join_exists) done instance star :: (meet_semilorder) meet_semilorder apply (intro_classes) apply (rule ex_star_fun2) apply (transfer is_meet_def) apply (rule meet_exists) done instance star :: (lorder) lorder .. lemma star_join_def [transfer_unfold]: "join ≡ *f2* join" apply (rule is_join_unique [OF is_join_join, THEN eq_reflection]) apply (transfer is_join_def, rule is_join_join) done lemma star_meet_def [transfer_unfold]: "meet ≡ *f2* meet" apply (rule is_meet_unique [OF is_meet_meet, THEN eq_reflection]) apply (transfer is_meet_def, rule is_meet_meet) done 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.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 .. 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 :: (axclass_0_neq_1) axclass_0_neq_1 by (intro_classes, transfer, rule zero_neq_one) instance star :: (semiring_1) semiring_1 .. instance star :: (comm_semiring_1) comm_semiring_1 .. instance star :: (axclass_no_zero_divisors) axclass_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 :: (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 pordered_comm_semiring_class.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.mult_strict_mono) instance star :: (pordered_ring) pordered_ring .. instance star :: (lordered_ring) lordered_ring .. instance star :: (axclass_abs_if) axclass_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 "!!a::'a star. a ^ 0 = 1" by transfer (rule power_0) next fix n show "!!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 (rule eq_reflection, induct_tac n, simp_all) lemma star_of_of_nat [simp]: "star_of (of_nat n) = of_nat n" by transfer (rule refl) lemma int_diff_cases: assumes prem: "!!m n. z = int m - int n ==> P" shows "P" apply (rule_tac z=z in int_cases) apply (rule_tac m=n and n=0 in prem, simp) apply (rule_tac m=0 and n="Suc n" in prem, simp) done -- "Belongs in Integ/IntDef.thy" lemma star_of_int_def [transfer_unfold]: "of_int z ≡ star_of (of_int z)" by (rule eq_reflection, rule_tac z=z in int_diff_cases, simp) 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 ==> *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
lemmas star_class_defs:
0 == star_of (0::'a)
1 == star_of (1::'a)
number_of b == star_of (number_of b)
op + == *f2* op +
op - == *f2* op -
uminus == *f* uminus
op * == *f2* op *
op / == *f2* op /
inverse == *f* inverse
op ≤ == *p2* op ≤
op < == *p2* op <
abs == *f* abs
op div == *f2* op div
op mod == *f2* op mod
op ^ == %x n. (*f* (%x. x ^ n)) x
lemmas star_class_defs:
0 == star_of (0::'a)
1 == star_of (1::'a)
number_of b == star_of (number_of b)
op + == *f2* op +
op - == *f2* op -
uminus == *f* uminus
op * == *f2* op *
op / == *f2* op /
inverse == *f* inverse
op ≤ == *p2* op ≤
op < == *p2* op <
abs == *f* abs
op div == *f2* op div
op mod == *f2* op mod
op ^ == %x n. (*f* (%x. x ^ n)) x
lemma star_of_add:
star_of (x + y) = star_of x + star_of y
lemma star_of_diff:
star_of (x - y) = star_of x - star_of y
lemma star_of_minus:
star_of (- x) = - star_of x
lemma star_of_mult:
star_of (x * y) = star_of x * star_of y
lemma star_of_divide:
star_of (x / y) = star_of x / star_of y
lemma star_of_inverse:
star_of (inverse x) = inverse (star_of x)
lemma star_of_div:
star_of (x div y) = star_of x div star_of y
lemma star_of_mod:
star_of (x mod y) = star_of x mod star_of y
lemma star_of_power:
star_of (x ^ n) = star_of x ^ n
lemma star_of_abs:
star_of ¦x¦ = ¦star_of x¦
lemma star_of_zero:
star_of (0::'a) = 0
lemma star_of_one:
star_of (1::'a) = 1
lemma star_of_number_of:
star_of (number_of x) = number_of x
lemma star_of_less:
(star_of x < star_of y) = (x < y)
lemma star_of_le:
(star_of x ≤ star_of y) = (x ≤ y)
lemma star_of_eq:
(star_of x = star_of y) = (x = y)
lemmas star_of_0_less:
(0 < star_of y) = ((0::'b) < y)
lemmas star_of_0_less:
(0 < star_of y) = ((0::'b) < y)
lemmas star_of_0_le:
(0 ≤ star_of y) = ((0::'b) ≤ y)
lemmas star_of_0_le:
(0 ≤ star_of y) = ((0::'b) ≤ y)
lemmas star_of_0_eq:
(0 = star_of y) = ((0::'b) = y)
lemmas star_of_0_eq:
(0 = star_of y) = ((0::'b) = y)
lemmas star_of_less_0:
(star_of x < 0) = (x < (0::'b))
lemmas star_of_less_0:
(star_of x < 0) = (x < (0::'b))
lemmas star_of_le_0:
(star_of x ≤ 0) = (x ≤ (0::'b))
lemmas star_of_le_0:
(star_of x ≤ 0) = (x ≤ (0::'b))
lemmas star_of_eq_0:
(star_of x = 0) = (x = (0::'b))
lemmas star_of_eq_0:
(star_of x = 0) = (x = (0::'b))
lemmas star_of_1_less:
(1 < star_of y) = ((1::'b) < y)
lemmas star_of_1_less:
(1 < star_of y) = ((1::'b) < y)
lemmas star_of_1_le:
(1 ≤ star_of y) = ((1::'b) ≤ y)
lemmas star_of_1_le:
(1 ≤ star_of y) = ((1::'b) ≤ y)
lemmas star_of_1_eq:
(1 = star_of y) = ((1::'b) = y)
lemmas star_of_1_eq:
(1 = star_of y) = ((1::'b) = y)
lemmas star_of_less_1:
(star_of x < 1) = (x < (1::'b))
lemmas star_of_less_1:
(star_of x < 1) = (x < (1::'b))
lemmas star_of_le_1:
(star_of x ≤ 1) = (x ≤ (1::'b))
lemmas star_of_le_1:
(star_of x ≤ 1) = (x ≤ (1::'b))
lemmas star_of_eq_1:
(star_of x = 1) = (x = (1::'b))
lemmas star_of_eq_1:
(star_of x = 1) = (x = (1::'b))
lemmas star_of_number_less:
(number_of w < star_of y) = (number_of w < y)
lemmas star_of_number_less:
(number_of w < star_of y) = (number_of w < y)
lemmas star_of_number_le:
(number_of w ≤ star_of y) = (number_of w ≤ y)
lemmas star_of_number_le:
(number_of w ≤ star_of y) = (number_of w ≤ y)
lemmas star_of_number_eq:
(number_of w = star_of y) = (number_of w = y)
lemmas star_of_number_eq:
(number_of w = star_of y) = (number_of w = y)
lemmas star_of_less_number:
(star_of x < number_of w) = (x < number_of w)
lemmas star_of_less_number:
(star_of x < number_of w) = (x < number_of w)
lemmas star_of_le_number:
(star_of x ≤ number_of w) = (x ≤ number_of w)
lemmas star_of_le_number:
(star_of x ≤ number_of w) = (x ≤ number_of w)
lemmas star_of_eq_number:
(star_of x = number_of w) = (x = number_of w)
lemmas star_of_eq_number:
(star_of x = number_of w) = (x = number_of w)
lemmas star_of_simps:
star_of (x + y) = star_of x + star_of y
star_of (x - y) = star_of x - star_of y
star_of (- x) = - star_of x
star_of (x * y) = star_of x * star_of y
star_of (x / y) = star_of x / star_of y
star_of (inverse x) = inverse (star_of x)
star_of (x div y) = star_of x div star_of y
star_of (x mod y) = star_of x mod star_of y
star_of (x ^ n) = star_of x ^ n
star_of ¦x¦ = ¦star_of x¦
star_of (0::'a) = 0
star_of (1::'a) = 1
star_of (number_of x) = number_of x
(star_of x < star_of y) = (x < y)
(star_of x ≤ star_of y) = (x ≤ y)
(star_of x = star_of y) = (x = y)
(0 < star_of y) = ((0::'b) < y)
(0 ≤ star_of y) = ((0::'b) ≤ y)
(0 = star_of y) = ((0::'b) = y)
(star_of x < 0) = (x < (0::'b))
(star_of x ≤ 0) = (x ≤ (0::'b))
(star_of x = 0) = (x = (0::'b))
(1 < star_of y) = ((1::'b) < y)
(1 ≤ star_of y) = ((1::'b) ≤ y)
(1 = star_of y) = ((1::'b) = y)
(star_of x < 1) = (x < (1::'b))
(star_of x ≤ 1) = (x ≤ (1::'b))
(star_of x = 1) = (x = (1::'b))
(number_of w < star_of y) = (number_of w < y)
(number_of w ≤ star_of y) = (number_of w ≤ y)
(number_of w = star_of y) = (number_of w = y)
(star_of x < number_of w) = (x < number_of w)
(star_of x ≤ number_of w) = (x ≤ number_of w)
(star_of x = number_of w) = (x = number_of w)
lemmas star_of_simps:
star_of (x + y) = star_of x + star_of y
star_of (x - y) = star_of x - star_of y
star_of (- x) = - star_of x
star_of (x * y) = star_of x * star_of y
star_of (x / y) = star_of x / star_of y
star_of (inverse x) = inverse (star_of x)
star_of (x div y) = star_of x div star_of y
star_of (x mod y) = star_of x mod star_of y
star_of (x ^ n) = star_of x ^ n
star_of ¦x¦ = ¦star_of x¦
star_of (0::'a) = 0
star_of (1::'a) = 1
star_of (number_of x) = number_of x
(star_of x < star_of y) = (x < y)
(star_of x ≤ star_of y) = (x ≤ y)
(star_of x = star_of y) = (x = y)
(0 < star_of y) = ((0::'b) < y)
(0 ≤ star_of y) = ((0::'b) ≤ y)
(0 = star_of y) = ((0::'b) = y)
(star_of x < 0) = (x < (0::'b))
(star_of x ≤ 0) = (x ≤ (0::'b))
(star_of x = 0) = (x = (0::'b))
(1 < star_of y) = ((1::'b) < y)
(1 ≤ star_of y) = ((1::'b) ≤ y)
(1 = star_of y) = ((1::'b) = y)
(star_of x < 1) = (x < (1::'b))
(star_of x ≤ 1) = (x ≤ (1::'b))
(star_of x = 1) = (x = (1::'b))
(number_of w < star_of y) = (number_of w < y)
(number_of w ≤ star_of y) = (number_of w ≤ y)
(number_of w = star_of y) = (number_of w = y)
(star_of x < number_of w) = (x < number_of w)
(star_of x ≤ number_of w) = (x ≤ number_of w)
(star_of x = number_of w) = (x = number_of w)
lemma ex_star_fun:
∃f. P (Ifun f) ==> ∃f. P f
lemma ex_star_fun2:
∃f. P (%x. Ifun (f ∗ x)) ==> ∃f. P f
lemma star_join_def:
join == *f2* join
lemma star_meet_def:
meet == *f2* meet
lemma star_of_nat_def:
of_nat n == star_of (of_nat n)
lemma star_of_of_nat:
star_of (of_nat n) = of_nat n
lemma int_diff_cases:
(!!m n. z = int m - int n ==> P) ==> P
lemma star_of_int_def:
of_int z == star_of (of_int z)
lemma star_of_of_int:
star_of (of_int z) = of_int z
lemma starset_finite:
finite A ==> *s* A = star_of ` A