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(* Title : HOL/Hyperreal/StarClasses.thy
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ID : $Id$
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Author : Brian Huffman
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*)
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header {* Class Instances *}
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theory StarClasses
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imports Transfer
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begin
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subsection "HOL.thy"
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instance star :: (order) order
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apply (intro_classes)
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apply (transfer, rule order_refl)
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apply (transfer, erule (1) order_trans)
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apply (transfer, erule (1) order_antisym)
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apply (transfer, rule order_less_le)
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done
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instance star :: (linorder) linorder
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by (intro_classes, transfer, rule linorder_linear)
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subsection "LOrder.thy"
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text {*
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Some extra trouble is necessary because the class axioms
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for @{term meet} and @{term join} use quantification over
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function spaces.
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*}
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lemma ex_star_fun:
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"\<exists>f::('a \<Rightarrow> 'b) star. P (Ifun f)
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\<Longrightarrow> \<exists>f::'a star \<Rightarrow> 'b star. P f"
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by (erule exE, erule exI)
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lemma ex_star_fun2:
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"\<exists>f::('a \<Rightarrow> 'b \<Rightarrow> 'c) star. P (Ifun2 f)
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\<Longrightarrow> \<exists>f::'a star \<Rightarrow> 'b star \<Rightarrow> 'c star. P f"
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by (erule exE, erule exI)
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instance star :: (join_semilorder) join_semilorder
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apply (intro_classes)
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apply (rule ex_star_fun2)
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apply (transfer is_join_def)
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apply (rule join_exists)
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done
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instance star :: (meet_semilorder) meet_semilorder
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apply (intro_classes)
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apply (rule ex_star_fun2)
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apply (transfer is_meet_def)
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apply (rule meet_exists)
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done
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instance star :: (lorder) lorder ..
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lemma star_join_def: "join \<equiv> Ifun2_of join"
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apply (rule is_join_unique[OF is_join_join, THEN eq_reflection])
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apply (transfer is_join_def, rule is_join_join)
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done
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lemma star_meet_def: "meet \<equiv> Ifun2_of meet"
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apply (rule is_meet_unique[OF is_meet_meet, THEN eq_reflection])
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apply (transfer is_meet_def, rule is_meet_meet)
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done
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subsection "OrderedGroup.thy"
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instance star :: (semigroup_add) semigroup_add
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by (intro_classes, transfer, rule add_assoc)
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instance star :: (ab_semigroup_add) ab_semigroup_add
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by (intro_classes, transfer, rule add_commute)
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instance star :: (semigroup_mult) semigroup_mult
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by (intro_classes, transfer, rule mult_assoc)
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instance star :: (ab_semigroup_mult) ab_semigroup_mult
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by (intro_classes, transfer, rule mult_commute)
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instance star :: (comm_monoid_add) comm_monoid_add
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by (intro_classes, transfer, rule comm_monoid_add_class.add_0)
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instance star :: (monoid_mult) monoid_mult
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apply (intro_classes)
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apply (transfer, rule mult_1_left)
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apply (transfer, rule mult_1_right)
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done
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instance star :: (comm_monoid_mult) comm_monoid_mult
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by (intro_classes, transfer, rule mult_1)
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instance star :: (cancel_semigroup_add) cancel_semigroup_add
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apply (intro_classes)
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apply (transfer, erule add_left_imp_eq)
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apply (transfer, erule add_right_imp_eq)
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done
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instance star :: (cancel_ab_semigroup_add) cancel_ab_semigroup_add
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by (intro_classes, transfer, rule add_imp_eq)
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instance star :: (ab_group_add) ab_group_add
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apply (intro_classes)
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apply (transfer, rule left_minus)
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apply (transfer, rule diff_minus)
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done
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instance star :: (pordered_ab_semigroup_add) pordered_ab_semigroup_add
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by (intro_classes, transfer, rule add_left_mono)
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instance star :: (pordered_cancel_ab_semigroup_add) pordered_cancel_ab_semigroup_add ..
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instance star :: (pordered_ab_semigroup_add_imp_le) pordered_ab_semigroup_add_imp_le
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by (intro_classes, transfer, rule add_le_imp_le_left)
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instance star :: (pordered_ab_group_add) pordered_ab_group_add ..
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instance star :: (ordered_cancel_ab_semigroup_add) ordered_cancel_ab_semigroup_add ..
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instance star :: (lordered_ab_group_meet) lordered_ab_group_meet ..
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instance star :: (lordered_ab_group_meet) lordered_ab_group_meet ..
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instance star :: (lordered_ab_group) lordered_ab_group ..
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instance star :: (lordered_ab_group_abs) lordered_ab_group_abs
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apply (intro_classes)
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apply (transfer star_join_def, rule abs_lattice)
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done
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text "Ring-and-Field.thy"
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instance star :: (semiring) semiring
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apply (intro_classes)
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apply (transfer, rule left_distrib)
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apply (transfer, rule right_distrib)
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done
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instance star :: (semiring_0) semiring_0 ..
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instance star :: (semiring_0_cancel) semiring_0_cancel ..
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instance star :: (comm_semiring) comm_semiring
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by (intro_classes, transfer, rule distrib)
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instance star :: (comm_semiring_0) comm_semiring_0 ..
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instance star :: (comm_semiring_0_cancel) comm_semiring_0_cancel ..
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instance star :: (axclass_0_neq_1) axclass_0_neq_1
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by (intro_classes, transfer, rule zero_neq_one)
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instance star :: (semiring_1) semiring_1 ..
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instance star :: (comm_semiring_1) comm_semiring_1 ..
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instance star :: (axclass_no_zero_divisors) axclass_no_zero_divisors
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by (intro_classes, transfer, rule no_zero_divisors)
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instance star :: (semiring_1_cancel) semiring_1_cancel ..
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instance star :: (comm_semiring_1_cancel) comm_semiring_1_cancel ..
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instance star :: (ring) ring ..
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instance star :: (comm_ring) comm_ring ..
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instance star :: (ring_1) ring_1 ..
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instance star :: (comm_ring_1) comm_ring_1 ..
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instance star :: (idom) idom ..
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instance star :: (field) field
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apply (intro_classes)
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apply (transfer, erule left_inverse)
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apply (transfer, rule divide_inverse)
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done
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instance star :: (division_by_zero) division_by_zero
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by (intro_classes, transfer, rule inverse_zero)
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instance star :: (pordered_semiring) pordered_semiring
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apply (intro_classes)
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apply (transfer, erule (1) mult_left_mono)
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apply (transfer, erule (1) mult_right_mono)
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done
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instance star :: (pordered_cancel_semiring) pordered_cancel_semiring ..
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instance star :: (ordered_semiring_strict) ordered_semiring_strict
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apply (intro_classes)
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apply (transfer, erule (1) mult_strict_left_mono)
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apply (transfer, erule (1) mult_strict_right_mono)
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done
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instance star :: (pordered_comm_semiring) pordered_comm_semiring
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by (intro_classes, transfer, rule pordered_comm_semiring_class.mult_mono)
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instance star :: (pordered_cancel_comm_semiring) pordered_cancel_comm_semiring ..
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instance star :: (ordered_comm_semiring_strict) ordered_comm_semiring_strict
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by (intro_classes, transfer, rule ordered_comm_semiring_strict_class.mult_strict_mono)
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instance star :: (pordered_ring) pordered_ring ..
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instance star :: (lordered_ring) lordered_ring ..
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instance star :: (axclass_abs_if) axclass_abs_if
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by (intro_classes, transfer, rule abs_if)
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instance star :: (ordered_ring_strict) ordered_ring_strict ..
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instance star :: (pordered_comm_ring) pordered_comm_ring ..
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instance star :: (ordered_semidom) ordered_semidom
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by (intro_classes, transfer, rule zero_less_one)
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instance star :: (ordered_idom) ordered_idom ..
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instance star :: (ordered_field) ordered_field ..
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subsection "Power.thy"
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text {*
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Proving the class axiom @{thm [source] power_Suc} for type
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@{typ "'a star"} is a little tricky, because it quantifies
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over values of type @{typ nat}. The transfer principle does
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not handle quantification over non-star types in general,
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but we can work around this by fixing an arbitrary @{typ nat}
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value, and then applying the transfer principle.
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*}
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instance star :: (recpower) recpower
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proof
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show "\<And>a::'a star. a ^ 0 = 1"
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by transfer (rule power_0)
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next
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fix n show "\<And>a::'a star. a ^ Suc n = a * a ^ n"
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by transfer (rule power_Suc)
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qed
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subsection "Integ/Number.thy"
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lemma star_of_nat_def: "of_nat n \<equiv> star_of (of_nat n)"
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by (rule eq_reflection, induct_tac n, simp_all)
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lemma int_diff_cases:
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assumes prem: "\<And>m n. z = int m - int n \<Longrightarrow> P" shows "P"
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apply (rule_tac z=z in int_cases)
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apply (rule_tac m=n and n=0 in prem, simp)
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apply (rule_tac m=0 and n="Suc n" in prem, simp)
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done -- "Belongs in Integ/IntDef.thy"
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lemma star_of_int_def: "of_int z \<equiv> star_of (of_int z)"
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apply (rule eq_reflection)
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apply (rule_tac z=z in int_diff_cases)
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apply (simp add: star_of_nat_def)
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done
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instance star :: (number_ring) number_ring
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by (intro_classes, simp only: star_number_def star_of_int_def number_of_eq)
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lemma star_of_of_nat [simp]: "star_of (of_nat n) = of_nat n"
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by (simp add: star_of_nat_def)
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lemma star_of_of_int [simp]: "star_of (of_int z) = of_int z"
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by (simp add: star_of_int_def)
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end
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