--- a/NEWS Mon Feb 08 11:01:47 2010 +0100
+++ b/NEWS Mon Feb 08 11:13:30 2010 +0100
@@ -6,12 +6,63 @@
*** Pure ***
-* Code generator: details of internal data cache have no impact on the
-user space functionality any longer.
+* Code generator: details of internal data cache have no impact on
+the user space functionality any longer.
*** HOL ***
+* more consistent naming of type classes involving orderings (and lattices):
+
+ lower_semilattice ~> semilattice_inf
+ upper_semilattice ~> semilattice_sup
+
+ dense_linear_order ~> dense_linorder
+
+ pordered_ab_group_add ~> ordered_ab_group_add
+ pordered_ab_group_add_abs ~> ordered_ab_group_add_abs
+ pordered_ab_semigroup_add ~> ordered_ab_semigroup_add
+ pordered_ab_semigroup_add_imp_le ~> ordered_ab_semigroup_add_imp_le
+ pordered_cancel_ab_semigroup_add ~> ordered_cancel_ab_semigroup_add
+ pordered_cancel_comm_semiring ~> ordered_cancel_comm_semiring
+ pordered_cancel_semiring ~> ordered_cancel_semiring
+ pordered_comm_monoid_add ~> ordered_comm_monoid_add
+ pordered_comm_ring ~> ordered_comm_ring
+ pordered_comm_semiring ~> ordered_comm_semiring
+ pordered_ring ~> ordered_ring
+ pordered_ring_abs ~> ordered_ring_abs
+ pordered_semiring ~> ordered_semiring
+
+ lordered_ab_group_add ~> lattice_ab_group_add
+ lordered_ab_group_add_abs ~> lattice_ab_group_add_abs
+ lordered_ab_group_add_meet ~> semilattice_inf_ab_group_add
+ lordered_ab_group_add_join ~> semilattice_sup_ab_group_add
+ lordered_ring ~> lattice_ring
+
+ ordered_ab_group_add ~> linordered_ab_group_add
+ ordered_ab_semigroup_add ~> linordered_ab_semigroup_add
+ ordered_cancel_ab_semigroup_add ~> linordered_cancel_ab_semigroup_add
+ ordered_comm_semiring_strict ~> linordered_comm_semiring_strict
+ ordered_field ~> linordered_field
+ ordered_field_no_lb ~> linordered_field_no_lb
+ ordered_field_no_ub ~> linordered_field_no_ub
+ ordered_field_dense_linear_order ~> dense_linordered_field
+ ordered_idom ~> linordered_idom
+ ordered_ring ~> linordered_ring
+ ordered_ring_le_cancel_factor ~> linordered_ring_le_cancel_factor
+ ordered_ring_less_cancel_factor ~> linordered_ring_less_cancel_factor
+ ordered_ring_strict ~> linordered_ring_strict
+ ordered_semidom ~> linordered_semidom
+ ordered_semiring ~> linordered_semiring
+ ordered_semiring_1 ~> linordered_semiring_1
+ ordered_semiring_1_strict ~> linordered_semiring_1_strict
+ ordered_semiring_strict ~> linordered_semiring_strict
+
+INCOMPATIBILITY.
+
+* Index syntax for structures must be imported explicitly from library
+theory Structure_Syntax. INCOMPATIBILITY.
+
* New theory Algebras contains generic algebraic structures and
generic algebraic operations. INCOMPATIBILTY: constants zero, one,
plus, minus, uminus, times, inverse, divide, abs, sgn, less_eq and
--- a/src/HOL/Algebras.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Algebras.thy Mon Feb 08 11:13:30 2010 +0100
@@ -10,15 +10,30 @@
subsection {* Generic algebraic structures *}
+text {*
+ These locales provide basic structures for interpretation into
+ bigger structures; extensions require careful thinking, otherwise
+ undesired effects may occur due to interpretation.
+*}
+
+ML {*
+structure Ac_Simps = Named_Thms(
+ val name = "ac_simps"
+ val description = "associativity and commutativity simplification rules"
+)
+*}
+
+setup Ac_Simps.setup
+
locale semigroup =
fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "*" 70)
- assumes assoc: "a * b * c = a * (b * c)"
+ assumes assoc [ac_simps]: "a * b * c = a * (b * c)"
locale abel_semigroup = semigroup +
- assumes commute: "a * b = b * a"
+ assumes commute [ac_simps]: "a * b = b * a"
begin
-lemma left_commute:
+lemma left_commute [ac_simps]:
"b * (a * c) = a * (b * c)"
proof -
have "(b * a) * c = (a * b) * c"
@@ -39,25 +54,6 @@
end
-locale lattice = inf!: abel_semigroup inf + sup!: abel_semigroup sup
- for inf (infixl "\<sqinter>" 70) and sup (infixl "\<squnion>" 70) +
- assumes sup_inf_absorb: "a \<squnion> (a \<sqinter> b) = a"
- and inf_sup_absorb: "a \<sqinter> (a \<squnion> b) = a"
-
-sublocale lattice < inf!: semilattice inf
-proof
- fix a
- have "a \<sqinter> (a \<squnion> (a \<sqinter> a)) = a" by (fact inf_sup_absorb)
- then show "a \<sqinter> a = a" by (simp add: sup_inf_absorb)
-qed
-
-sublocale lattice < sup!: semilattice sup
-proof
- fix a
- have "a \<squnion> (a \<sqinter> (a \<squnion> a)) = a" by (fact sup_inf_absorb)
- then show "a \<squnion> a = a" by (simp add: inf_sup_absorb)
-qed
-
subsection {* Generic algebraic operations *}
@@ -158,9 +154,4 @@
end
-syntax
- "_index1" :: index ("\<^sub>1")
-translations
- (index) "\<^sub>1" => (index) "\<^bsub>\<struct>\<^esub>"
-
end
\ No newline at end of file
--- a/src/HOL/Archimedean_Field.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Archimedean_Field.thy Mon Feb 08 11:13:30 2010 +0100
@@ -12,7 +12,7 @@
text {* Archimedean fields have no infinite elements. *}
-class archimedean_field = ordered_field + number_ring +
+class archimedean_field = linordered_field + number_ring +
assumes ex_le_of_int: "\<exists>z. x \<le> of_int z"
lemma ex_less_of_int:
--- a/src/HOL/Code_Numeral.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Code_Numeral.thy Mon Feb 08 11:13:30 2010 +0100
@@ -144,7 +144,7 @@
subsection {* Basic arithmetic *}
-instantiation code_numeral :: "{minus, ordered_semidom, semiring_div, linorder}"
+instantiation code_numeral :: "{minus, linordered_semidom, semiring_div, linorder}"
begin
definition [simp, code del]:
--- a/src/HOL/Decision_Procs/Approximation.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Decision_Procs/Approximation.thy Mon Feb 08 11:13:30 2010 +0100
@@ -1431,7 +1431,7 @@
moreover have "0 \<le> exp t / real (fact (get_even n)) * (real x) ^ (get_even n)"
by (auto intro!: mult_nonneg_nonneg divide_nonneg_pos simp add: get_even zero_le_even_power exp_gt_zero)
ultimately show ?thesis
- using get_odd exp_gt_zero by (auto intro!: pordered_cancel_semiring_class.mult_nonneg_nonneg)
+ using get_odd exp_gt_zero by (auto intro!: mult_nonneg_nonneg)
qed
finally have "real (lb_exp_horner prec (get_even n) 1 1 x) \<le> exp (real x)" .
} moreover
@@ -1451,7 +1451,7 @@
moreover have "exp t / real (fact (get_odd n)) * (real x) ^ (get_odd n) \<le> 0"
by (auto intro!: mult_nonneg_nonpos divide_nonpos_pos simp add: x_less_zero exp_gt_zero)
ultimately have "exp (real x) \<le> (\<Sum>j = 0..<get_odd n. 1 / real (fact j) * real x ^ j)"
- using get_odd exp_gt_zero by (auto intro!: pordered_cancel_semiring_class.mult_nonneg_nonneg)
+ using get_odd exp_gt_zero by (auto intro!: mult_nonneg_nonneg)
also have "\<dots> \<le> real (ub_exp_horner prec (get_odd n) 1 1 x)"
using bounds(2) by auto
finally have "exp (real x) \<le> real (ub_exp_horner prec (get_odd n) 1 1 x)" .
--- a/src/HOL/Decision_Procs/Dense_Linear_Order.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Decision_Procs/Dense_Linear_Order.thy Mon Feb 08 11:13:30 2010 +0100
@@ -208,7 +208,7 @@
section {* The classical QE after Langford for dense linear orders *}
-context dense_linear_order
+context dense_linorder
begin
lemma interval_empty_iff:
@@ -265,7 +265,7 @@
lemmas dlo_simps[noatp] = order_refl less_irrefl not_less not_le exists_neq
le_less neq_iff linear less_not_permute
-lemma axiom[noatp]: "dense_linear_order (op \<le>) (op <)" by (rule dense_linear_order_axioms)
+lemma axiom[noatp]: "dense_linorder (op \<le>) (op <)" by (rule dense_linorder_axioms)
lemma atoms[noatp]:
shows "TERM (less :: 'a \<Rightarrow> _)"
and "TERM (less_eq :: 'a \<Rightarrow> _)"
@@ -409,17 +409,17 @@
end
-locale constr_dense_linear_order = linorder_no_lb + linorder_no_ub +
+locale constr_dense_linorder = linorder_no_lb + linorder_no_ub +
fixes between
assumes between_less: "less x y \<Longrightarrow> less x (between x y) \<and> less (between x y) y"
and between_same: "between x x = x"
-sublocale constr_dense_linear_order < dense_linear_order
+sublocale constr_dense_linorder < dense_linorder
apply unfold_locales
using gt_ex lt_ex between_less
by (auto, rule_tac x="between x y" in exI, simp)
-context constr_dense_linear_order
+context constr_dense_linorder
begin
lemma rinf_U[noatp]:
@@ -500,8 +500,8 @@
lemmas npi_thms[noatp] = npi_conj npi_disj npi_eq npi_neq npi_lt npi_le npi_gt npi_ge npi_P
lemmas lin_dense_thms[noatp] = lin_dense_conj lin_dense_disj lin_dense_eq lin_dense_neq lin_dense_lt lin_dense_le lin_dense_gt lin_dense_ge lin_dense_P
-lemma ferrack_axiom[noatp]: "constr_dense_linear_order less_eq less between"
- by (rule constr_dense_linear_order_axioms)
+lemma ferrack_axiom[noatp]: "constr_dense_linorder less_eq less between"
+ by (rule constr_dense_linorder_axioms)
lemma atoms[noatp]:
shows "TERM (less :: 'a \<Rightarrow> _)"
and "TERM (less_eq :: 'a \<Rightarrow> _)"
@@ -551,7 +551,7 @@
subsection {* Ferrante and Rackoff algorithm over ordered fields *}
-lemma neg_prod_lt:"(c\<Colon>'a\<Colon>ordered_field) < 0 \<Longrightarrow> ((c*x < 0) == (x > 0))"
+lemma neg_prod_lt:"(c\<Colon>'a\<Colon>linordered_field) < 0 \<Longrightarrow> ((c*x < 0) == (x > 0))"
proof-
assume H: "c < 0"
have "c*x < 0 = (0/c < x)" by (simp only: neg_divide_less_eq[OF H] algebra_simps)
@@ -559,7 +559,7 @@
finally show "(c*x < 0) == (x > 0)" by simp
qed
-lemma pos_prod_lt:"(c\<Colon>'a\<Colon>ordered_field) > 0 \<Longrightarrow> ((c*x < 0) == (x < 0))"
+lemma pos_prod_lt:"(c\<Colon>'a\<Colon>linordered_field) > 0 \<Longrightarrow> ((c*x < 0) == (x < 0))"
proof-
assume H: "c > 0"
hence "c*x < 0 = (0/c > x)" by (simp only: pos_less_divide_eq[OF H] algebra_simps)
@@ -567,7 +567,7 @@
finally show "(c*x < 0) == (x < 0)" by simp
qed
-lemma neg_prod_sum_lt: "(c\<Colon>'a\<Colon>ordered_field) < 0 \<Longrightarrow> ((c*x + t< 0) == (x > (- 1/c)*t))"
+lemma neg_prod_sum_lt: "(c\<Colon>'a\<Colon>linordered_field) < 0 \<Longrightarrow> ((c*x + t< 0) == (x > (- 1/c)*t))"
proof-
assume H: "c < 0"
have "c*x + t< 0 = (c*x < -t)" by (subst less_iff_diff_less_0 [of "c*x" "-t"], simp)
@@ -576,7 +576,7 @@
finally show "(c*x + t < 0) == (x > (- 1/c)*t)" by simp
qed
-lemma pos_prod_sum_lt:"(c\<Colon>'a\<Colon>ordered_field) > 0 \<Longrightarrow> ((c*x + t < 0) == (x < (- 1/c)*t))"
+lemma pos_prod_sum_lt:"(c\<Colon>'a\<Colon>linordered_field) > 0 \<Longrightarrow> ((c*x + t < 0) == (x < (- 1/c)*t))"
proof-
assume H: "c > 0"
have "c*x + t< 0 = (c*x < -t)" by (subst less_iff_diff_less_0 [of "c*x" "-t"], simp)
@@ -585,10 +585,10 @@
finally show "(c*x + t < 0) == (x < (- 1/c)*t)" by simp
qed
-lemma sum_lt:"((x::'a::pordered_ab_group_add) + t < 0) == (x < - t)"
+lemma sum_lt:"((x::'a::ordered_ab_group_add) + t < 0) == (x < - t)"
using less_diff_eq[where a= x and b=t and c=0] by simp
-lemma neg_prod_le:"(c\<Colon>'a\<Colon>ordered_field) < 0 \<Longrightarrow> ((c*x <= 0) == (x >= 0))"
+lemma neg_prod_le:"(c\<Colon>'a\<Colon>linordered_field) < 0 \<Longrightarrow> ((c*x <= 0) == (x >= 0))"
proof-
assume H: "c < 0"
have "c*x <= 0 = (0/c <= x)" by (simp only: neg_divide_le_eq[OF H] algebra_simps)
@@ -596,7 +596,7 @@
finally show "(c*x <= 0) == (x >= 0)" by simp
qed
-lemma pos_prod_le:"(c\<Colon>'a\<Colon>ordered_field) > 0 \<Longrightarrow> ((c*x <= 0) == (x <= 0))"
+lemma pos_prod_le:"(c\<Colon>'a\<Colon>linordered_field) > 0 \<Longrightarrow> ((c*x <= 0) == (x <= 0))"
proof-
assume H: "c > 0"
hence "c*x <= 0 = (0/c >= x)" by (simp only: pos_le_divide_eq[OF H] algebra_simps)
@@ -604,7 +604,7 @@
finally show "(c*x <= 0) == (x <= 0)" by simp
qed
-lemma neg_prod_sum_le: "(c\<Colon>'a\<Colon>ordered_field) < 0 \<Longrightarrow> ((c*x + t <= 0) == (x >= (- 1/c)*t))"
+lemma neg_prod_sum_le: "(c\<Colon>'a\<Colon>linordered_field) < 0 \<Longrightarrow> ((c*x + t <= 0) == (x >= (- 1/c)*t))"
proof-
assume H: "c < 0"
have "c*x + t <= 0 = (c*x <= -t)" by (subst le_iff_diff_le_0 [of "c*x" "-t"], simp)
@@ -613,7 +613,7 @@
finally show "(c*x + t <= 0) == (x >= (- 1/c)*t)" by simp
qed
-lemma pos_prod_sum_le:"(c\<Colon>'a\<Colon>ordered_field) > 0 \<Longrightarrow> ((c*x + t <= 0) == (x <= (- 1/c)*t))"
+lemma pos_prod_sum_le:"(c\<Colon>'a\<Colon>linordered_field) > 0 \<Longrightarrow> ((c*x + t <= 0) == (x <= (- 1/c)*t))"
proof-
assume H: "c > 0"
have "c*x + t <= 0 = (c*x <= -t)" by (subst le_iff_diff_le_0 [of "c*x" "-t"], simp)
@@ -622,24 +622,24 @@
finally show "(c*x + t <= 0) == (x <= (- 1/c)*t)" by simp
qed
-lemma sum_le:"((x::'a::pordered_ab_group_add) + t <= 0) == (x <= - t)"
+lemma sum_le:"((x::'a::ordered_ab_group_add) + t <= 0) == (x <= - t)"
using le_diff_eq[where a= x and b=t and c=0] by simp
-lemma nz_prod_eq:"(c\<Colon>'a\<Colon>ordered_field) \<noteq> 0 \<Longrightarrow> ((c*x = 0) == (x = 0))" by simp
-lemma nz_prod_sum_eq: "(c\<Colon>'a\<Colon>ordered_field) \<noteq> 0 \<Longrightarrow> ((c*x + t = 0) == (x = (- 1/c)*t))"
+lemma nz_prod_eq:"(c\<Colon>'a\<Colon>linordered_field) \<noteq> 0 \<Longrightarrow> ((c*x = 0) == (x = 0))" by simp
+lemma nz_prod_sum_eq: "(c\<Colon>'a\<Colon>linordered_field) \<noteq> 0 \<Longrightarrow> ((c*x + t = 0) == (x = (- 1/c)*t))"
proof-
assume H: "c \<noteq> 0"
have "c*x + t = 0 = (c*x = -t)" by (subst eq_iff_diff_eq_0 [of "c*x" "-t"], simp)
also have "\<dots> = (x = -t/c)" by (simp only: nonzero_eq_divide_eq[OF H] algebra_simps)
finally show "(c*x + t = 0) == (x = (- 1/c)*t)" by simp
qed
-lemma sum_eq:"((x::'a::pordered_ab_group_add) + t = 0) == (x = - t)"
+lemma sum_eq:"((x::'a::ordered_ab_group_add) + t = 0) == (x = - t)"
using eq_diff_eq[where a= x and b=t and c=0] by simp
-interpretation class_ordered_field_dense_linear_order: constr_dense_linear_order
+interpretation class_dense_linlinordered_field: constr_dense_linorder
"op <=" "op <"
- "\<lambda> x y. 1/2 * ((x::'a::{ordered_field,number_ring}) + y)"
+ "\<lambda> x y. 1/2 * ((x::'a::{linordered_field,number_ring}) + y)"
proof (unfold_locales, dlo, dlo, auto)
fix x y::'a assume lt: "x < y"
from less_half_sum[OF lt] show "x < (x + y) /2" by simp
@@ -871,7 +871,7 @@
addsplits [@{thm "abs_split"},@{thm "split_max"}, @{thm "split_min"}]
in
-Ferrante_Rackoff_Data.funs @{thm "class_ordered_field_dense_linear_order.ferrack_axiom"}
+Ferrante_Rackoff_Data.funs @{thm "class_dense_linlinordered_field.ferrack_axiom"}
{isolate_conv = field_isolate_conv, whatis = classfield_whatis, simpset = class_field_ss}
end
*}
--- a/src/HOL/Decision_Procs/MIR.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Decision_Procs/MIR.thy Mon Feb 08 11:13:30 2010 +0100
@@ -54,7 +54,7 @@
by clarsimp
-lemma myl: "\<forall> (a::'a::{pordered_ab_group_add}) (b::'a). (a \<le> b) = (0 \<le> b - a)"
+lemma myl: "\<forall> (a::'a::{ordered_ab_group_add}) (b::'a). (a \<le> b) = (0 \<le> b - a)"
proof(clarify)
fix x y ::"'a"
have "(x \<le> y) = (x - y \<le> 0)" by (simp only: le_iff_diff_le_0[where a="x" and b="y"])
@@ -63,7 +63,7 @@
finally show "(x \<le> y) = (0 \<le> y - x)" .
qed
-lemma myless: "\<forall> (a::'a::{pordered_ab_group_add}) (b::'a). (a < b) = (0 < b - a)"
+lemma myless: "\<forall> (a::'a::{ordered_ab_group_add}) (b::'a). (a < b) = (0 < b - a)"
proof(clarify)
fix x y ::"'a"
have "(x < y) = (x - y < 0)" by (simp only: less_iff_diff_less_0[where a="x" and b="y"])
@@ -72,7 +72,7 @@
finally show "(x < y) = (0 < y - x)" .
qed
-lemma myeq: "\<forall> (a::'a::{pordered_ab_group_add}) (b::'a). (a = b) = (0 = b - a)"
+lemma myeq: "\<forall> (a::'a::{ordered_ab_group_add}) (b::'a). (a = b) = (0 = b - a)"
by auto
(* Maybe should be added to the library \<dots> *)
--- a/src/HOL/Decision_Procs/Parametric_Ferrante_Rackoff.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Decision_Procs/Parametric_Ferrante_Rackoff.thy Mon Feb 08 11:13:30 2010 +0100
@@ -447,7 +447,7 @@
by (induct p rule: fmsize.induct) simp_all
(* Semantics of formulae (fm) *)
-consts Ifm ::"'a::{division_by_zero,ordered_field} list \<Rightarrow> 'a list \<Rightarrow> fm \<Rightarrow> bool"
+consts Ifm ::"'a::{division_by_zero,linordered_field} list \<Rightarrow> 'a list \<Rightarrow> fm \<Rightarrow> bool"
primrec
"Ifm vs bs T = True"
"Ifm vs bs F = False"
@@ -1833,16 +1833,16 @@
ultimately show ?case by blast
qed (auto simp add: nth_pos2 tmbound0_I[where vs=vs and bs="bs" and b="y" and b'="x"] bound0_I[where vs=vs and bs="bs" and b="y" and b'="x"])
-lemma one_plus_one_pos[simp]: "(1::'a::{ordered_field}) + 1 > 0"
+lemma one_plus_one_pos[simp]: "(1::'a::{linordered_field}) + 1 > 0"
proof-
have op: "(1::'a) > 0" by simp
from add_pos_pos[OF op op] show ?thesis .
qed
-lemma one_plus_one_nonzero[simp]: "(1::'a::{ordered_field}) + 1 \<noteq> 0"
+lemma one_plus_one_nonzero[simp]: "(1::'a::{linordered_field}) + 1 \<noteq> 0"
using one_plus_one_pos[where ?'a = 'a] by (simp add: less_le)
-lemma half_sum_eq: "(u + u) / (1+1) = (u::'a::{ordered_field})"
+lemma half_sum_eq: "(u + u) / (1+1) = (u::'a::{linordered_field})"
proof-
have "(u + u) = (1 + 1) * u" by (simp add: ring_simps)
hence "(u + u) / (1+1) = (1 + 1)*u / (1 + 1)" by simp
@@ -3172,54 +3172,54 @@
*} "Parametric QE for linear Arithmetic over fields, Version 2"
-lemma "\<exists>(x::'a::{division_by_zero,ordered_field,number_ring}). y \<noteq> -1 \<longrightarrow> (y + 1)*x < 0"
- apply (frpar type: "'a::{division_by_zero,ordered_field,number_ring}" pars: "y::'a::{division_by_zero,ordered_field,number_ring}")
+lemma "\<exists>(x::'a::{division_by_zero,linordered_field,number_ring}). y \<noteq> -1 \<longrightarrow> (y + 1)*x < 0"
+ apply (frpar type: "'a::{division_by_zero,linordered_field,number_ring}" pars: "y::'a::{division_by_zero,linordered_field,number_ring}")
apply (simp add: ring_simps)
apply (rule spec[where x=y])
- apply (frpar type: "'a::{division_by_zero,ordered_field,number_ring}" pars: "z::'a::{division_by_zero,ordered_field,number_ring}")
+ apply (frpar type: "'a::{division_by_zero,linordered_field,number_ring}" pars: "z::'a::{division_by_zero,linordered_field,number_ring}")
by simp
text{* Collins/Jones Problem *}
(*
-lemma "\<exists>(r::'a::{division_by_zero,ordered_field,number_ring}). 0 < r \<and> r < 1 \<and> 0 < (2 - 3*r) *(a^2 + b^2) + (2*a)*r \<and> (2 - 3*r) *(a^2 + b^2) + 4*a*r - 2*a - r < 0"
+lemma "\<exists>(r::'a::{division_by_zero,linordered_field,number_ring}). 0 < r \<and> r < 1 \<and> 0 < (2 - 3*r) *(a^2 + b^2) + (2*a)*r \<and> (2 - 3*r) *(a^2 + b^2) + 4*a*r - 2*a - r < 0"
proof-
- have "(\<exists>(r::'a::{division_by_zero,ordered_field,number_ring}). 0 < r \<and> r < 1 \<and> 0 < (2 - 3*r) *(a^2 + b^2) + (2*a)*r \<and> (2 - 3*r) *(a^2 + b^2) + 4*a*r - 2*a - r < 0) \<longleftrightarrow> (\<exists>(r::'a::{division_by_zero,ordered_field,number_ring}). 0 < r \<and> r < 1 \<and> 0 < 2 *(a^2 + b^2) - (3*(a^2 + b^2)) * r + (2*a)*r \<and> 2*(a^2 + b^2) - (3*(a^2 + b^2) - 4*a + 1)*r - 2*a < 0)" (is "?lhs \<longleftrightarrow> ?rhs")
+ have "(\<exists>(r::'a::{division_by_zero,linordered_field,number_ring}). 0 < r \<and> r < 1 \<and> 0 < (2 - 3*r) *(a^2 + b^2) + (2*a)*r \<and> (2 - 3*r) *(a^2 + b^2) + 4*a*r - 2*a - r < 0) \<longleftrightarrow> (\<exists>(r::'a::{division_by_zero,linordered_field,number_ring}). 0 < r \<and> r < 1 \<and> 0 < 2 *(a^2 + b^2) - (3*(a^2 + b^2)) * r + (2*a)*r \<and> 2*(a^2 + b^2) - (3*(a^2 + b^2) - 4*a + 1)*r - 2*a < 0)" (is "?lhs \<longleftrightarrow> ?rhs")
by (simp add: ring_simps)
have "?rhs"
- apply (frpar type: "'a::{division_by_zero,ordered_field,number_ring}" pars: "a::'a::{division_by_zero,ordered_field,number_ring}" "b::'a::{division_by_zero,ordered_field,number_ring}")
+ apply (frpar type: "'a::{division_by_zero,linordered_field,number_ring}" pars: "a::'a::{division_by_zero,linordered_field,number_ring}" "b::'a::{division_by_zero,linordered_field,number_ring}")
apply (simp add: ring_simps)
oops
*)
(*
-lemma "ALL (x::'a::{division_by_zero,ordered_field,number_ring}) y. (1 - t)*x \<le> (1+t)*y \<and> (1 - t)*y \<le> (1+t)*x --> 0 \<le> y"
-apply (frpar type: "'a::{division_by_zero,ordered_field,number_ring}" pars: "t::'a::{division_by_zero,ordered_field,number_ring}")
+lemma "ALL (x::'a::{division_by_zero,linordered_field,number_ring}) y. (1 - t)*x \<le> (1+t)*y \<and> (1 - t)*y \<le> (1+t)*x --> 0 \<le> y"
+apply (frpar type: "'a::{division_by_zero,linordered_field,number_ring}" pars: "t::'a::{division_by_zero,linordered_field,number_ring}")
oops
*)
-lemma "\<exists>(x::'a::{division_by_zero,ordered_field,number_ring}). y \<noteq> -1 \<longrightarrow> (y + 1)*x < 0"
- apply (frpar2 type: "'a::{division_by_zero,ordered_field,number_ring}" pars: "y::'a::{division_by_zero,ordered_field,number_ring}")
+lemma "\<exists>(x::'a::{division_by_zero,linordered_field,number_ring}). y \<noteq> -1 \<longrightarrow> (y + 1)*x < 0"
+ apply (frpar2 type: "'a::{division_by_zero,linordered_field,number_ring}" pars: "y::'a::{division_by_zero,linordered_field,number_ring}")
apply (simp add: ring_simps)
apply (rule spec[where x=y])
- apply (frpar2 type: "'a::{division_by_zero,ordered_field,number_ring}" pars: "z::'a::{division_by_zero,ordered_field,number_ring}")
+ apply (frpar2 type: "'a::{division_by_zero,linordered_field,number_ring}" pars: "z::'a::{division_by_zero,linordered_field,number_ring}")
by simp
text{* Collins/Jones Problem *}
(*
-lemma "\<exists>(r::'a::{division_by_zero,ordered_field,number_ring}). 0 < r \<and> r < 1 \<and> 0 < (2 - 3*r) *(a^2 + b^2) + (2*a)*r \<and> (2 - 3*r) *(a^2 + b^2) + 4*a*r - 2*a - r < 0"
+lemma "\<exists>(r::'a::{division_by_zero,linordered_field,number_ring}). 0 < r \<and> r < 1 \<and> 0 < (2 - 3*r) *(a^2 + b^2) + (2*a)*r \<and> (2 - 3*r) *(a^2 + b^2) + 4*a*r - 2*a - r < 0"
proof-
- have "(\<exists>(r::'a::{division_by_zero,ordered_field,number_ring}). 0 < r \<and> r < 1 \<and> 0 < (2 - 3*r) *(a^2 + b^2) + (2*a)*r \<and> (2 - 3*r) *(a^2 + b^2) + 4*a*r - 2*a - r < 0) \<longleftrightarrow> (\<exists>(r::'a::{division_by_zero,ordered_field,number_ring}). 0 < r \<and> r < 1 \<and> 0 < 2 *(a^2 + b^2) - (3*(a^2 + b^2)) * r + (2*a)*r \<and> 2*(a^2 + b^2) - (3*(a^2 + b^2) - 4*a + 1)*r - 2*a < 0)" (is "?lhs \<longleftrightarrow> ?rhs")
+ have "(\<exists>(r::'a::{division_by_zero,linordered_field,number_ring}). 0 < r \<and> r < 1 \<and> 0 < (2 - 3*r) *(a^2 + b^2) + (2*a)*r \<and> (2 - 3*r) *(a^2 + b^2) + 4*a*r - 2*a - r < 0) \<longleftrightarrow> (\<exists>(r::'a::{division_by_zero,linordered_field,number_ring}). 0 < r \<and> r < 1 \<and> 0 < 2 *(a^2 + b^2) - (3*(a^2 + b^2)) * r + (2*a)*r \<and> 2*(a^2 + b^2) - (3*(a^2 + b^2) - 4*a + 1)*r - 2*a < 0)" (is "?lhs \<longleftrightarrow> ?rhs")
by (simp add: ring_simps)
have "?rhs"
- apply (frpar2 type: "'a::{division_by_zero,ordered_field,number_ring}" pars: "a::'a::{division_by_zero,ordered_field,number_ring}" "b::'a::{division_by_zero,ordered_field,number_ring}")
+ apply (frpar2 type: "'a::{division_by_zero,linordered_field,number_ring}" pars: "a::'a::{division_by_zero,linordered_field,number_ring}" "b::'a::{division_by_zero,linordered_field,number_ring}")
apply simp
oops
*)
(*
-lemma "ALL (x::'a::{division_by_zero,ordered_field,number_ring}) y. (1 - t)*x \<le> (1+t)*y \<and> (1 - t)*y \<le> (1+t)*x --> 0 \<le> y"
-apply (frpar2 type: "'a::{division_by_zero,ordered_field,number_ring}" pars: "t::'a::{division_by_zero,ordered_field,number_ring}")
+lemma "ALL (x::'a::{division_by_zero,linordered_field,number_ring}) y. (1 - t)*x \<le> (1+t)*y \<and> (1 - t)*y \<le> (1+t)*x --> 0 \<le> y"
+apply (frpar2 type: "'a::{division_by_zero,linordered_field,number_ring}" pars: "t::'a::{division_by_zero,linordered_field,number_ring}")
apply (simp add: field_simps linorder_neq_iff[symmetric])
apply ferrack
oops
--- a/src/HOL/Decision_Procs/Polynomial_List.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Decision_Procs/Polynomial_List.thy Mon Feb 08 11:13:30 2010 +0100
@@ -772,7 +772,7 @@
text{*bound for polynomial.*}
-lemma poly_mono: "abs(x) \<le> k ==> abs(poly p (x::'a::{ordered_idom})) \<le> poly (map abs p) k"
+lemma poly_mono: "abs(x) \<le> k ==> abs(poly p (x::'a::{linordered_idom})) \<le> poly (map abs p) k"
apply (induct "p", auto)
apply (rule_tac y = "abs a + abs (x * poly p x)" in order_trans)
apply (rule abs_triangle_ineq)
--- a/src/HOL/Decision_Procs/ex/Dense_Linear_Order_Ex.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Decision_Procs/ex/Dense_Linear_Order_Ex.thy Mon Feb 08 11:13:30 2010 +0100
@@ -7,147 +7,147 @@
begin
lemma
- "\<exists>(y::'a::{ordered_field,number_ring, division_by_zero}) <2. x + 3* y < 0 \<and> x - y >0"
+ "\<exists>(y::'a::{linordered_field,number_ring, division_by_zero}) <2. x + 3* y < 0 \<and> x - y >0"
by ferrack
-lemma "~ (ALL x (y::'a::{ordered_field,number_ring, division_by_zero}). x < y --> 10*x < 11*y)"
+lemma "~ (ALL x (y::'a::{linordered_field,number_ring, division_by_zero}). x < y --> 10*x < 11*y)"
by ferrack
-lemma "ALL (x::'a::{ordered_field,number_ring, division_by_zero}) y. x < y --> (10*(x + 5*y + -1) < 60*y)"
+lemma "ALL (x::'a::{linordered_field,number_ring, division_by_zero}) y. x < y --> (10*(x + 5*y + -1) < 60*y)"
by ferrack
-lemma "EX (x::'a::{ordered_field,number_ring, division_by_zero}) y. x ~= y --> x < y"
+lemma "EX (x::'a::{linordered_field,number_ring, division_by_zero}) y. x ~= y --> x < y"
by ferrack
-lemma "EX (x::'a::{ordered_field,number_ring, division_by_zero}) y. (x ~= y & 10*x ~= 9*y & 10*x < y) --> x < y"
+lemma "EX (x::'a::{linordered_field,number_ring, division_by_zero}) y. (x ~= y & 10*x ~= 9*y & 10*x < y) --> x < y"
by ferrack
-lemma "ALL (x::'a::{ordered_field,number_ring, division_by_zero}) y. (x ~= y & 5*x <= y) --> 500*x <= 100*y"
+lemma "ALL (x::'a::{linordered_field,number_ring, division_by_zero}) y. (x ~= y & 5*x <= y) --> 500*x <= 100*y"
by ferrack
-lemma "ALL (x::'a::{ordered_field,number_ring, division_by_zero}). (EX (y::'a::{ordered_field,number_ring, division_by_zero}). 4*x + 3*y <= 0 & 4*x + 3*y >= -1)"
+lemma "ALL (x::'a::{linordered_field,number_ring, division_by_zero}). (EX (y::'a::{linordered_field,number_ring, division_by_zero}). 4*x + 3*y <= 0 & 4*x + 3*y >= -1)"
by ferrack
-lemma "ALL (x::'a::{ordered_field,number_ring, division_by_zero}) < 0. (EX (y::'a::{ordered_field,number_ring, division_by_zero}) > 0. 7*x + y > 0 & x - y <= 9)"
+lemma "ALL (x::'a::{linordered_field,number_ring, division_by_zero}) < 0. (EX (y::'a::{linordered_field,number_ring, division_by_zero}) > 0. 7*x + y > 0 & x - y <= 9)"
by ferrack
-lemma "EX (x::'a::{ordered_field,number_ring, division_by_zero}). (0 < x & x < 1) --> (ALL y > 1. x + y ~= 1)"
+lemma "EX (x::'a::{linordered_field,number_ring, division_by_zero}). (0 < x & x < 1) --> (ALL y > 1. x + y ~= 1)"
by ferrack
-lemma "EX x. (ALL (y::'a::{ordered_field,number_ring, division_by_zero}). y < 2 --> 2*(y - x) \<le> 0 )"
+lemma "EX x. (ALL (y::'a::{linordered_field,number_ring, division_by_zero}). y < 2 --> 2*(y - x) \<le> 0 )"
by ferrack
-lemma "ALL (x::'a::{ordered_field,number_ring, division_by_zero}). x < 10 | x > 20 | (EX y. y>= 0 & y <= 10 & x+y = 20)"
+lemma "ALL (x::'a::{linordered_field,number_ring, division_by_zero}). x < 10 | x > 20 | (EX y. y>= 0 & y <= 10 & x+y = 20)"
by ferrack
-lemma "ALL (x::'a::{ordered_field,number_ring, division_by_zero}) y z. x + y < z --> y >= z --> x < 0"
+lemma "ALL (x::'a::{linordered_field,number_ring, division_by_zero}) y z. x + y < z --> y >= z --> x < 0"
by ferrack
-lemma "EX (x::'a::{ordered_field,number_ring, division_by_zero}) y z. x + 7*y < 5* z & 5*y >= 7*z & x < 0"
+lemma "EX (x::'a::{linordered_field,number_ring, division_by_zero}) y z. x + 7*y < 5* z & 5*y >= 7*z & x < 0"
by ferrack
-lemma "ALL (x::'a::{ordered_field,number_ring, division_by_zero}) y z. abs (x + y) <= z --> (abs z = z)"
+lemma "ALL (x::'a::{linordered_field,number_ring, division_by_zero}) y z. abs (x + y) <= z --> (abs z = z)"
by ferrack
-lemma "EX (x::'a::{ordered_field,number_ring, division_by_zero}) y z. x + 7*y - 5* z < 0 & 5*y + 7*z + 3*x < 0"
+lemma "EX (x::'a::{linordered_field,number_ring, division_by_zero}) y z. x + 7*y - 5* z < 0 & 5*y + 7*z + 3*x < 0"
by ferrack
-lemma "ALL (x::'a::{ordered_field,number_ring, division_by_zero}) y z. (abs (5*x+3*y+z) <= 5*x+3*y+z & abs (5*x+3*y+z) >= - (5*x+3*y+z)) | (abs (5*x+3*y+z) >= 5*x+3*y+z & abs (5*x+3*y+z) <= - (5*x+3*y+z))"
+lemma "ALL (x::'a::{linordered_field,number_ring, division_by_zero}) y z. (abs (5*x+3*y+z) <= 5*x+3*y+z & abs (5*x+3*y+z) >= - (5*x+3*y+z)) | (abs (5*x+3*y+z) >= 5*x+3*y+z & abs (5*x+3*y+z) <= - (5*x+3*y+z))"
by ferrack
-lemma "ALL (x::'a::{ordered_field,number_ring, division_by_zero}) y. x < y --> (EX z>0. x+z = y)"
+lemma "ALL (x::'a::{linordered_field,number_ring, division_by_zero}) y. x < y --> (EX z>0. x+z = y)"
by ferrack
-lemma "ALL (x::'a::{ordered_field,number_ring, division_by_zero}) y. x < y --> (EX z>0. x+z = y)"
+lemma "ALL (x::'a::{linordered_field,number_ring, division_by_zero}) y. x < y --> (EX z>0. x+z = y)"
by ferrack
-lemma "ALL (x::'a::{ordered_field,number_ring, division_by_zero}) y. (EX z>0. abs (x - y) <= z )"
+lemma "ALL (x::'a::{linordered_field,number_ring, division_by_zero}) y. (EX z>0. abs (x - y) <= z )"
by ferrack
-lemma "EX (x::'a::{ordered_field,number_ring, division_by_zero}) y. (ALL z<0. (z < x --> z <= y) & (z > y --> z >= x))"
+lemma "EX (x::'a::{linordered_field,number_ring, division_by_zero}) y. (ALL z<0. (z < x --> z <= y) & (z > y --> z >= x))"
by ferrack
-lemma "EX (x::'a::{ordered_field,number_ring, division_by_zero}) y. (ALL z>=0. abs (3*x+7*y) <= 2*z + 1)"
+lemma "EX (x::'a::{linordered_field,number_ring, division_by_zero}) y. (ALL z>=0. abs (3*x+7*y) <= 2*z + 1)"
by ferrack
-lemma "EX (x::'a::{ordered_field,number_ring, division_by_zero}) y. (ALL z<0. (z < x --> z <= y) & (z > y --> z >= x))"
+lemma "EX (x::'a::{linordered_field,number_ring, division_by_zero}) y. (ALL z<0. (z < x --> z <= y) & (z > y --> z >= x))"
by ferrack
-lemma "EX (x::'a::{ordered_field,number_ring, division_by_zero})>0. (ALL y. (EX z. 13* abs z \<noteq> abs (12*y - x) & 5*x - 3*(abs y) <= 7*z))"
+lemma "EX (x::'a::{linordered_field,number_ring, division_by_zero})>0. (ALL y. (EX z. 13* abs z \<noteq> abs (12*y - x) & 5*x - 3*(abs y) <= 7*z))"
by ferrack
-lemma "EX (x::'a::{ordered_field,number_ring, division_by_zero}). abs (4*x + 17) < 4 & (ALL y . abs (x*34 - 34*y - 9) \<noteq> 0 \<longrightarrow> (EX z. 5*x - 3*abs y <= 7*z))"
+lemma "EX (x::'a::{linordered_field,number_ring, division_by_zero}). abs (4*x + 17) < 4 & (ALL y . abs (x*34 - 34*y - 9) \<noteq> 0 \<longrightarrow> (EX z. 5*x - 3*abs y <= 7*z))"
by ferrack
-lemma "ALL (x::'a::{ordered_field,number_ring, division_by_zero}). (EX y > abs (23*x - 9). (ALL z > abs (3*y - 19* abs x). x+z > 2*y))"
+lemma "ALL (x::'a::{linordered_field,number_ring, division_by_zero}). (EX y > abs (23*x - 9). (ALL z > abs (3*y - 19* abs x). x+z > 2*y))"
by ferrack
-lemma "ALL (x::'a::{ordered_field,number_ring, division_by_zero}). (EX y< abs (3*x - 1). (ALL z >= (3*abs x - 1). abs (12*x - 13*y + 19*z) > abs (23*x) ))"
+lemma "ALL (x::'a::{linordered_field,number_ring, division_by_zero}). (EX y< abs (3*x - 1). (ALL z >= (3*abs x - 1). abs (12*x - 13*y + 19*z) > abs (23*x) ))"
by ferrack
-lemma "EX (x::'a::{ordered_field,number_ring, division_by_zero}). abs x < 100 & (ALL y > x. (EX z<2*y - x. 5*x - 3*y <= 7*z))"
+lemma "EX (x::'a::{linordered_field,number_ring, division_by_zero}). abs x < 100 & (ALL y > x. (EX z<2*y - x. 5*x - 3*y <= 7*z))"
by ferrack
-lemma "ALL (x::'a::{ordered_field,number_ring, division_by_zero}) y z w. 7*x<3*y --> 5*y < 7*z --> z < 2*w --> 7*(2*w-x) > 2*y"
+lemma "ALL (x::'a::{linordered_field,number_ring, division_by_zero}) y z w. 7*x<3*y --> 5*y < 7*z --> z < 2*w --> 7*(2*w-x) > 2*y"
by ferrack
-lemma "EX (x::'a::{ordered_field,number_ring, division_by_zero}) y z w. 5*x + 3*z - 17*w + abs (y - 8*x + z) <= 89"
+lemma "EX (x::'a::{linordered_field,number_ring, division_by_zero}) y z w. 5*x + 3*z - 17*w + abs (y - 8*x + z) <= 89"
by ferrack
-lemma "EX (x::'a::{ordered_field,number_ring, division_by_zero}) y z w. 5*x + 3*z - 17*w + 7* (y - 8*x + z) <= max y (7*z - x + w)"
+lemma "EX (x::'a::{linordered_field,number_ring, division_by_zero}) y z w. 5*x + 3*z - 17*w + 7* (y - 8*x + z) <= max y (7*z - x + w)"
by ferrack
-lemma "EX (x::'a::{ordered_field,number_ring, division_by_zero}) y z w. min (5*x + 3*z) (17*w) + 5* abs (y - 8*x + z) <= max y (7*z - x + w)"
+lemma "EX (x::'a::{linordered_field,number_ring, division_by_zero}) y z w. min (5*x + 3*z) (17*w) + 5* abs (y - 8*x + z) <= max y (7*z - x + w)"
by ferrack
-lemma "ALL (x::'a::{ordered_field,number_ring, division_by_zero}) y z. (EX w >= (x+y+z). w <= abs x + abs y + abs z)"
+lemma "ALL (x::'a::{linordered_field,number_ring, division_by_zero}) y z. (EX w >= (x+y+z). w <= abs x + abs y + abs z)"
by ferrack
-lemma "~(ALL (x::'a::{ordered_field,number_ring, division_by_zero}). (EX y z w. 3* x + z*4 = 3*y & x + y < z & x> w & 3*x < w + y))"
+lemma "~(ALL (x::'a::{linordered_field,number_ring, division_by_zero}). (EX y z w. 3* x + z*4 = 3*y & x + y < z & x> w & 3*x < w + y))"
by ferrack
-lemma "ALL (x::'a::{ordered_field,number_ring, division_by_zero}) y. (EX z w. abs (x-y) = (z-w) & z*1234 < 233*x & w ~= y)"
+lemma "ALL (x::'a::{linordered_field,number_ring, division_by_zero}) y. (EX z w. abs (x-y) = (z-w) & z*1234 < 233*x & w ~= y)"
by ferrack
-lemma "ALL (x::'a::{ordered_field,number_ring, division_by_zero}). (EX y z w. min (5*x + 3*z) (17*w) + 5* abs (y - 8*x + z) <= max y (7*z - x + w))"
+lemma "ALL (x::'a::{linordered_field,number_ring, division_by_zero}). (EX y z w. min (5*x + 3*z) (17*w) + 5* abs (y - 8*x + z) <= max y (7*z - x + w))"
by ferrack
-lemma "EX (x::'a::{ordered_field,number_ring, division_by_zero}) y z. (ALL w >= abs (x+y+z). w >= abs x + abs y + abs z)"
+lemma "EX (x::'a::{linordered_field,number_ring, division_by_zero}) y z. (ALL w >= abs (x+y+z). w >= abs x + abs y + abs z)"
by ferrack
-lemma "EX z. (ALL (x::'a::{ordered_field,number_ring, division_by_zero}) y. (EX w >= (x+y+z). w <= abs x + abs y + abs z))"
+lemma "EX z. (ALL (x::'a::{linordered_field,number_ring, division_by_zero}) y. (EX w >= (x+y+z). w <= abs x + abs y + abs z))"
by ferrack
-lemma "EX z. (ALL (x::'a::{ordered_field,number_ring, division_by_zero}) < abs z. (EX y w. x< y & x < z & x> w & 3*x < w + y))"
+lemma "EX z. (ALL (x::'a::{linordered_field,number_ring, division_by_zero}) < abs z. (EX y w. x< y & x < z & x> w & 3*x < w + y))"
by ferrack
-lemma "ALL (x::'a::{ordered_field,number_ring, division_by_zero}) y. (EX z. (ALL w. abs (x-y) = abs (z-w) --> z < x & w ~= y))"
+lemma "ALL (x::'a::{linordered_field,number_ring, division_by_zero}) y. (EX z. (ALL w. abs (x-y) = abs (z-w) --> z < x & w ~= y))"
by ferrack
-lemma "EX y. (ALL (x::'a::{ordered_field,number_ring, division_by_zero}). (EX z w. min (5*x + 3*z) (17*w) + 5* abs (y - 8*x + z) <= max y (7*z - x + w)))"
+lemma "EX y. (ALL (x::'a::{linordered_field,number_ring, division_by_zero}). (EX z w. min (5*x + 3*z) (17*w) + 5* abs (y - 8*x + z) <= max y (7*z - x + w)))"
by ferrack
-lemma "EX (x::'a::{ordered_field,number_ring, division_by_zero}) z. (ALL w >= 13*x - 4*z. (EX y. w >= abs x + abs y + z))"
+lemma "EX (x::'a::{linordered_field,number_ring, division_by_zero}) z. (ALL w >= 13*x - 4*z. (EX y. w >= abs x + abs y + z))"
by ferrack
-lemma "EX (x::'a::{ordered_field,number_ring, division_by_zero}). (ALL y < x. (EX z > (x+y).
+lemma "EX (x::'a::{linordered_field,number_ring, division_by_zero}). (ALL y < x. (EX z > (x+y).
(ALL w. 5*w + 10*x - z >= y --> w + 7*x + 3*z >= 2*y)))"
by ferrack
-lemma "EX (x::'a::{ordered_field,number_ring, division_by_zero}). (ALL y. (EX z > y.
+lemma "EX (x::'a::{linordered_field,number_ring, division_by_zero}). (ALL y. (EX z > y.
(ALL w . w < 13 --> w + 10*x - z >= y --> 5*w + 7*x + 13*z >= 2*y)))"
by ferrack
-lemma "EX (x::'a::{ordered_field,number_ring, division_by_zero}) y z w. min (5*x + 3*z) (17*w) + 5* abs (y - 8*x + z) <= max y (7*z - x + w)"
+lemma "EX (x::'a::{linordered_field,number_ring, division_by_zero}) y z w. min (5*x + 3*z) (17*w) + 5* abs (y - 8*x + z) <= max y (7*z - x + w)"
by ferrack
-lemma "ALL (x::'a::{ordered_field,number_ring, division_by_zero}). (EX y. (ALL z>19. y <= x + z & (EX w. abs (y - x) < w)))"
+lemma "ALL (x::'a::{linordered_field,number_ring, division_by_zero}). (EX y. (ALL z>19. y <= x + z & (EX w. abs (y - x) < w)))"
by ferrack
-lemma "ALL (x::'a::{ordered_field,number_ring, division_by_zero}). (EX y. (ALL z>19. y <= x + z & (EX w. abs (x + z) < w - y)))"
+lemma "ALL (x::'a::{linordered_field,number_ring, division_by_zero}). (EX y. (ALL z>19. y <= x + z & (EX w. abs (x + z) < w - y)))"
by ferrack
-lemma "ALL (x::'a::{ordered_field,number_ring, division_by_zero}). (EX y. abs y ~= abs x & (ALL z> max x y. (EX w. w ~= y & w ~= z & 3*w - z >= x + y)))"
+lemma "ALL (x::'a::{linordered_field,number_ring, division_by_zero}). (EX y. abs y ~= abs x & (ALL z> max x y. (EX w. w ~= y & w ~= z & 3*w - z >= x + y)))"
by ferrack
end
--- a/src/HOL/Fact.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Fact.thy Mon Feb 08 11:13:30 2010 +0100
@@ -266,15 +266,15 @@
lemma of_nat_fact_not_zero [simp]: "of_nat (fact n) \<noteq> (0::'a::semiring_char_0)"
by auto
-lemma of_nat_fact_gt_zero [simp]: "(0::'a::{ordered_semidom}) < of_nat(fact n)" by auto
+lemma of_nat_fact_gt_zero [simp]: "(0::'a::{linordered_semidom}) < of_nat(fact n)" by auto
-lemma of_nat_fact_ge_zero [simp]: "(0::'a::ordered_semidom) \<le> of_nat(fact n)"
+lemma of_nat_fact_ge_zero [simp]: "(0::'a::linordered_semidom) \<le> of_nat(fact n)"
by simp
-lemma inv_of_nat_fact_gt_zero [simp]: "(0::'a::ordered_field) < inverse (of_nat (fact n))"
+lemma inv_of_nat_fact_gt_zero [simp]: "(0::'a::linordered_field) < inverse (of_nat (fact n))"
by (auto simp add: positive_imp_inverse_positive)
-lemma inv_of_nat_fact_ge_zero [simp]: "(0::'a::ordered_field) \<le> inverse (of_nat (fact n))"
+lemma inv_of_nat_fact_ge_zero [simp]: "(0::'a::linordered_field) \<le> inverse (of_nat (fact n))"
by (auto intro: order_less_imp_le)
end
--- a/src/HOL/Finite_Set.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Finite_Set.thy Mon Feb 08 11:13:30 2010 +0100
@@ -830,7 +830,7 @@
end
-context lower_semilattice
+context semilattice_inf
begin
lemma ab_semigroup_idem_mult_inf: "ab_semigroup_idem_mult inf"
@@ -857,20 +857,20 @@
end
-context upper_semilattice
+context semilattice_sup
begin
lemma ab_semigroup_idem_mult_sup: "ab_semigroup_idem_mult sup"
-by (rule lower_semilattice.ab_semigroup_idem_mult_inf)(rule dual_semilattice)
+by (rule semilattice_inf.ab_semigroup_idem_mult_inf)(rule dual_semilattice)
lemma fold_sup_insert[simp]: "finite A \<Longrightarrow> fold sup b (insert a A) = sup a (fold sup b A)"
-by(rule lower_semilattice.fold_inf_insert)(rule dual_semilattice)
+by(rule semilattice_inf.fold_inf_insert)(rule dual_semilattice)
lemma fold_sup_le_sup: "finite A \<Longrightarrow> ALL a:A. a \<le> b \<Longrightarrow> fold sup c A \<le> sup b c"
-by(rule lower_semilattice.inf_le_fold_inf)(rule dual_semilattice)
+by(rule semilattice_inf.inf_le_fold_inf)(rule dual_semilattice)
lemma sup_le_fold_sup: "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> sup a b \<le> fold sup b A"
-by(rule lower_semilattice.fold_inf_le_inf)(rule dual_semilattice)
+by(rule semilattice_inf.fold_inf_le_inf)(rule dual_semilattice)
end
@@ -1486,7 +1486,7 @@
qed
lemma setsum_mono:
- assumes le: "\<And>i. i\<in>K \<Longrightarrow> f (i::'a) \<le> ((g i)::('b::{comm_monoid_add, pordered_ab_semigroup_add}))"
+ assumes le: "\<And>i. i\<in>K \<Longrightarrow> f (i::'a) \<le> ((g i)::('b::{comm_monoid_add, ordered_ab_semigroup_add}))"
shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
proof (cases "finite K")
case True
@@ -1505,7 +1505,7 @@
qed
lemma setsum_strict_mono:
- fixes f :: "'a \<Rightarrow> 'b::{pordered_cancel_ab_semigroup_add,comm_monoid_add}"
+ fixes f :: "'a \<Rightarrow> 'b::{ordered_cancel_ab_semigroup_add,comm_monoid_add}"
assumes "finite A" "A \<noteq> {}"
and "!!x. x:A \<Longrightarrow> f x < g x"
shows "setsum f A < setsum g A"
@@ -1534,7 +1534,7 @@
qed
lemma setsum_nonneg:
- assumes nn: "\<forall>x\<in>A. (0::'a::{pordered_ab_semigroup_add,comm_monoid_add}) \<le> f x"
+ assumes nn: "\<forall>x\<in>A. (0::'a::{ordered_ab_semigroup_add,comm_monoid_add}) \<le> f x"
shows "0 \<le> setsum f A"
proof (cases "finite A")
case True thus ?thesis using nn
@@ -1550,7 +1550,7 @@
qed
lemma setsum_nonpos:
- assumes np: "\<forall>x\<in>A. f x \<le> (0::'a::{pordered_ab_semigroup_add,comm_monoid_add})"
+ assumes np: "\<forall>x\<in>A. f x \<le> (0::'a::{ordered_ab_semigroup_add,comm_monoid_add})"
shows "setsum f A \<le> 0"
proof (cases "finite A")
case True thus ?thesis using np
@@ -1566,7 +1566,7 @@
qed
lemma setsum_mono2:
-fixes f :: "'a \<Rightarrow> 'b :: {pordered_ab_semigroup_add_imp_le,comm_monoid_add}"
+fixes f :: "'a \<Rightarrow> 'b :: {ordered_ab_semigroup_add_imp_le,comm_monoid_add}"
assumes fin: "finite B" and sub: "A \<subseteq> B" and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b"
shows "setsum f A \<le> setsum f B"
proof -
@@ -1580,7 +1580,7 @@
lemma setsum_mono3: "finite B ==> A <= B ==>
ALL x: B - A.
- 0 <= ((f x)::'a::{comm_monoid_add,pordered_ab_semigroup_add}) ==>
+ 0 <= ((f x)::'a::{comm_monoid_add,ordered_ab_semigroup_add}) ==>
setsum f A <= setsum f B"
apply (subgoal_tac "setsum f B = setsum f A + setsum f (B - A)")
apply (erule ssubst)
@@ -1640,7 +1640,7 @@
qed
lemma setsum_abs[iff]:
- fixes f :: "'a => ('b::pordered_ab_group_add_abs)"
+ fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) A"
proof (cases "finite A")
case True
@@ -1656,7 +1656,7 @@
qed
lemma setsum_abs_ge_zero[iff]:
- fixes f :: "'a => ('b::pordered_ab_group_add_abs)"
+ fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
shows "0 \<le> setsum (%i. abs(f i)) A"
proof (cases "finite A")
case True
@@ -1671,7 +1671,7 @@
qed
lemma abs_setsum_abs[simp]:
- fixes f :: "'a => ('b::pordered_ab_group_add_abs)"
+ fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
shows "abs (\<Sum>a\<in>A. abs(f a)) = (\<Sum>a\<in>A. abs(f a))"
proof (cases "finite A")
case True
@@ -1946,10 +1946,10 @@
done
lemma setprod_nonneg [rule_format]:
- "(ALL x: A. (0::'a::ordered_semidom) \<le> f x) --> 0 \<le> setprod f A"
+ "(ALL x: A. (0::'a::linordered_semidom) \<le> f x) --> 0 \<le> setprod f A"
by (cases "finite A", induct set: finite, simp_all add: mult_nonneg_nonneg)
-lemma setprod_pos [rule_format]: "(ALL x: A. (0::'a::ordered_semidom) < f x)
+lemma setprod_pos [rule_format]: "(ALL x: A. (0::'a::linordered_semidom) < f x)
--> 0 < setprod f A"
by (cases "finite A", induct set: finite, simp_all add: mult_pos_pos)
@@ -2289,7 +2289,7 @@
lemma setsum_bounded:
- assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> (K::'a::{semiring_1, pordered_ab_semigroup_add})"
+ assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> (K::'a::{semiring_1, ordered_ab_semigroup_add})"
shows "setsum f A \<le> of_nat(card A) * K"
proof (cases "finite A")
case True
@@ -2791,7 +2791,7 @@
over (non-empty) sets by means of @{text fold1}.
*}
-context lower_semilattice
+context semilattice_inf
begin
lemma below_fold1_iff:
@@ -2859,7 +2859,7 @@
apply(erule exE)
apply(rule order_trans)
apply(erule (1) fold1_belowI)
-apply(erule (1) lower_semilattice.fold1_belowI [OF dual_semilattice])
+apply(erule (1) semilattice_inf.fold1_belowI [OF dual_semilattice])
done
lemma sup_Inf_absorb [simp]:
@@ -2871,7 +2871,7 @@
lemma inf_Sup_absorb [simp]:
"finite A \<Longrightarrow> a \<in> A \<Longrightarrow> inf a (\<Squnion>\<^bsub>fin\<^esub>A) = a"
by (simp add: Sup_fin_def inf_absorb1
- lower_semilattice.fold1_belowI [OF dual_semilattice])
+ semilattice_inf.fold1_belowI [OF dual_semilattice])
end
@@ -2991,7 +2991,7 @@
proof qed (auto simp add: max_def)
lemma max_lattice:
- "lower_semilattice (op \<ge>) (op >) max"
+ "semilattice_inf (op \<ge>) (op >) max"
by (fact min_max.dual_semilattice)
lemma dual_max:
@@ -3158,7 +3158,7 @@
assumes "finite A" and "x \<in> A"
shows "x \<le> Max A"
proof -
- interpret lower_semilattice "op \<ge>" "op >" max
+ interpret semilattice_inf "op \<ge>" "op >" max
by (rule max_lattice)
from assms show ?thesis by (simp add: Max_def fold1_belowI)
qed
@@ -3172,7 +3172,7 @@
assumes "finite A" and "A \<noteq> {}"
shows "Max A \<le> x \<longleftrightarrow> (\<forall>a\<in>A. a \<le> x)"
proof -
- interpret lower_semilattice "op \<ge>" "op >" max
+ interpret semilattice_inf "op \<ge>" "op >" max
by (rule max_lattice)
from assms show ?thesis by (simp add: Max_def below_fold1_iff)
qed
@@ -3293,7 +3293,7 @@
end
-context ordered_ab_semigroup_add
+context linordered_ab_semigroup_add
begin
lemma add_Min_commute:
@@ -3347,12 +3347,12 @@
proof
qed auto
-lemma (in lower_semilattice) fun_left_comm_idem_inf:
+lemma (in semilattice_inf) fun_left_comm_idem_inf:
"fun_left_comm_idem inf"
proof
qed (auto simp add: inf_left_commute)
-lemma (in upper_semilattice) fun_left_comm_idem_sup:
+lemma (in semilattice_sup) fun_left_comm_idem_sup:
"fun_left_comm_idem sup"
proof
qed (auto simp add: sup_left_commute)
--- a/src/HOL/GCD.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/GCD.thy Mon Feb 08 11:13:30 2010 +0100
@@ -1445,12 +1445,12 @@
subsubsection {* The complete divisibility lattice *}
-interpretation gcd_semilattice_nat: lower_semilattice "op dvd" "(%m n::nat. m dvd n & ~ n dvd m)" gcd
+interpretation gcd_semilattice_nat: semilattice_inf "op dvd" "(%m n::nat. m dvd n & ~ n dvd m)" gcd
proof
case goal3 thus ?case by(metis gcd_unique_nat)
qed auto
-interpretation lcm_semilattice_nat: upper_semilattice "op dvd" "(%m n::nat. m dvd n & ~ n dvd m)" lcm
+interpretation lcm_semilattice_nat: semilattice_sup "op dvd" "(%m n::nat. m dvd n & ~ n dvd m)" lcm
proof
case goal3 thus ?case by(metis lcm_unique_nat)
qed auto
--- a/src/HOL/Import/HOL/real.imp Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Import/HOL/real.imp Mon Feb 08 11:13:30 2010 +0100
@@ -147,7 +147,7 @@
"REAL_LT_TOTAL" > "HOL4Compat.REAL_LT_TOTAL"
"REAL_LT_SUB_RADD" > "OrderedGroup.compare_rls_6"
"REAL_LT_SUB_LADD" > "OrderedGroup.compare_rls_7"
- "REAL_LT_RMUL_IMP" > "Ring_and_Field.ordered_semiring_strict_class.mult_strict_right_mono"
+ "REAL_LT_RMUL_IMP" > "Ring_and_Field.mult_strict_right_mono"
"REAL_LT_RMUL_0" > "HOL4Real.real.REAL_LT_RMUL_0"
"REAL_LT_RMUL" > "RealDef.real_mult_less_iff1"
"REAL_LT_REFL" > "Orderings.order_less_irrefl"
@@ -161,7 +161,7 @@
"REAL_LT_MULTIPLE" > "HOL4Real.real.REAL_LT_MULTIPLE"
"REAL_LT_MUL2" > "Ring_and_Field.mult_strict_mono'"
"REAL_LT_MUL" > "Ring_and_Field.mult_pos_pos"
- "REAL_LT_LMUL_IMP" > "Ring_and_Field.ordered_comm_semiring_strict_class.mult_strict_mono"
+ "REAL_LT_LMUL_IMP" > "Ring_and_Field.linordered_comm_semiring_strict_class.mult_strict_mono"
"REAL_LT_LMUL_0" > "HOL4Real.real.REAL_LT_LMUL_0"
"REAL_LT_LMUL" > "HOL4Real.real.REAL_LT_LMUL"
"REAL_LT_LE" > "Orderings.order_class.order_less_le"
@@ -188,7 +188,7 @@
"REAL_LT_ADD1" > "HOL4Real.real.REAL_LT_ADD1"
"REAL_LT_ADD" > "OrderedGroup.add_pos_pos"
"REAL_LT_1" > "HOL4Real.real.REAL_LT_1"
- "REAL_LT_01" > "Ring_and_Field.ordered_semidom_class.zero_less_one"
+ "REAL_LT_01" > "Ring_and_Field.zero_less_one"
"REAL_LTE_TRANS" > "Set.basic_trans_rules_24"
"REAL_LTE_TOTAL" > "HOL4Real.real.REAL_LTE_TOTAL"
"REAL_LTE_ANTSYM" > "HOL4Real.real.REAL_LTE_ANTSYM"
@@ -204,7 +204,7 @@
"REAL_LE_SUB_LADD" > "OrderedGroup.compare_rls_9"
"REAL_LE_SQUARE" > "Ring_and_Field.zero_le_square"
"REAL_LE_RNEG" > "OrderedGroup.le_eq_neg"
- "REAL_LE_RMUL_IMP" > "Ring_and_Field.pordered_semiring_class.mult_right_mono"
+ "REAL_LE_RMUL_IMP" > "Ring_and_Field.mult_right_mono"
"REAL_LE_RMUL" > "RealDef.real_mult_le_cancel_iff1"
"REAL_LE_REFL" > "Finite_Set.max.f_below.below_refl"
"REAL_LE_RDIV_EQ" > "Ring_and_Field.pos_le_divide_eq"
@@ -220,11 +220,11 @@
"REAL_LE_MUL" > "Ring_and_Field.mult_nonneg_nonneg"
"REAL_LE_LT" > "Orderings.order_le_less"
"REAL_LE_LNEG" > "RealDef.real_0_le_add_iff"
- "REAL_LE_LMUL_IMP" > "Ring_and_Field.pordered_comm_semiring_class.mult_mono"
+ "REAL_LE_LMUL_IMP" > "Ring_and_Field.mult_mono"
"REAL_LE_LMUL" > "RealDef.real_mult_le_cancel_iff2"
"REAL_LE_LDIV_EQ" > "Ring_and_Field.pos_divide_le_eq"
"REAL_LE_LDIV" > "Ring_and_Field.mult_imp_div_pos_le"
- "REAL_LE_LADD_IMP" > "OrderedGroup.pordered_ab_semigroup_add_class.add_left_mono"
+ "REAL_LE_LADD_IMP" > "OrderedGroup.add_left_mono"
"REAL_LE_LADD" > "OrderedGroup.add_le_cancel_left"
"REAL_LE_INV_EQ" > "Ring_and_Field.inverse_nonnegative_iff_nonnegative"
"REAL_LE_INV" > "HOL4Real.real.REAL_LE_INV"
--- a/src/HOL/Int.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Int.thy Mon Feb 08 11:13:30 2010 +0100
@@ -208,7 +208,7 @@
end
-instance int :: pordered_cancel_ab_semigroup_add
+instance int :: ordered_cancel_ab_semigroup_add
proof
fix i j k :: int
show "i \<le> j \<Longrightarrow> k + i \<le> k + j"
@@ -245,7 +245,7 @@
done
text{*The integers form an ordered integral domain*}
-instance int :: ordered_idom
+instance int :: linordered_idom
proof
fix i j k :: int
show "i < j \<Longrightarrow> 0 < k \<Longrightarrow> k * i < k * j"
@@ -256,7 +256,7 @@
by (simp only: zsgn_def)
qed
-instance int :: lordered_ring
+instance int :: lattice_ring
proof
fix k :: int
show "abs k = sup k (- k)"
@@ -331,7 +331,7 @@
end
-context ordered_idom
+context linordered_idom
begin
lemma of_int_le_iff [simp]:
@@ -370,8 +370,8 @@
end
-text{*Every @{text ordered_idom} has characteristic zero.*}
-subclass (in ordered_idom) ring_char_0 by intro_locales
+text{*Every @{text linordered_idom} has characteristic zero.*}
+subclass (in linordered_idom) ring_char_0 by intro_locales
lemma of_int_eq_id [simp]: "of_int = id"
proof
@@ -529,7 +529,7 @@
in theory @{text Ring_and_Field}.
But is it really better than just rewriting with @{text abs_if}?*}
lemma abs_split [arith_split,noatp]:
- "P(abs(a::'a::ordered_idom)) = ((0 \<le> a --> P a) & (a < 0 --> P(-a)))"
+ "P(abs(a::'a::linordered_idom)) = ((0 \<le> a --> P a) & (a < 0 --> P(-a)))"
by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
lemma negD: "(x \<Colon> int) < 0 \<Longrightarrow> \<exists>n. x = - (of_nat (Suc n))"
@@ -804,7 +804,7 @@
text {* Preliminaries *}
lemma even_less_0_iff:
- "a + a < 0 \<longleftrightarrow> a < (0::'a::ordered_idom)"
+ "a + a < 0 \<longleftrightarrow> a < (0::'a::linordered_idom)"
proof -
have "a + a < 0 \<longleftrightarrow> (1+1)*a < 0" by (simp add: left_distrib)
also have "(1+1)*a < 0 \<longleftrightarrow> a < 0"
@@ -1147,7 +1147,7 @@
subsubsection {* The Less-Than Relation *}
lemma double_less_0_iff:
- "(a + a < 0) = (a < (0::'a::ordered_idom))"
+ "(a + a < 0) = (a < (0::'a::linordered_idom))"
proof -
have "(a + a < 0) = ((1+1)*a < 0)" by (simp add: left_distrib)
also have "... = (a < 0)"
@@ -1180,7 +1180,7 @@
text {* Absolute value (@{term abs}) *}
lemma abs_number_of:
- "abs(number_of x::'a::{ordered_idom,number_ring}) =
+ "abs(number_of x::'a::{linordered_idom,number_ring}) =
(if number_of x < (0::'a) then -number_of x else number_of x)"
by (simp add: abs_if)
@@ -1214,11 +1214,11 @@
text {* Simplification of relational operations *}
lemma less_number_of [simp]:
- "(number_of x::'a::{ordered_idom,number_ring}) < number_of y \<longleftrightarrow> x < y"
+ "(number_of x::'a::{linordered_idom,number_ring}) < number_of y \<longleftrightarrow> x < y"
unfolding number_of_eq by (rule of_int_less_iff)
lemma le_number_of [simp]:
- "(number_of x::'a::{ordered_idom,number_ring}) \<le> number_of y \<longleftrightarrow> x \<le> y"
+ "(number_of x::'a::{linordered_idom,number_ring}) \<le> number_of y \<longleftrightarrow> x \<le> y"
unfolding number_of_eq by (rule of_int_le_iff)
lemma eq_number_of [simp]:
@@ -1362,7 +1362,7 @@
lemma Ints_odd_less_0:
assumes in_Ints: "a \<in> Ints"
- shows "(1 + a + a < 0) = (a < (0::'a::ordered_idom))"
+ shows "(1 + a + a < 0) = (a < (0::'a::linordered_idom))"
proof -
from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
then obtain z where a: "a = of_int z" ..
@@ -1519,11 +1519,11 @@
finally show ?thesis .
qed
-lemma abs_minus_one [simp]: "abs (-1) = (1::'a::{ordered_idom,number_ring})"
+lemma abs_minus_one [simp]: "abs (-1) = (1::'a::{linordered_idom,number_ring})"
by (simp add: abs_if)
lemma abs_power_minus_one [simp]:
- "abs(-1 ^ n) = (1::'a::{ordered_idom,number_ring})"
+ "abs(-1 ^ n) = (1::'a::{linordered_idom,number_ring})"
by (simp add: power_abs)
lemma of_int_number_of_eq [simp]:
@@ -1906,12 +1906,12 @@
text{*To Simplify Inequalities Where One Side is the Constant 1*}
lemma less_minus_iff_1 [simp,noatp]:
- fixes b::"'b::{ordered_idom,number_ring}"
+ fixes b::"'b::{linordered_idom,number_ring}"
shows "(1 < - b) = (b < -1)"
by auto
lemma le_minus_iff_1 [simp,noatp]:
- fixes b::"'b::{ordered_idom,number_ring}"
+ fixes b::"'b::{linordered_idom,number_ring}"
shows "(1 \<le> - b) = (b \<le> -1)"
by auto
@@ -1921,12 +1921,12 @@
by (subst equation_minus_iff, auto)
lemma minus_less_iff_1 [simp,noatp]:
- fixes a::"'b::{ordered_idom,number_ring}"
+ fixes a::"'b::{linordered_idom,number_ring}"
shows "(- a < 1) = (-1 < a)"
by auto
lemma minus_le_iff_1 [simp,noatp]:
- fixes a::"'b::{ordered_idom,number_ring}"
+ fixes a::"'b::{linordered_idom,number_ring}"
shows "(- a \<le> 1) = (-1 \<le> a)"
by auto
@@ -1990,7 +1990,7 @@
by (simp add: divide_inverse inverse_minus_eq)
lemma half_gt_zero_iff:
- "(0 < r/2) = (0 < (r::'a::{ordered_field,division_by_zero,number_ring}))"
+ "(0 < r/2) = (0 < (r::'a::{linordered_field,division_by_zero,number_ring}))"
by auto
lemmas half_gt_zero [simp] = half_gt_zero_iff [THEN iffD2, standard]
--- a/src/HOL/IsaMakefile Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/IsaMakefile Mon Feb 08 11:13:30 2010 +0100
@@ -394,7 +394,7 @@
Library/Library/document/root.tex Library/Library/document/root.bib \
Library/Transitive_Closure_Table.thy Library/While_Combinator.thy \
Library/Product_ord.thy Library/Char_nat.thy \
- Library/Char_ord.thy Library/Option_ord.thy \
+ Library/Structure_Syntax.thy \
Library/Sublist_Order.thy Library/List_lexord.thy \
Library/Coinductive_List.thy Library/AssocList.thy \
Library/Formal_Power_Series.thy Library/Binomial.thy \
--- a/src/HOL/Lattices.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Lattices.thy Mon Feb 08 11:13:30 2010 +0100
@@ -16,13 +16,13 @@
top ("\<top>") and
bot ("\<bottom>")
-class lower_semilattice = order +
+class semilattice_inf = order +
fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<sqinter>" 70)
assumes inf_le1 [simp]: "x \<sqinter> y \<sqsubseteq> x"
and inf_le2 [simp]: "x \<sqinter> y \<sqsubseteq> y"
and inf_greatest: "x \<sqsubseteq> y \<Longrightarrow> x \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> y \<sqinter> z"
-class upper_semilattice = order +
+class semilattice_sup = order +
fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<squnion>" 65)
assumes sup_ge1 [simp]: "x \<sqsubseteq> x \<squnion> y"
and sup_ge2 [simp]: "y \<sqsubseteq> x \<squnion> y"
@@ -32,18 +32,18 @@
text {* Dual lattice *}
lemma dual_semilattice:
- "lower_semilattice (op \<ge>) (op >) sup"
-by (rule lower_semilattice.intro, rule dual_order)
+ "semilattice_inf (op \<ge>) (op >) sup"
+by (rule semilattice_inf.intro, rule dual_order)
(unfold_locales, simp_all add: sup_least)
end
-class lattice = lower_semilattice + upper_semilattice
+class lattice = semilattice_inf + semilattice_sup
subsubsection {* Intro and elim rules*}
-context lower_semilattice
+context semilattice_inf
begin
lemma le_infI1:
@@ -69,13 +69,13 @@
by (auto intro: le_infI1 antisym dest: eq_iff [THEN iffD1])
lemma mono_inf:
- fixes f :: "'a \<Rightarrow> 'b\<Colon>lower_semilattice"
+ fixes f :: "'a \<Rightarrow> 'b\<Colon>semilattice_inf"
shows "mono f \<Longrightarrow> f (A \<sqinter> B) \<sqsubseteq> f A \<sqinter> f B"
by (auto simp add: mono_def intro: Lattices.inf_greatest)
end
-context upper_semilattice
+context semilattice_sup
begin
lemma le_supI1:
@@ -103,7 +103,7 @@
by (auto intro: le_supI2 antisym dest: eq_iff [THEN iffD1])
lemma mono_sup:
- fixes f :: "'a \<Rightarrow> 'b\<Colon>upper_semilattice"
+ fixes f :: "'a \<Rightarrow> 'b\<Colon>semilattice_sup"
shows "mono f \<Longrightarrow> f A \<squnion> f B \<sqsubseteq> f (A \<squnion> B)"
by (auto simp add: mono_def intro: Lattices.sup_least)
@@ -112,7 +112,7 @@
subsubsection {* Equational laws *}
-sublocale lower_semilattice < inf!: semilattice inf
+sublocale semilattice_inf < inf!: semilattice inf
proof
fix a b c
show "(a \<sqinter> b) \<sqinter> c = a \<sqinter> (b \<sqinter> c)"
@@ -123,7 +123,7 @@
by (rule antisym) auto
qed
-context lower_semilattice
+context semilattice_inf
begin
lemma inf_assoc: "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
@@ -151,7 +151,7 @@
end
-sublocale upper_semilattice < sup!: semilattice sup
+sublocale semilattice_sup < sup!: semilattice sup
proof
fix a b c
show "(a \<squnion> b) \<squnion> c = a \<squnion> (b \<squnion> c)"
@@ -162,7 +162,7 @@
by (rule antisym) auto
qed
-context upper_semilattice
+context semilattice_sup
begin
lemma sup_assoc: "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
@@ -195,7 +195,7 @@
lemma dual_lattice:
"lattice (op \<ge>) (op >) sup inf"
- by (rule lattice.intro, rule dual_semilattice, rule upper_semilattice.intro, rule dual_order)
+ by (rule lattice.intro, rule dual_semilattice, rule semilattice_sup.intro, rule dual_order)
(unfold_locales, auto)
lemma inf_sup_absorb: "x \<sqinter> (x \<squnion> y) = x"
@@ -246,7 +246,7 @@
subsubsection {* Strict order *}
-context lower_semilattice
+context semilattice_inf
begin
lemma less_infI1:
@@ -259,13 +259,13 @@
end
-context upper_semilattice
+context semilattice_sup
begin
lemma less_supI1:
"x \<sqsubset> a \<Longrightarrow> x \<sqsubset> a \<squnion> b"
proof -
- interpret dual: lower_semilattice "op \<ge>" "op >" sup
+ interpret dual: semilattice_inf "op \<ge>" "op >" sup
by (fact dual_semilattice)
assume "x \<sqsubset> a"
then show "x \<sqsubset> a \<squnion> b"
@@ -275,7 +275,7 @@
lemma less_supI2:
"x \<sqsubset> b \<Longrightarrow> x \<sqsubset> a \<squnion> b"
proof -
- interpret dual: lower_semilattice "op \<ge>" "op >" sup
+ interpret dual: semilattice_inf "op \<ge>" "op >" sup
by (fact dual_semilattice)
assume "x \<sqsubset> b"
then show "x \<sqsubset> a \<squnion> b"
@@ -492,7 +492,7 @@
subsection {* Uniqueness of inf and sup *}
-lemma (in lower_semilattice) inf_unique:
+lemma (in semilattice_inf) inf_unique:
fixes f (infixl "\<triangle>" 70)
assumes le1: "\<And>x y. x \<triangle> y \<sqsubseteq> x" and le2: "\<And>x y. x \<triangle> y \<sqsubseteq> y"
and greatest: "\<And>x y z. x \<sqsubseteq> y \<Longrightarrow> x \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> y \<triangle> z"
@@ -504,7 +504,7 @@
show "x \<sqinter> y \<sqsubseteq> x \<triangle> y" by (rule leI) simp_all
qed
-lemma (in upper_semilattice) sup_unique:
+lemma (in semilattice_sup) sup_unique:
fixes f (infixl "\<nabla>" 70)
assumes ge1 [simp]: "\<And>x y. x \<sqsubseteq> x \<nabla> y" and ge2: "\<And>x y. y \<sqsubseteq> x \<nabla> y"
and least: "\<And>x y z. y \<sqsubseteq> x \<Longrightarrow> z \<sqsubseteq> x \<Longrightarrow> y \<nabla> z \<sqsubseteq> x"
@@ -527,10 +527,10 @@
by (auto simp add: min_def max_def)
qed (auto simp add: min_def max_def not_le less_imp_le)
-lemma inf_min: "inf = (min \<Colon> 'a\<Colon>{lower_semilattice, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)"
+lemma inf_min: "inf = (min \<Colon> 'a\<Colon>{semilattice_inf, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)"
by (rule ext)+ (auto intro: antisym)
-lemma sup_max: "sup = (max \<Colon> 'a\<Colon>{upper_semilattice, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)"
+lemma sup_max: "sup = (max \<Colon> 'a\<Colon>{semilattice_sup, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)"
by (rule ext)+ (auto intro: antisym)
lemmas le_maxI1 = min_max.sup_ge1
--- a/src/HOL/Library/Abstract_Rat.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Library/Abstract_Rat.thy Mon Feb 08 11:13:30 2010 +0100
@@ -332,7 +332,7 @@
lemma Ndiv[simp]: "INum (x \<div>\<^sub>N y) = INum x / (INum y ::'a :: {ring_char_0, division_by_zero,field})" by (simp add: Ndiv_def)
lemma Nlt0_iff[simp]: assumes nx: "isnormNum x"
- shows "((INum x :: 'a :: {ring_char_0,division_by_zero,ordered_field})< 0) = 0>\<^sub>N x "
+ shows "((INum x :: 'a :: {ring_char_0,division_by_zero,linordered_field})< 0) = 0>\<^sub>N x "
proof-
have " \<exists> a b. x = (a,b)" by simp
then obtain a b where x[simp]:"x = (a,b)" by blast
@@ -345,7 +345,7 @@
qed
lemma Nle0_iff[simp]:assumes nx: "isnormNum x"
- shows "((INum x :: 'a :: {ring_char_0,division_by_zero,ordered_field}) \<le> 0) = 0\<ge>\<^sub>N x"
+ shows "((INum x :: 'a :: {ring_char_0,division_by_zero,linordered_field}) \<le> 0) = 0\<ge>\<^sub>N x"
proof-
have " \<exists> a b. x = (a,b)" by simp
then obtain a b where x[simp]:"x = (a,b)" by blast
@@ -357,7 +357,7 @@
ultimately show ?thesis by blast
qed
-lemma Ngt0_iff[simp]:assumes nx: "isnormNum x" shows "((INum x :: 'a :: {ring_char_0,division_by_zero,ordered_field})> 0) = 0<\<^sub>N x"
+lemma Ngt0_iff[simp]:assumes nx: "isnormNum x" shows "((INum x :: 'a :: {ring_char_0,division_by_zero,linordered_field})> 0) = 0<\<^sub>N x"
proof-
have " \<exists> a b. x = (a,b)" by simp
then obtain a b where x[simp]:"x = (a,b)" by blast
@@ -369,7 +369,7 @@
ultimately show ?thesis by blast
qed
lemma Nge0_iff[simp]:assumes nx: "isnormNum x"
- shows "((INum x :: 'a :: {ring_char_0,division_by_zero,ordered_field}) \<ge> 0) = 0\<le>\<^sub>N x"
+ shows "((INum x :: 'a :: {ring_char_0,division_by_zero,linordered_field}) \<ge> 0) = 0\<le>\<^sub>N x"
proof-
have " \<exists> a b. x = (a,b)" by simp
then obtain a b where x[simp]:"x = (a,b)" by blast
@@ -382,7 +382,7 @@
qed
lemma Nlt_iff[simp]: assumes nx: "isnormNum x" and ny: "isnormNum y"
- shows "((INum x :: 'a :: {ring_char_0,division_by_zero,ordered_field}) < INum y) = (x <\<^sub>N y)"
+ shows "((INum x :: 'a :: {ring_char_0,division_by_zero,linordered_field}) < INum y) = (x <\<^sub>N y)"
proof-
let ?z = "0::'a"
have "((INum x ::'a) < INum y) = (INum (x -\<^sub>N y) < ?z)" using nx ny by simp
@@ -391,7 +391,7 @@
qed
lemma Nle_iff[simp]: assumes nx: "isnormNum x" and ny: "isnormNum y"
- shows "((INum x :: 'a :: {ring_char_0,division_by_zero,ordered_field})\<le> INum y) = (x \<le>\<^sub>N y)"
+ shows "((INum x :: 'a :: {ring_char_0,division_by_zero,linordered_field})\<le> INum y) = (x \<le>\<^sub>N y)"
proof-
have "((INum x ::'a) \<le> INum y) = (INum (x -\<^sub>N y) \<le> (0::'a))" using nx ny by simp
also have "\<dots> = (0\<ge>\<^sub>N (x -\<^sub>N y))" using Nle0_iff[OF Nsub_normN[OF ny]] by simp
--- a/src/HOL/Library/BigO.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Library/BigO.thy Mon Feb 08 11:13:30 2010 +0100
@@ -38,11 +38,11 @@
subsection {* Definitions *}
definition
- bigo :: "('a => 'b::ordered_idom) => ('a => 'b) set" ("(1O'(_'))") where
+ bigo :: "('a => 'b::linordered_idom) => ('a => 'b) set" ("(1O'(_'))") where
"O(f::('a => 'b)) =
{h. EX c. ALL x. abs (h x) <= c * abs (f x)}"
-lemma bigo_pos_const: "(EX (c::'a::ordered_idom).
+lemma bigo_pos_const: "(EX (c::'a::linordered_idom).
ALL x. (abs (h x)) <= (c * (abs (f x))))
= (EX c. 0 < c & (ALL x. (abs(h x)) <= (c * (abs (f x)))))"
apply auto
@@ -352,7 +352,7 @@
done
lemma bigo_mult5: "ALL x. f x ~= 0 ==>
- O(f * g) <= (f::'a => ('b::ordered_field)) *o O(g)"
+ O(f * g) <= (f::'a => ('b::linordered_field)) *o O(g)"
proof -
assume "ALL x. f x ~= 0"
show "O(f * g) <= f *o O(g)"
@@ -381,14 +381,14 @@
qed
lemma bigo_mult6: "ALL x. f x ~= 0 ==>
- O(f * g) = (f::'a => ('b::ordered_field)) *o O(g)"
+ O(f * g) = (f::'a => ('b::linordered_field)) *o O(g)"
apply (rule equalityI)
apply (erule bigo_mult5)
apply (rule bigo_mult2)
done
lemma bigo_mult7: "ALL x. f x ~= 0 ==>
- O(f * g) <= O(f::'a => ('b::ordered_field)) \<otimes> O(g)"
+ O(f * g) <= O(f::'a => ('b::linordered_field)) \<otimes> O(g)"
apply (subst bigo_mult6)
apply assumption
apply (rule set_times_mono3)
@@ -396,7 +396,7 @@
done
lemma bigo_mult8: "ALL x. f x ~= 0 ==>
- O(f * g) = O(f::'a => ('b::ordered_field)) \<otimes> O(g)"
+ O(f * g) = O(f::'a => ('b::linordered_field)) \<otimes> O(g)"
apply (rule equalityI)
apply (erule bigo_mult7)
apply (rule bigo_mult)
@@ -481,16 +481,16 @@
apply (rule bigo_const1)
done
-lemma bigo_const3: "(c::'a::ordered_field) ~= 0 ==> (%x. 1) : O(%x. c)"
+lemma bigo_const3: "(c::'a::linordered_field) ~= 0 ==> (%x. 1) : O(%x. c)"
apply (simp add: bigo_def)
apply (rule_tac x = "abs(inverse c)" in exI)
apply (simp add: abs_mult [symmetric])
done
-lemma bigo_const4: "(c::'a::ordered_field) ~= 0 ==> O(%x. 1) <= O(%x. c)"
+lemma bigo_const4: "(c::'a::linordered_field) ~= 0 ==> O(%x. 1) <= O(%x. c)"
by (rule bigo_elt_subset, rule bigo_const3, assumption)
-lemma bigo_const [simp]: "(c::'a::ordered_field) ~= 0 ==>
+lemma bigo_const [simp]: "(c::'a::linordered_field) ~= 0 ==>
O(%x. c) = O(%x. 1)"
by (rule equalityI, rule bigo_const2, rule bigo_const4, assumption)
@@ -503,21 +503,21 @@
lemma bigo_const_mult2: "O(%x. c * f x) <= O(f)"
by (rule bigo_elt_subset, rule bigo_const_mult1)
-lemma bigo_const_mult3: "(c::'a::ordered_field) ~= 0 ==> f : O(%x. c * f x)"
+lemma bigo_const_mult3: "(c::'a::linordered_field) ~= 0 ==> f : O(%x. c * f x)"
apply (simp add: bigo_def)
apply (rule_tac x = "abs(inverse c)" in exI)
apply (simp add: abs_mult [symmetric] mult_assoc [symmetric])
done
-lemma bigo_const_mult4: "(c::'a::ordered_field) ~= 0 ==>
+lemma bigo_const_mult4: "(c::'a::linordered_field) ~= 0 ==>
O(f) <= O(%x. c * f x)"
by (rule bigo_elt_subset, rule bigo_const_mult3, assumption)
-lemma bigo_const_mult [simp]: "(c::'a::ordered_field) ~= 0 ==>
+lemma bigo_const_mult [simp]: "(c::'a::linordered_field) ~= 0 ==>
O(%x. c * f x) = O(f)"
by (rule equalityI, rule bigo_const_mult2, erule bigo_const_mult4)
-lemma bigo_const_mult5 [simp]: "(c::'a::ordered_field) ~= 0 ==>
+lemma bigo_const_mult5 [simp]: "(c::'a::linordered_field) ~= 0 ==>
(%x. c) *o O(f) = O(f)"
apply (auto del: subsetI)
apply (rule order_trans)
@@ -688,7 +688,7 @@
apply assumption+
done
-lemma bigo_useful_const_mult: "(c::'a::ordered_field) ~= 0 ==>
+lemma bigo_useful_const_mult: "(c::'a::linordered_field) ~= 0 ==>
(%x. c) * f =o O(h) ==> f =o O(h)"
apply (rule subsetD)
apply (subgoal_tac "(%x. 1 / c) *o O(h) <= O(h)")
@@ -733,7 +733,7 @@
subsection {* Less than or equal to *}
definition
- lesso :: "('a => 'b::ordered_idom) => ('a => 'b) => ('a => 'b)"
+ lesso :: "('a => 'b::linordered_idom) => ('a => 'b) => ('a => 'b)"
(infixl "<o" 70) where
"f <o g = (%x. max (f x - g x) 0)"
@@ -833,7 +833,7 @@
apply (simp add: algebra_simps)
done
-lemma bigo_lesso4: "f <o g =o O(k::'a=>'b::ordered_field) ==>
+lemma bigo_lesso4: "f <o g =o O(k::'a=>'b::linordered_field) ==>
g =o h +o O(k) ==> f <o h =o O(k)"
apply (unfold lesso_def)
apply (drule set_plus_imp_minus)
--- a/src/HOL/Library/Kleene_Algebra.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Library/Kleene_Algebra.thy Mon Feb 08 11:13:30 2010 +0100
@@ -72,7 +72,7 @@
class pre_kleene = semiring_1 + order_by_add
begin
-subclass pordered_semiring proof
+subclass ordered_semiring proof
fix x y z :: 'a
assume "x \<le> y"
--- a/src/HOL/Library/Library.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Library/Library.thy Mon Feb 08 11:13:30 2010 +0100
@@ -50,6 +50,7 @@
RBT
SML_Quickcheck
State_Monad
+ Structure_Syntax
Sum_Of_Squares
Transitive_Closure_Table
Univ_Poly
--- a/src/HOL/Library/Multiset.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Library/Multiset.thy Mon Feb 08 11:13:30 2010 +0100
@@ -415,11 +415,11 @@
mset_le_trans simp: mset_less_def)
interpretation mset_order_cancel_semigroup:
- pordered_cancel_ab_semigroup_add "op +" "op \<le>#" "op <#"
+ ordered_cancel_ab_semigroup_add "op +" "op \<le>#" "op <#"
proof qed (erule mset_le_mono_add [OF mset_le_refl])
interpretation mset_order_semigroup_cancel:
- pordered_ab_semigroup_add_imp_le "op +" "op \<le>#" "op <#"
+ ordered_ab_semigroup_add_imp_le "op +" "op \<le>#" "op <#"
proof qed simp
lemma mset_lessD: "A \<subset># B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
@@ -1348,7 +1348,7 @@
lemma union_upper2: "B <= A + (B::'a::order multiset)"
by (subst add_commute) (rule union_upper1)
-instance multiset :: (order) pordered_ab_semigroup_add
+instance multiset :: (order) ordered_ab_semigroup_add
apply intro_classes
apply (erule union_le_mono[OF mult_le_refl])
done
--- a/src/HOL/Library/Nat_Infinity.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Library/Nat_Infinity.thy Mon Feb 08 11:13:30 2010 +0100
@@ -234,7 +234,7 @@
subsection {* Ordering *}
-instantiation inat :: ordered_ab_semigroup_add
+instantiation inat :: linordered_ab_semigroup_add
begin
definition
@@ -268,7 +268,7 @@
end
-instance inat :: pordered_comm_semiring
+instance inat :: ordered_comm_semiring
proof
fix a b c :: inat
assume "a \<le> b" and "0 \<le> c"
--- a/src/HOL/Library/Polynomial.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Library/Polynomial.thy Mon Feb 08 11:13:30 2010 +0100
@@ -706,7 +706,7 @@
subsection {* Polynomials form an ordered integral domain *}
definition
- pos_poly :: "'a::ordered_idom poly \<Rightarrow> bool"
+ pos_poly :: "'a::linordered_idom poly \<Rightarrow> bool"
where
"pos_poly p \<longleftrightarrow> 0 < coeff p (degree p)"
@@ -732,7 +732,7 @@
lemma pos_poly_total: "p = 0 \<or> pos_poly p \<or> pos_poly (- p)"
by (induct p) (auto simp add: pos_poly_pCons)
-instantiation poly :: (ordered_idom) ordered_idom
+instantiation poly :: (linordered_idom) linordered_idom
begin
definition
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Structure_Syntax.thy Mon Feb 08 11:13:30 2010 +0100
@@ -0,0 +1,14 @@
+(* Author: Florian Haftmann, TU Muenchen *)
+
+header {* Index syntax for structures *}
+
+theory Structure_Syntax
+imports Pure
+begin
+
+syntax
+ "_index1" :: index ("\<^sub>1")
+translations
+ (index) "\<^sub>1" => (index) "\<^bsub>\<struct>\<^esub>"
+
+end
--- a/src/HOL/Library/Univ_Poly.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Library/Univ_Poly.thy Mon Feb 08 11:13:30 2010 +0100
@@ -990,7 +990,7 @@
text{*bound for polynomial.*}
-lemma poly_mono: "abs(x) \<le> k ==> abs(poly p (x::'a::{ordered_idom})) \<le> poly (map abs p) k"
+lemma poly_mono: "abs(x) \<le> k ==> abs(poly p (x::'a::{linordered_idom})) \<le> poly (map abs p) k"
apply (induct "p", auto)
apply (rule_tac y = "abs a + abs (x * poly p x)" in order_trans)
apply (rule abs_triangle_ineq)
--- a/src/HOL/Library/positivstellensatz.ML Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Library/positivstellensatz.ML Mon Feb 08 11:13:30 2010 +0100
@@ -275,7 +275,7 @@
"((a + b + min x y + c > r) = (a + b + x + c > r & a + b + y + c > r))"
by auto};
-val abs_split' = @{lemma "P (abs (x::'a::ordered_idom)) == (x >= 0 & P x | x < 0 & P (-x))"
+val abs_split' = @{lemma "P (abs (x::'a::linordered_idom)) == (x >= 0 & P x | x < 0 & P (-x))"
by (atomize (full)) (auto split add: abs_split)};
val max_split = @{lemma "P (max x y) == ((x::'a::linorder) <= y & P y | x > y & P x)"
--- a/src/HOL/List.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/List.thy Mon Feb 08 11:13:30 2010 +0100
@@ -2500,12 +2500,12 @@
by (induct xs, simp_all add: algebra_simps)
lemma listsum_abs:
- fixes xs :: "'a::pordered_ab_group_add_abs list"
+ fixes xs :: "'a::ordered_ab_group_add_abs list"
shows "\<bar>listsum xs\<bar> \<le> listsum (map abs xs)"
by (induct xs, simp, simp add: order_trans [OF abs_triangle_ineq])
lemma listsum_mono:
- fixes f g :: "'a \<Rightarrow> 'b::{comm_monoid_add, pordered_ab_semigroup_add}"
+ fixes f g :: "'a \<Rightarrow> 'b::{comm_monoid_add, ordered_ab_semigroup_add}"
shows "(\<And>x. x \<in> set xs \<Longrightarrow> f x \<le> g x) \<Longrightarrow> (\<Sum>x\<leftarrow>xs. f x) \<le> (\<Sum>x\<leftarrow>xs. g x)"
by (induct xs, simp, simp add: add_mono)
--- a/src/HOL/Matrix/ComputeNumeral.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Matrix/ComputeNumeral.thy Mon Feb 08 11:13:30 2010 +0100
@@ -109,22 +109,22 @@
lemmas compute_natarith = bitarith natnorm natsuc natadd natsub natmul nateq natless natle natfac.simps
-lemma number_eq: "(((number_of x)::'a::{number_ring, ordered_idom}) = (number_of y)) = (x = y)"
+lemma number_eq: "(((number_of x)::'a::{number_ring, linordered_idom}) = (number_of y)) = (x = y)"
unfolding number_of_eq
apply simp
done
-lemma number_le: "(((number_of x)::'a::{number_ring, ordered_idom}) \<le> (number_of y)) = (x \<le> y)"
+lemma number_le: "(((number_of x)::'a::{number_ring, linordered_idom}) \<le> (number_of y)) = (x \<le> y)"
unfolding number_of_eq
apply simp
done
-lemma number_less: "(((number_of x)::'a::{number_ring, ordered_idom}) < (number_of y)) = (x < y)"
+lemma number_less: "(((number_of x)::'a::{number_ring, linordered_idom}) < (number_of y)) = (x < y)"
unfolding number_of_eq
apply simp
done
-lemma number_diff: "((number_of x)::'a::{number_ring, ordered_idom}) - number_of y = number_of (x + (- y))"
+lemma number_diff: "((number_of x)::'a::{number_ring, linordered_idom}) - number_of y = number_of (x + (- y))"
apply (subst diff_number_of_eq)
apply simp
done
--- a/src/HOL/Matrix/LP.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Matrix/LP.thy Mon Feb 08 11:13:30 2010 +0100
@@ -8,7 +8,7 @@
lemma linprog_dual_estimate:
assumes
- "A * x \<le> (b::'a::lordered_ring)"
+ "A * x \<le> (b::'a::lattice_ring)"
"0 \<le> y"
"abs (A - A') \<le> \<delta>A"
"b \<le> b'"
@@ -57,7 +57,7 @@
lemma le_ge_imp_abs_diff_1:
assumes
- "A1 <= (A::'a::lordered_ring)"
+ "A1 <= (A::'a::lattice_ring)"
"A <= A2"
shows "abs (A-A1) <= A2-A1"
proof -
@@ -72,7 +72,7 @@
lemma mult_le_prts:
assumes
- "a1 <= (a::'a::lordered_ring)"
+ "a1 <= (a::'a::lattice_ring)"
"a <= a2"
"b1 <= b"
"b <= b2"
@@ -120,7 +120,7 @@
lemma mult_le_dual_prts:
assumes
- "A * x \<le> (b::'a::lordered_ring)"
+ "A * x \<le> (b::'a::lattice_ring)"
"0 \<le> y"
"A1 \<le> A"
"A \<le> A2"
--- a/src/HOL/Matrix/Matrix.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Matrix/Matrix.thy Mon Feb 08 11:13:30 2010 +0100
@@ -1545,7 +1545,7 @@
by (simp add: plus_matrix_def diff_matrix_def minus_matrix_def Rep_matrix_inject[symmetric] ext)
qed
-instance matrix :: (pordered_ab_group_add) pordered_ab_group_add
+instance matrix :: (ordered_ab_group_add) ordered_ab_group_add
proof
fix A B C :: "'a matrix"
assume "A <= B"
@@ -1556,8 +1556,8 @@
done
qed
-instance matrix :: (lordered_ab_group_add) lordered_ab_group_add_meet ..
-instance matrix :: (lordered_ab_group_add) lordered_ab_group_add_join ..
+instance matrix :: (lattice_ab_group_add) semilattice_inf_ab_group_add ..
+instance matrix :: (lattice_ab_group_add) semilattice_sup_ab_group_add ..
instance matrix :: (semiring_0) semiring_0
proof
@@ -1583,7 +1583,7 @@
instance matrix :: (ring) ring ..
-instance matrix :: (pordered_ring) pordered_ring
+instance matrix :: (ordered_ring) ordered_ring
proof
fix A B C :: "'a matrix"
assume a: "A \<le> B"
@@ -1600,9 +1600,9 @@
done
qed
-instance matrix :: (lordered_ring) lordered_ring
+instance matrix :: (lattice_ring) lattice_ring
proof
- fix A B C :: "('a :: lordered_ring) matrix"
+ fix A B C :: "('a :: lattice_ring) matrix"
show "abs A = sup A (-A)"
by (simp add: abs_matrix_def)
qed
@@ -1738,7 +1738,7 @@
by auto
lemma Rep_matrix_zero_imp_mult_zero:
- "! j i k. (Rep_matrix A j k = 0) | (Rep_matrix B k i) = 0 \<Longrightarrow> A * B = (0::('a::lordered_ring) matrix)"
+ "! j i k. (Rep_matrix A j k = 0) | (Rep_matrix B k i) = 0 \<Longrightarrow> A * B = (0::('a::lattice_ring) matrix)"
apply (subst Rep_matrix_inject[symmetric])
apply (rule ext)+
apply (auto simp add: Rep_matrix_mult foldseq_zero zero_imp_mult_zero)
@@ -1803,7 +1803,7 @@
lemma Rep_minus[simp]: "Rep_matrix (-(A::_::group_add)) x y = - (Rep_matrix A x y)"
by (simp add: minus_matrix_def)
-lemma Rep_abs[simp]: "Rep_matrix (abs (A::_::lordered_ab_group_add)) x y = abs (Rep_matrix A x y)"
+lemma Rep_abs[simp]: "Rep_matrix (abs (A::_::lattice_ab_group_add)) x y = abs (Rep_matrix A x y)"
by (simp add: abs_lattice sup_matrix_def)
end
--- a/src/HOL/Matrix/SparseMatrix.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Matrix/SparseMatrix.thy Mon Feb 08 11:13:30 2010 +0100
@@ -103,7 +103,7 @@
"minus_spvec [] = []"
| "minus_spvec (a#as) = (fst a, -(snd a))#(minus_spvec as)"
-primrec abs_spvec :: "('a::lordered_ab_group_add_abs) spvec \<Rightarrow> 'a spvec" where
+primrec abs_spvec :: "('a::lattice_ab_group_add_abs) spvec \<Rightarrow> 'a spvec" where
"abs_spvec [] = []"
| "abs_spvec (a#as) = (fst a, abs (snd a))#(abs_spvec as)"
@@ -116,12 +116,12 @@
apply simp
done
-instance matrix :: (lordered_ab_group_add_abs) lordered_ab_group_add_abs
+instance matrix :: (lattice_ab_group_add_abs) lattice_ab_group_add_abs
apply default
unfolding abs_matrix_def .. (*FIXME move*)
lemma sparse_row_vector_abs:
- "sorted_spvec (v :: 'a::lordered_ring spvec) \<Longrightarrow> sparse_row_vector (abs_spvec v) = abs (sparse_row_vector v)"
+ "sorted_spvec (v :: 'a::lattice_ring spvec) \<Longrightarrow> sparse_row_vector (abs_spvec v) = abs (sparse_row_vector v)"
apply (induct v)
apply simp_all
apply (frule_tac sorted_spvec_cons1, simp)
@@ -174,7 +174,7 @@
lemma addmult_spvec_empty2[simp]: "addmult_spvec y a [] = a"
by (induct a) auto
-lemma sparse_row_vector_map: "(! x y. f (x+y) = (f x) + (f y)) \<Longrightarrow> (f::'a\<Rightarrow>('a::lordered_ring)) 0 = 0 \<Longrightarrow>
+lemma sparse_row_vector_map: "(! x y. f (x+y) = (f x) + (f y)) \<Longrightarrow> (f::'a\<Rightarrow>('a::lattice_ring)) 0 = 0 \<Longrightarrow>
sparse_row_vector (map (% x. (fst x, f (snd x))) a) = apply_matrix f (sparse_row_vector a)"
apply (induct a)
apply (simp_all add: apply_matrix_add)
@@ -185,7 +185,7 @@
apply (simp_all add: smult_spvec_cons scalar_mult_add)
done
-lemma sparse_row_vector_addmult_spvec: "sparse_row_vector (addmult_spvec (y::'a::lordered_ring) a b) =
+lemma sparse_row_vector_addmult_spvec: "sparse_row_vector (addmult_spvec (y::'a::lattice_ring) a b) =
(sparse_row_vector a) + (scalar_mult y (sparse_row_vector b))"
apply (induct y a b rule: addmult_spvec.induct)
apply (simp add: scalar_mult_add smult_spvec_cons sparse_row_vector_smult singleton_matrix_add)+
@@ -235,7 +235,7 @@
apply (simp_all add: sorted_spvec_addmult_spvec_helper3)
done
-fun mult_spvec_spmat :: "('a::lordered_ring) spvec \<Rightarrow> 'a spvec \<Rightarrow> 'a spmat \<Rightarrow> 'a spvec" where
+fun mult_spvec_spmat :: "('a::lattice_ring) spvec \<Rightarrow> 'a spvec \<Rightarrow> 'a spmat \<Rightarrow> 'a spvec" where
(* recdef mult_spvec_spmat "measure (% (c, arr, brr). (length arr) + (length brr))" *)
"mult_spvec_spmat c [] brr = c" |
"mult_spvec_spmat c arr [] = c" |
@@ -244,7 +244,7 @@
else if (j < i) then mult_spvec_spmat c ((i,a)#arr) brr
else mult_spvec_spmat (addmult_spvec a c b) arr brr)"
-lemma sparse_row_mult_spvec_spmat[rule_format]: "sorted_spvec (a::('a::lordered_ring) spvec) \<longrightarrow> sorted_spvec B \<longrightarrow>
+lemma sparse_row_mult_spvec_spmat[rule_format]: "sorted_spvec (a::('a::lattice_ring) spvec) \<longrightarrow> sorted_spvec B \<longrightarrow>
sparse_row_vector (mult_spvec_spmat c a B) = (sparse_row_vector c) + (sparse_row_vector a) * (sparse_row_matrix B)"
proof -
have comp_1: "!! a b. a < b \<Longrightarrow> Suc 0 <= nat ((int b)-(int a))" by arith
@@ -337,13 +337,13 @@
qed
lemma sorted_mult_spvec_spmat[rule_format]:
- "sorted_spvec (c::('a::lordered_ring) spvec) \<longrightarrow> sorted_spmat B \<longrightarrow> sorted_spvec (mult_spvec_spmat c a B)"
+ "sorted_spvec (c::('a::lattice_ring) spvec) \<longrightarrow> sorted_spmat B \<longrightarrow> sorted_spvec (mult_spvec_spmat c a B)"
apply (induct c a B rule: mult_spvec_spmat.induct)
apply (simp_all add: sorted_addmult_spvec)
done
consts
- mult_spmat :: "('a::lordered_ring) spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat"
+ mult_spmat :: "('a::lattice_ring) spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat"
primrec
"mult_spmat [] A = []"
@@ -357,7 +357,7 @@
done
lemma sorted_spvec_mult_spmat[rule_format]:
- "sorted_spvec (A::('a::lordered_ring) spmat) \<longrightarrow> sorted_spvec (mult_spmat A B)"
+ "sorted_spvec (A::('a::lattice_ring) spmat) \<longrightarrow> sorted_spvec (mult_spmat A B)"
apply (induct A)
apply (auto)
apply (drule sorted_spvec_cons1, simp)
@@ -366,13 +366,13 @@
done
lemma sorted_spmat_mult_spmat:
- "sorted_spmat (B::('a::lordered_ring) spmat) \<Longrightarrow> sorted_spmat (mult_spmat A B)"
+ "sorted_spmat (B::('a::lattice_ring) spmat) \<Longrightarrow> sorted_spmat (mult_spmat A B)"
apply (induct A)
apply (auto simp add: sorted_mult_spvec_spmat)
done
-fun add_spvec :: "('a::lordered_ab_group_add) spvec \<Rightarrow> 'a spvec \<Rightarrow> 'a spvec" where
+fun add_spvec :: "('a::lattice_ab_group_add) spvec \<Rightarrow> 'a spvec \<Rightarrow> 'a spvec" where
(* "measure (% (a, b). length a + (length b))" *)
"add_spvec arr [] = arr" |
"add_spvec [] brr = brr" |
@@ -389,7 +389,7 @@
apply (simp_all add: singleton_matrix_add)
done
-fun add_spmat :: "('a::lordered_ab_group_add) spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat" where
+fun add_spmat :: "('a::lattice_ab_group_add) spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat" where
(* "measure (% (A,B). (length A)+(length B))" *)
"add_spmat [] bs = bs" |
"add_spmat as [] = as" |
@@ -532,7 +532,7 @@
apply (simp_all add: sorted_spvec_add_spvec)
done
-fun le_spvec :: "('a::lordered_ab_group_add) spvec \<Rightarrow> 'a spvec \<Rightarrow> bool" where
+fun le_spvec :: "('a::lattice_ab_group_add) spvec \<Rightarrow> 'a spvec \<Rightarrow> bool" where
(* "measure (% (a,b). (length a) + (length b))" *)
"le_spvec [] [] = True" |
"le_spvec ((_,a)#as) [] = (a <= 0 & le_spvec as [])" |
@@ -542,7 +542,7 @@
else if (j < i) then 0 <= b & le_spvec ((i,a)#as) bs
else a <= b & le_spvec as bs)"
-fun le_spmat :: "('a::lordered_ab_group_add) spmat \<Rightarrow> 'a spmat \<Rightarrow> bool" where
+fun le_spmat :: "('a::lattice_ab_group_add) spmat \<Rightarrow> 'a spmat \<Rightarrow> bool" where
(* "measure (% (a,b). (length a) + (length b))" *)
"le_spmat [] [] = True" |
"le_spmat ((i,a)#as) [] = (le_spvec a [] & le_spmat as [])" |
@@ -566,7 +566,7 @@
lemma disj_matrices_add: "disj_matrices A B \<Longrightarrow> disj_matrices C D \<Longrightarrow> disj_matrices A D \<Longrightarrow> disj_matrices B C \<Longrightarrow>
- (A + B <= C + D) = (A <= C & B <= (D::('a::lordered_ab_group_add) matrix))"
+ (A + B <= C + D) = (A <= C & B <= (D::('a::lattice_ab_group_add) matrix))"
apply (auto)
apply (simp (no_asm_use) only: le_matrix_def disj_matrices_def)
apply (intro strip)
@@ -596,19 +596,19 @@
by (auto simp add: disj_matrices_def)
lemma disj_matrices_add_le_zero: "disj_matrices A B \<Longrightarrow>
- (A + B <= 0) = (A <= 0 & (B::('a::lordered_ab_group_add) matrix) <= 0)"
+ (A + B <= 0) = (A <= 0 & (B::('a::lattice_ab_group_add) matrix) <= 0)"
by (rule disj_matrices_add[of A B 0 0, simplified])
lemma disj_matrices_add_zero_le: "disj_matrices A B \<Longrightarrow>
- (0 <= A + B) = (0 <= A & 0 <= (B::('a::lordered_ab_group_add) matrix))"
+ (0 <= A + B) = (0 <= A & 0 <= (B::('a::lattice_ab_group_add) matrix))"
by (rule disj_matrices_add[of 0 0 A B, simplified])
lemma disj_matrices_add_x_le: "disj_matrices A B \<Longrightarrow> disj_matrices B C \<Longrightarrow>
- (A <= B + C) = (A <= C & 0 <= (B::('a::lordered_ab_group_add) matrix))"
+ (A <= B + C) = (A <= C & 0 <= (B::('a::lattice_ab_group_add) matrix))"
by (auto simp add: disj_matrices_add[of 0 A B C, simplified])
lemma disj_matrices_add_le_x: "disj_matrices A B \<Longrightarrow> disj_matrices B C \<Longrightarrow>
- (B + A <= C) = (A <= C & (B::('a::lordered_ab_group_add) matrix) <= 0)"
+ (B + A <= C) = (A <= C & (B::('a::lattice_ab_group_add) matrix) <= 0)"
by (auto simp add: disj_matrices_add[of B A 0 C,simplified] disj_matrices_commute)
lemma disj_sparse_row_singleton: "i <= j \<Longrightarrow> sorted_spvec((j,y)#v) \<Longrightarrow> disj_matrices (sparse_row_vector v) (singleton_matrix 0 i x)"
@@ -624,7 +624,7 @@
apply (simp_all)
done
-lemma disj_matrices_x_add: "disj_matrices A B \<Longrightarrow> disj_matrices A C \<Longrightarrow> disj_matrices (A::('a::lordered_ab_group_add) matrix) (B+C)"
+lemma disj_matrices_x_add: "disj_matrices A B \<Longrightarrow> disj_matrices A C \<Longrightarrow> disj_matrices (A::('a::lattice_ab_group_add) matrix) (B+C)"
apply (simp add: disj_matrices_def)
apply (auto)
apply (drule_tac j=j and i=i in spec2)+
@@ -633,7 +633,7 @@
apply (simp_all)
done
-lemma disj_matrices_add_x: "disj_matrices A B \<Longrightarrow> disj_matrices A C \<Longrightarrow> disj_matrices (B+C) (A::('a::lordered_ab_group_add) matrix)"
+lemma disj_matrices_add_x: "disj_matrices A B \<Longrightarrow> disj_matrices A C \<Longrightarrow> disj_matrices (B+C) (A::('a::lattice_ab_group_add) matrix)"
by (simp add: disj_matrices_x_add disj_matrices_commute)
lemma disj_singleton_matrices[simp]: "disj_matrices (singleton_matrix j i x) (singleton_matrix u v y) = (j \<noteq> u | i \<noteq> v | x = 0 | y = 0)"
@@ -731,11 +731,11 @@
declare [[simp_depth_limit = 999]]
-primrec abs_spmat :: "('a::lordered_ring) spmat \<Rightarrow> 'a spmat" where
+primrec abs_spmat :: "('a::lattice_ring) spmat \<Rightarrow> 'a spmat" where
"abs_spmat [] = []" |
"abs_spmat (a#as) = (fst a, abs_spvec (snd a))#(abs_spmat as)"
-primrec minus_spmat :: "('a::lordered_ring) spmat \<Rightarrow> 'a spmat" where
+primrec minus_spmat :: "('a::lattice_ring) spmat \<Rightarrow> 'a spmat" where
"minus_spmat [] = []" |
"minus_spmat (a#as) = (fst a, minus_spvec (snd a))#(minus_spmat as)"
@@ -803,7 +803,7 @@
done
constdefs
- diff_spmat :: "('a::lordered_ring) spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat"
+ diff_spmat :: "('a::lattice_ring) spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat"
"diff_spmat A B == add_spmat A (minus_spmat B)"
lemma sorted_spmat_diff_spmat: "sorted_spmat A \<Longrightarrow> sorted_spmat B \<Longrightarrow> sorted_spmat (diff_spmat A B)"
@@ -845,10 +845,10 @@
lemma if_case_eq: "(if b then x else y) = (case b of True => x | False => y)" by simp
consts
- pprt_spvec :: "('a::{lordered_ab_group_add}) spvec \<Rightarrow> 'a spvec"
- nprt_spvec :: "('a::{lordered_ab_group_add}) spvec \<Rightarrow> 'a spvec"
- pprt_spmat :: "('a::{lordered_ab_group_add}) spmat \<Rightarrow> 'a spmat"
- nprt_spmat :: "('a::{lordered_ab_group_add}) spmat \<Rightarrow> 'a spmat"
+ pprt_spvec :: "('a::{lattice_ab_group_add}) spvec \<Rightarrow> 'a spvec"
+ nprt_spvec :: "('a::{lattice_ab_group_add}) spvec \<Rightarrow> 'a spvec"
+ pprt_spmat :: "('a::{lattice_ab_group_add}) spmat \<Rightarrow> 'a spmat"
+ nprt_spmat :: "('a::{lattice_ab_group_add}) spmat \<Rightarrow> 'a spmat"
primrec
"pprt_spvec [] = []"
@@ -869,7 +869,7 @@
(*case (nprt_spvec (snd a)) of [] \<Rightarrow> (nprt_spmat as) | y#ys \<Rightarrow> (fst a, y#ys)#(nprt_spmat as))"*)
-lemma pprt_add: "disj_matrices A (B::(_::lordered_ring) matrix) \<Longrightarrow> pprt (A+B) = pprt A + pprt B"
+lemma pprt_add: "disj_matrices A (B::(_::lattice_ring) matrix) \<Longrightarrow> pprt (A+B) = pprt A + pprt B"
apply (simp add: pprt_def sup_matrix_def)
apply (simp add: Rep_matrix_inject[symmetric])
apply (rule ext)+
@@ -878,7 +878,7 @@
apply (simp_all add: disj_matrices_contr1)
done
-lemma nprt_add: "disj_matrices A (B::(_::lordered_ring) matrix) \<Longrightarrow> nprt (A+B) = nprt A + nprt B"
+lemma nprt_add: "disj_matrices A (B::(_::lattice_ring) matrix) \<Longrightarrow> nprt (A+B) = nprt A + nprt B"
apply (simp add: nprt_def inf_matrix_def)
apply (simp add: Rep_matrix_inject[symmetric])
apply (rule ext)+
@@ -887,14 +887,14 @@
apply (simp_all add: disj_matrices_contr1)
done
-lemma pprt_singleton[simp]: "pprt (singleton_matrix j i (x::_::lordered_ring)) = singleton_matrix j i (pprt x)"
+lemma pprt_singleton[simp]: "pprt (singleton_matrix j i (x::_::lattice_ring)) = singleton_matrix j i (pprt x)"
apply (simp add: pprt_def sup_matrix_def)
apply (simp add: Rep_matrix_inject[symmetric])
apply (rule ext)+
apply simp
done
-lemma nprt_singleton[simp]: "nprt (singleton_matrix j i (x::_::lordered_ring)) = singleton_matrix j i (nprt x)"
+lemma nprt_singleton[simp]: "nprt (singleton_matrix j i (x::_::lattice_ring)) = singleton_matrix j i (nprt x)"
apply (simp add: nprt_def inf_matrix_def)
apply (simp add: Rep_matrix_inject[symmetric])
apply (rule ext)+
@@ -903,7 +903,7 @@
lemma less_imp_le: "a < b \<Longrightarrow> a <= (b::_::order)" by (simp add: less_def)
-lemma sparse_row_vector_pprt: "sorted_spvec (v :: 'a::lordered_ring spvec) \<Longrightarrow> sparse_row_vector (pprt_spvec v) = pprt (sparse_row_vector v)"
+lemma sparse_row_vector_pprt: "sorted_spvec (v :: 'a::lattice_ring spvec) \<Longrightarrow> sparse_row_vector (pprt_spvec v) = pprt (sparse_row_vector v)"
apply (induct v)
apply (simp_all)
apply (frule sorted_spvec_cons1, auto)
@@ -913,7 +913,7 @@
apply auto
done
-lemma sparse_row_vector_nprt: "sorted_spvec (v :: 'a::lordered_ring spvec) \<Longrightarrow> sparse_row_vector (nprt_spvec v) = nprt (sparse_row_vector v)"
+lemma sparse_row_vector_nprt: "sorted_spvec (v :: 'a::lattice_ring spvec) \<Longrightarrow> sparse_row_vector (nprt_spvec v) = nprt (sparse_row_vector v)"
apply (induct v)
apply (simp_all)
apply (frule sorted_spvec_cons1, auto)
@@ -924,7 +924,7 @@
done
-lemma pprt_move_matrix: "pprt (move_matrix (A::('a::lordered_ring) matrix) j i) = move_matrix (pprt A) j i"
+lemma pprt_move_matrix: "pprt (move_matrix (A::('a::lattice_ring) matrix) j i) = move_matrix (pprt A) j i"
apply (simp add: pprt_def)
apply (simp add: sup_matrix_def)
apply (simp add: Rep_matrix_inject[symmetric])
@@ -932,7 +932,7 @@
apply (simp)
done
-lemma nprt_move_matrix: "nprt (move_matrix (A::('a::lordered_ring) matrix) j i) = move_matrix (nprt A) j i"
+lemma nprt_move_matrix: "nprt (move_matrix (A::('a::lattice_ring) matrix) j i) = move_matrix (nprt A) j i"
apply (simp add: nprt_def)
apply (simp add: inf_matrix_def)
apply (simp add: Rep_matrix_inject[symmetric])
@@ -940,7 +940,7 @@
apply (simp)
done
-lemma sparse_row_matrix_pprt: "sorted_spvec (m :: 'a::lordered_ring spmat) \<Longrightarrow> sorted_spmat m \<Longrightarrow> sparse_row_matrix (pprt_spmat m) = pprt (sparse_row_matrix m)"
+lemma sparse_row_matrix_pprt: "sorted_spvec (m :: 'a::lattice_ring spmat) \<Longrightarrow> sorted_spmat m \<Longrightarrow> sparse_row_matrix (pprt_spmat m) = pprt (sparse_row_matrix m)"
apply (induct m)
apply simp
apply simp
@@ -956,7 +956,7 @@
apply (simp add: pprt_move_matrix)
done
-lemma sparse_row_matrix_nprt: "sorted_spvec (m :: 'a::lordered_ring spmat) \<Longrightarrow> sorted_spmat m \<Longrightarrow> sparse_row_matrix (nprt_spmat m) = nprt (sparse_row_matrix m)"
+lemma sparse_row_matrix_nprt: "sorted_spvec (m :: 'a::lattice_ring spmat) \<Longrightarrow> sorted_spmat m \<Longrightarrow> sparse_row_matrix (nprt_spmat m) = nprt (sparse_row_matrix m)"
apply (induct m)
apply simp
apply simp
@@ -1015,7 +1015,7 @@
done
constdefs
- mult_est_spmat :: "('a::lordered_ring) spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat"
+ mult_est_spmat :: "('a::lattice_ring) spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat"
"mult_est_spmat r1 r2 s1 s2 ==
add_spmat (mult_spmat (pprt_spmat s2) (pprt_spmat r2)) (add_spmat (mult_spmat (pprt_spmat s1) (nprt_spmat r2))
(add_spmat (mult_spmat (nprt_spmat s2) (pprt_spmat r1)) (mult_spmat (nprt_spmat s1) (nprt_spmat r1))))"
@@ -1057,7 +1057,7 @@
"sorted_spvec b"
"sorted_spvec r"
"le_spmat ([], y)"
- "A * x \<le> sparse_row_matrix (b::('a::lordered_ring) spmat)"
+ "A * x \<le> sparse_row_matrix (b::('a::lattice_ring) spmat)"
"sparse_row_matrix A1 <= A"
"A <= sparse_row_matrix A2"
"sparse_row_matrix c1 <= c"
--- a/src/HOL/Matrix/cplex/Cplex.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Matrix/cplex/Cplex.thy Mon Feb 08 11:13:30 2010 +0100
@@ -25,7 +25,7 @@
"c \<le> sparse_row_matrix c2"
"sparse_row_matrix r1 \<le> x"
"x \<le> sparse_row_matrix r2"
- "A * x \<le> sparse_row_matrix (b::('a::lordered_ring) spmat)"
+ "A * x \<le> sparse_row_matrix (b::('a::lattice_ring) spmat)"
shows
"c * x \<le> sparse_row_matrix (add_spmat (mult_spmat y b)
(let s1 = diff_spmat c1 (mult_spmat y A2); s2 = diff_spmat c2 (mult_spmat y A1) in
@@ -55,7 +55,7 @@
"c \<le> sparse_row_matrix c2"
"sparse_row_matrix r1 \<le> x"
"x \<le> sparse_row_matrix r2"
- "A * x \<le> sparse_row_matrix (b::('a::lordered_ring) spmat)"
+ "A * x \<le> sparse_row_matrix (b::('a::lattice_ring) spmat)"
shows
"c * x \<le> sparse_row_matrix (add_spmat (mult_spmat y b)
(mult_est_spmat r1 r2 (diff_spmat c1 (mult_spmat y A2)) (diff_spmat c2 (mult_spmat y A1))))"
--- a/src/HOL/Metis_Examples/BigO.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Metis_Examples/BigO.thy Mon Feb 08 11:13:30 2010 +0100
@@ -12,11 +12,11 @@
subsection {* Definitions *}
-definition bigo :: "('a => 'b::ordered_idom) => ('a => 'b) set" ("(1O'(_'))") where
+definition bigo :: "('a => 'b::linordered_idom) => ('a => 'b) set" ("(1O'(_'))") where
"O(f::('a => 'b)) == {h. EX c. ALL x. abs (h x) <= c * abs (f x)}"
declare [[ atp_problem_prefix = "BigO__bigo_pos_const" ]]
-lemma bigo_pos_const: "(EX (c::'a::ordered_idom).
+lemma bigo_pos_const: "(EX (c::'a::linordered_idom).
ALL x. (abs (h x)) <= (c * (abs (f x))))
= (EX c. 0 < c & (ALL x. (abs(h x)) <= (c * (abs (f x)))))"
apply auto
@@ -30,7 +30,7 @@
declare [[sledgehammer_modulus = 1]]
-lemma (*bigo_pos_const:*) "(EX (c::'a::ordered_idom).
+lemma (*bigo_pos_const:*) "(EX (c::'a::linordered_idom).
ALL x. (abs (h x)) <= (c * (abs (f x))))
= (EX c. 0 < c & (ALL x. (abs(h x)) <= (c * (abs (f x)))))"
apply auto
@@ -39,59 +39,59 @@
apply (rule_tac x = "abs c" in exI, auto)
proof (neg_clausify)
fix c x
-have 0: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2\<Colon>'a\<Colon>ordered_idom. \<bar>X1 * X2\<bar> = \<bar>X2 * X1\<bar>"
+have 0: "\<And>(X1\<Colon>'a\<Colon>linordered_idom) X2\<Colon>'a\<Colon>linordered_idom. \<bar>X1 * X2\<bar> = \<bar>X2 * X1\<bar>"
by (metis abs_mult mult_commute)
-have 1: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2\<Colon>'a\<Colon>ordered_idom.
- X1 \<le> (0\<Colon>'a\<Colon>ordered_idom) \<or> \<bar>X2\<bar> * X1 = \<bar>X2 * X1\<bar>"
+have 1: "\<And>(X1\<Colon>'a\<Colon>linordered_idom) X2\<Colon>'a\<Colon>linordered_idom.
+ X1 \<le> (0\<Colon>'a\<Colon>linordered_idom) \<or> \<bar>X2\<bar> * X1 = \<bar>X2 * X1\<bar>"
by (metis abs_mult_pos linorder_linear)
-have 2: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2\<Colon>'a\<Colon>ordered_idom.
- \<not> (0\<Colon>'a\<Colon>ordered_idom) < X1 * X2 \<or>
- \<not> (0\<Colon>'a\<Colon>ordered_idom) \<le> X2 \<or> \<not> X1 \<le> (0\<Colon>'a\<Colon>ordered_idom)"
+have 2: "\<And>(X1\<Colon>'a\<Colon>linordered_idom) X2\<Colon>'a\<Colon>linordered_idom.
+ \<not> (0\<Colon>'a\<Colon>linordered_idom) < X1 * X2 \<or>
+ \<not> (0\<Colon>'a\<Colon>linordered_idom) \<le> X2 \<or> \<not> X1 \<le> (0\<Colon>'a\<Colon>linordered_idom)"
by (metis linorder_not_less mult_nonneg_nonpos2)
assume 3: "\<And>x\<Colon>'b\<Colon>type.
- \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) x\<bar>
- \<le> (c\<Colon>'a\<Colon>ordered_idom) * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) x\<bar>"
-assume 4: "\<not> \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) (x\<Colon>'b\<Colon>type)\<bar>
- \<le> \<bar>c\<Colon>'a\<Colon>ordered_idom\<bar> * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) x\<bar>"
-have 5: "\<not> \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) (x\<Colon>'b\<Colon>type)\<bar>
- \<le> \<bar>(c\<Colon>'a\<Colon>ordered_idom) * (f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) x\<bar>"
+ \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>linordered_idom) x\<bar>
+ \<le> (c\<Colon>'a\<Colon>linordered_idom) * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>linordered_idom) x\<bar>"
+assume 4: "\<not> \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>linordered_idom) (x\<Colon>'b\<Colon>type)\<bar>
+ \<le> \<bar>c\<Colon>'a\<Colon>linordered_idom\<bar> * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>linordered_idom) x\<bar>"
+have 5: "\<not> \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>linordered_idom) (x\<Colon>'b\<Colon>type)\<bar>
+ \<le> \<bar>(c\<Colon>'a\<Colon>linordered_idom) * (f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>linordered_idom) x\<bar>"
by (metis 4 abs_mult)
-have 6: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2\<Colon>'a\<Colon>ordered_idom.
- \<not> X1 \<le> (0\<Colon>'a\<Colon>ordered_idom) \<or> X1 \<le> \<bar>X2\<bar>"
+have 6: "\<And>(X1\<Colon>'a\<Colon>linordered_idom) X2\<Colon>'a\<Colon>linordered_idom.
+ \<not> X1 \<le> (0\<Colon>'a\<Colon>linordered_idom) \<or> X1 \<le> \<bar>X2\<bar>"
by (metis abs_ge_zero xt1(6))
-have 7: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2\<Colon>'a\<Colon>ordered_idom.
- X1 \<le> \<bar>X2\<bar> \<or> (0\<Colon>'a\<Colon>ordered_idom) < X1"
+have 7: "\<And>(X1\<Colon>'a\<Colon>linordered_idom) X2\<Colon>'a\<Colon>linordered_idom.
+ X1 \<le> \<bar>X2\<bar> \<or> (0\<Colon>'a\<Colon>linordered_idom) < X1"
by (metis not_leE 6)
-have 8: "(0\<Colon>'a\<Colon>ordered_idom) < \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) (x\<Colon>'b\<Colon>type)\<bar>"
+have 8: "(0\<Colon>'a\<Colon>linordered_idom) < \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>linordered_idom) (x\<Colon>'b\<Colon>type)\<bar>"
by (metis 5 7)
-have 9: "\<And>X1\<Colon>'a\<Colon>ordered_idom.
- \<not> \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) (x\<Colon>'b\<Colon>type)\<bar> \<le> X1 \<or>
- (0\<Colon>'a\<Colon>ordered_idom) < X1"
+have 9: "\<And>X1\<Colon>'a\<Colon>linordered_idom.
+ \<not> \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>linordered_idom) (x\<Colon>'b\<Colon>type)\<bar> \<le> X1 \<or>
+ (0\<Colon>'a\<Colon>linordered_idom) < X1"
by (metis 8 order_less_le_trans)
-have 10: "(0\<Colon>'a\<Colon>ordered_idom)
-< (c\<Colon>'a\<Colon>ordered_idom) * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) (x\<Colon>'b\<Colon>type)\<bar>"
+have 10: "(0\<Colon>'a\<Colon>linordered_idom)
+< (c\<Colon>'a\<Colon>linordered_idom) * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>linordered_idom) (x\<Colon>'b\<Colon>type)\<bar>"
by (metis 3 9)
-have 11: "\<not> (c\<Colon>'a\<Colon>ordered_idom) \<le> (0\<Colon>'a\<Colon>ordered_idom)"
+have 11: "\<not> (c\<Colon>'a\<Colon>linordered_idom) \<le> (0\<Colon>'a\<Colon>linordered_idom)"
by (metis abs_ge_zero 2 10)
-have 12: "\<And>X1\<Colon>'a\<Colon>ordered_idom. (c\<Colon>'a\<Colon>ordered_idom) * \<bar>X1\<bar> = \<bar>X1 * c\<bar>"
+have 12: "\<And>X1\<Colon>'a\<Colon>linordered_idom. (c\<Colon>'a\<Colon>linordered_idom) * \<bar>X1\<bar> = \<bar>X1 * c\<bar>"
by (metis mult_commute 1 11)
have 13: "\<And>X1\<Colon>'b\<Colon>type.
- - (h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) X1
- \<le> (c\<Colon>'a\<Colon>ordered_idom) * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) X1\<bar>"
+ - (h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>linordered_idom) X1
+ \<le> (c\<Colon>'a\<Colon>linordered_idom) * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>linordered_idom) X1\<bar>"
by (metis 3 abs_le_D2)
have 14: "\<And>X1\<Colon>'b\<Colon>type.
- - (h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) X1
- \<le> \<bar>(c\<Colon>'a\<Colon>ordered_idom) * (f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) X1\<bar>"
+ - (h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>linordered_idom) X1
+ \<le> \<bar>(c\<Colon>'a\<Colon>linordered_idom) * (f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>linordered_idom) X1\<bar>"
by (metis 0 12 13)
-have 15: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2\<Colon>'a\<Colon>ordered_idom. \<bar>X1 * \<bar>X2\<bar>\<bar> = \<bar>X1 * X2\<bar>"
+have 15: "\<And>(X1\<Colon>'a\<Colon>linordered_idom) X2\<Colon>'a\<Colon>linordered_idom. \<bar>X1 * \<bar>X2\<bar>\<bar> = \<bar>X1 * X2\<bar>"
by (metis abs_mult abs_mult_pos abs_ge_zero)
-have 16: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2\<Colon>'a\<Colon>ordered_idom. X1 \<le> \<bar>X2\<bar> \<or> \<not> X1 \<le> X2"
+have 16: "\<And>(X1\<Colon>'a\<Colon>linordered_idom) X2\<Colon>'a\<Colon>linordered_idom. X1 \<le> \<bar>X2\<bar> \<or> \<not> X1 \<le> X2"
by (metis xt1(6) abs_ge_self)
-have 17: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2\<Colon>'a\<Colon>ordered_idom. \<not> \<bar>X1\<bar> \<le> X2 \<or> X1 \<le> \<bar>X2\<bar>"
+have 17: "\<And>(X1\<Colon>'a\<Colon>linordered_idom) X2\<Colon>'a\<Colon>linordered_idom. \<not> \<bar>X1\<bar> \<le> X2 \<or> X1 \<le> \<bar>X2\<bar>"
by (metis 16 abs_le_D1)
have 18: "\<And>X1\<Colon>'b\<Colon>type.
- (h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) X1
- \<le> \<bar>(c\<Colon>'a\<Colon>ordered_idom) * (f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) X1\<bar>"
+ (h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>linordered_idom) X1
+ \<le> \<bar>(c\<Colon>'a\<Colon>linordered_idom) * (f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>linordered_idom) X1\<bar>"
by (metis 17 3 15)
show "False"
by (metis abs_le_iff 5 18 14)
@@ -99,7 +99,7 @@
declare [[sledgehammer_modulus = 2]]
-lemma (*bigo_pos_const:*) "(EX (c::'a::ordered_idom).
+lemma (*bigo_pos_const:*) "(EX (c::'a::linordered_idom).
ALL x. (abs (h x)) <= (c * (abs (f x))))
= (EX c. 0 < c & (ALL x. (abs(h x)) <= (c * (abs (f x)))))"
apply auto
@@ -108,31 +108,31 @@
apply (rule_tac x = "abs c" in exI, auto);
proof (neg_clausify)
fix c x
-have 0: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2\<Colon>'a\<Colon>ordered_idom. \<bar>X1 * X2\<bar> = \<bar>X2 * X1\<bar>"
+have 0: "\<And>(X1\<Colon>'a\<Colon>linordered_idom) X2\<Colon>'a\<Colon>linordered_idom. \<bar>X1 * X2\<bar> = \<bar>X2 * X1\<bar>"
by (metis abs_mult mult_commute)
assume 1: "\<And>x\<Colon>'b\<Colon>type.
- \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) x\<bar>
- \<le> (c\<Colon>'a\<Colon>ordered_idom) * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) x\<bar>"
-assume 2: "\<not> \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) (x\<Colon>'b\<Colon>type)\<bar>
- \<le> \<bar>c\<Colon>'a\<Colon>ordered_idom\<bar> * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) x\<bar>"
-have 3: "\<not> \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) (x\<Colon>'b\<Colon>type)\<bar>
- \<le> \<bar>(c\<Colon>'a\<Colon>ordered_idom) * (f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) x\<bar>"
+ \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>linordered_idom) x\<bar>
+ \<le> (c\<Colon>'a\<Colon>linordered_idom) * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>linordered_idom) x\<bar>"
+assume 2: "\<not> \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>linordered_idom) (x\<Colon>'b\<Colon>type)\<bar>
+ \<le> \<bar>c\<Colon>'a\<Colon>linordered_idom\<bar> * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>linordered_idom) x\<bar>"
+have 3: "\<not> \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>linordered_idom) (x\<Colon>'b\<Colon>type)\<bar>
+ \<le> \<bar>(c\<Colon>'a\<Colon>linordered_idom) * (f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>linordered_idom) x\<bar>"
by (metis 2 abs_mult)
-have 4: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2\<Colon>'a\<Colon>ordered_idom.
- \<not> X1 \<le> (0\<Colon>'a\<Colon>ordered_idom) \<or> X1 \<le> \<bar>X2\<bar>"
+have 4: "\<And>(X1\<Colon>'a\<Colon>linordered_idom) X2\<Colon>'a\<Colon>linordered_idom.
+ \<not> X1 \<le> (0\<Colon>'a\<Colon>linordered_idom) \<or> X1 \<le> \<bar>X2\<bar>"
by (metis abs_ge_zero xt1(6))
-have 5: "(0\<Colon>'a\<Colon>ordered_idom) < \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) (x\<Colon>'b\<Colon>type)\<bar>"
+have 5: "(0\<Colon>'a\<Colon>linordered_idom) < \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>linordered_idom) (x\<Colon>'b\<Colon>type)\<bar>"
by (metis not_leE 4 3)
-have 6: "(0\<Colon>'a\<Colon>ordered_idom)
-< (c\<Colon>'a\<Colon>ordered_idom) * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) (x\<Colon>'b\<Colon>type)\<bar>"
+have 6: "(0\<Colon>'a\<Colon>linordered_idom)
+< (c\<Colon>'a\<Colon>linordered_idom) * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>linordered_idom) (x\<Colon>'b\<Colon>type)\<bar>"
by (metis 1 order_less_le_trans 5)
-have 7: "\<And>X1\<Colon>'a\<Colon>ordered_idom. (c\<Colon>'a\<Colon>ordered_idom) * \<bar>X1\<bar> = \<bar>X1 * c\<bar>"
+have 7: "\<And>X1\<Colon>'a\<Colon>linordered_idom. (c\<Colon>'a\<Colon>linordered_idom) * \<bar>X1\<bar> = \<bar>X1 * c\<bar>"
by (metis abs_ge_zero linorder_not_less mult_nonneg_nonpos2 6 linorder_linear abs_mult_pos mult_commute)
have 8: "\<And>X1\<Colon>'b\<Colon>type.
- - (h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) X1
- \<le> \<bar>(c\<Colon>'a\<Colon>ordered_idom) * (f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) X1\<bar>"
+ - (h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>linordered_idom) X1
+ \<le> \<bar>(c\<Colon>'a\<Colon>linordered_idom) * (f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>linordered_idom) X1\<bar>"
by (metis 0 7 abs_le_D2 1)
-have 9: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2\<Colon>'a\<Colon>ordered_idom. \<not> \<bar>X1\<bar> \<le> X2 \<or> X1 \<le> \<bar>X2\<bar>"
+have 9: "\<And>(X1\<Colon>'a\<Colon>linordered_idom) X2\<Colon>'a\<Colon>linordered_idom. \<not> \<bar>X1\<bar> \<le> X2 \<or> X1 \<le> \<bar>X2\<bar>"
by (metis abs_ge_self xt1(6) abs_le_D1)
show "False"
by (metis 8 abs_ge_zero abs_mult_pos abs_mult 1 9 3 abs_le_iff)
@@ -140,7 +140,7 @@
declare [[sledgehammer_modulus = 3]]
-lemma (*bigo_pos_const:*) "(EX (c::'a::ordered_idom).
+lemma (*bigo_pos_const:*) "(EX (c::'a::linordered_idom).
ALL x. (abs (h x)) <= (c * (abs (f x))))
= (EX c. 0 < c & (ALL x. (abs(h x)) <= (c * (abs (f x)))))"
apply auto
@@ -150,20 +150,20 @@
proof (neg_clausify)
fix c x
assume 0: "\<And>x\<Colon>'b\<Colon>type.
- \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) x\<bar>
- \<le> (c\<Colon>'a\<Colon>ordered_idom) * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) x\<bar>"
-assume 1: "\<not> \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) (x\<Colon>'b\<Colon>type)\<bar>
- \<le> \<bar>c\<Colon>'a\<Colon>ordered_idom\<bar> * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) x\<bar>"
-have 2: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2\<Colon>'a\<Colon>ordered_idom.
- X1 \<le> \<bar>X2\<bar> \<or> (0\<Colon>'a\<Colon>ordered_idom) < X1"
+ \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>linordered_idom) x\<bar>
+ \<le> (c\<Colon>'a\<Colon>linordered_idom) * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>linordered_idom) x\<bar>"
+assume 1: "\<not> \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>linordered_idom) (x\<Colon>'b\<Colon>type)\<bar>
+ \<le> \<bar>c\<Colon>'a\<Colon>linordered_idom\<bar> * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>linordered_idom) x\<bar>"
+have 2: "\<And>(X1\<Colon>'a\<Colon>linordered_idom) X2\<Colon>'a\<Colon>linordered_idom.
+ X1 \<le> \<bar>X2\<bar> \<or> (0\<Colon>'a\<Colon>linordered_idom) < X1"
by (metis abs_ge_zero xt1(6) not_leE)
-have 3: "\<not> (c\<Colon>'a\<Colon>ordered_idom) \<le> (0\<Colon>'a\<Colon>ordered_idom)"
+have 3: "\<not> (c\<Colon>'a\<Colon>linordered_idom) \<le> (0\<Colon>'a\<Colon>linordered_idom)"
by (metis abs_ge_zero mult_nonneg_nonpos2 linorder_not_less order_less_le_trans 1 abs_mult 2 0)
-have 4: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2\<Colon>'a\<Colon>ordered_idom. \<bar>X1 * \<bar>X2\<bar>\<bar> = \<bar>X1 * X2\<bar>"
+have 4: "\<And>(X1\<Colon>'a\<Colon>linordered_idom) X2\<Colon>'a\<Colon>linordered_idom. \<bar>X1 * \<bar>X2\<bar>\<bar> = \<bar>X1 * X2\<bar>"
by (metis abs_ge_zero abs_mult_pos abs_mult)
have 5: "\<And>X1\<Colon>'b\<Colon>type.
- (h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) X1
- \<le> \<bar>(c\<Colon>'a\<Colon>ordered_idom) * (f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) X1\<bar>"
+ (h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>linordered_idom) X1
+ \<le> \<bar>(c\<Colon>'a\<Colon>linordered_idom) * (f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>linordered_idom) X1\<bar>"
by (metis 4 0 xt1(6) abs_ge_self abs_le_D1)
show "False"
by (metis abs_mult mult_commute 3 abs_mult_pos linorder_linear 0 abs_le_D2 5 1 abs_le_iff)
@@ -172,7 +172,7 @@
declare [[sledgehammer_modulus = 1]]
-lemma (*bigo_pos_const:*) "(EX (c::'a::ordered_idom).
+lemma (*bigo_pos_const:*) "(EX (c::'a::linordered_idom).
ALL x. (abs (h x)) <= (c * (abs (f x))))
= (EX c. 0 < c & (ALL x. (abs(h x)) <= (c * (abs (f x)))))"
apply auto
@@ -181,7 +181,7 @@
apply (rule_tac x = "abs c" in exI, auto);
proof (neg_clausify)
fix c x (*sort/type constraint inserted by hand!*)
-have 0: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2. \<bar>X1 * \<bar>X2\<bar>\<bar> = \<bar>X1 * X2\<bar>"
+have 0: "\<And>(X1\<Colon>'a\<Colon>linordered_idom) X2. \<bar>X1 * \<bar>X2\<bar>\<bar> = \<bar>X1 * X2\<bar>"
by (metis abs_ge_zero abs_mult_pos abs_mult)
assume 1: "\<And>A. \<bar>h A\<bar> \<le> c * \<bar>f A\<bar>"
have 2: "\<And>X1 X2. \<not> \<bar>X1\<bar> \<le> X2 \<or> (0\<Colon>'a) \<le> X2"
@@ -383,11 +383,11 @@
by (metis 0 order_antisym_conv)
have 3: "\<And>X3. \<not> f (x \<bar>X3\<bar>) \<le> \<bar>X3 * g (x \<bar>X3\<bar>)\<bar>"
by (metis 1 abs_mult)
-have 4: "\<And>X1 X3\<Colon>'b\<Colon>ordered_idom. X3 \<le> X1 \<or> X1 \<le> \<bar>X3\<bar>"
+have 4: "\<And>X1 X3\<Colon>'b\<Colon>linordered_idom. X3 \<le> X1 \<or> X1 \<le> \<bar>X3\<bar>"
by (metis linorder_linear abs_le_D1)
have 5: "\<And>X3::'b. \<bar>X3\<bar> * \<bar>X3\<bar> = X3 * X3"
by (metis abs_mult_self)
-have 6: "\<And>X3. \<not> X3 * X3 < (0\<Colon>'b\<Colon>ordered_idom)"
+have 6: "\<And>X3. \<not> X3 * X3 < (0\<Colon>'b\<Colon>linordered_idom)"
by (metis not_square_less_zero)
have 7: "\<And>X1 X3::'b. \<bar>X1\<bar> * \<bar>X3\<bar> = \<bar>X3 * X1\<bar>"
by (metis abs_mult mult_commute)
@@ -438,26 +438,26 @@
proof (neg_clausify)
fix x
assume 0: "\<And>A\<Colon>'a\<Colon>type.
- (f\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) A
- \<le> (c\<Colon>'b\<Colon>ordered_idom) * (g\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) A"
-assume 1: "\<And>A\<Colon>'b\<Colon>ordered_idom.
- \<not> (f\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) ((x\<Colon>'b\<Colon>ordered_idom \<Rightarrow> 'a\<Colon>type) A)
- \<le> A * \<bar>(g\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) (x A)\<bar>"
+ (f\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>linordered_idom) A
+ \<le> (c\<Colon>'b\<Colon>linordered_idom) * (g\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>linordered_idom) A"
+assume 1: "\<And>A\<Colon>'b\<Colon>linordered_idom.
+ \<not> (f\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>linordered_idom) ((x\<Colon>'b\<Colon>linordered_idom \<Rightarrow> 'a\<Colon>type) A)
+ \<le> A * \<bar>(g\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>linordered_idom) (x A)\<bar>"
have 2: "\<And>X2\<Colon>'a\<Colon>type.
- \<not> (c\<Colon>'b\<Colon>ordered_idom) * (g\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) X2
- < (f\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) X2"
+ \<not> (c\<Colon>'b\<Colon>linordered_idom) * (g\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>linordered_idom) X2
+ < (f\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>linordered_idom) X2"
by (metis 0 linorder_not_le)
-have 3: "\<And>X2\<Colon>'b\<Colon>ordered_idom.
- \<not> (f\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) ((x\<Colon>'b\<Colon>ordered_idom \<Rightarrow> 'a\<Colon>type) \<bar>X2\<bar>)
- \<le> \<bar>X2 * (g\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) (x \<bar>X2\<bar>)\<bar>"
+have 3: "\<And>X2\<Colon>'b\<Colon>linordered_idom.
+ \<not> (f\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>linordered_idom) ((x\<Colon>'b\<Colon>linordered_idom \<Rightarrow> 'a\<Colon>type) \<bar>X2\<bar>)
+ \<le> \<bar>X2 * (g\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>linordered_idom) (x \<bar>X2\<bar>)\<bar>"
by (metis abs_mult 1)
-have 4: "\<And>X2\<Colon>'b\<Colon>ordered_idom.
- \<bar>X2 * (g\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) ((x\<Colon>'b\<Colon>ordered_idom \<Rightarrow> 'a\<Colon>type) \<bar>X2\<bar>)\<bar>
- < (f\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) (x \<bar>X2\<bar>)"
+have 4: "\<And>X2\<Colon>'b\<Colon>linordered_idom.
+ \<bar>X2 * (g\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>linordered_idom) ((x\<Colon>'b\<Colon>linordered_idom \<Rightarrow> 'a\<Colon>type) \<bar>X2\<bar>)\<bar>
+ < (f\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>linordered_idom) (x \<bar>X2\<bar>)"
by (metis 3 linorder_not_less)
-have 5: "\<And>X2\<Colon>'b\<Colon>ordered_idom.
- X2 * (g\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) ((x\<Colon>'b\<Colon>ordered_idom \<Rightarrow> 'a\<Colon>type) \<bar>X2\<bar>)
- < (f\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) (x \<bar>X2\<bar>)"
+have 5: "\<And>X2\<Colon>'b\<Colon>linordered_idom.
+ X2 * (g\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>linordered_idom) ((x\<Colon>'b\<Colon>linordered_idom \<Rightarrow> 'a\<Colon>type) \<bar>X2\<bar>)
+ < (f\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>linordered_idom) (x \<bar>X2\<bar>)"
by (metis abs_less_iff 4)
show "False"
by (metis 2 5)
@@ -603,54 +603,54 @@
just been done*)
proof (neg_clausify)
fix a c b ca x
-assume 0: "(0\<Colon>'b\<Colon>ordered_idom) < (c\<Colon>'b\<Colon>ordered_idom)"
-assume 1: "\<bar>(a\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) (x\<Colon>'a)\<bar>
-\<le> (c\<Colon>'b\<Colon>ordered_idom) * \<bar>(f\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar>"
-assume 2: "\<bar>(b\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) (x\<Colon>'a)\<bar>
-\<le> (ca\<Colon>'b\<Colon>ordered_idom) * \<bar>(g\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar>"
-assume 3: "\<not> \<bar>(a\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) (x\<Colon>'a)\<bar> *
- \<bar>(b\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar>
- \<le> (c\<Colon>'b\<Colon>ordered_idom) * \<bar>(f\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar> *
- ((ca\<Colon>'b\<Colon>ordered_idom) * \<bar>(g\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar>)"
-have 4: "\<bar>c\<Colon>'b\<Colon>ordered_idom\<bar> = c"
+assume 0: "(0\<Colon>'b\<Colon>linordered_idom) < (c\<Colon>'b\<Colon>linordered_idom)"
+assume 1: "\<bar>(a\<Colon>'a \<Rightarrow> 'b\<Colon>linordered_idom) (x\<Colon>'a)\<bar>
+\<le> (c\<Colon>'b\<Colon>linordered_idom) * \<bar>(f\<Colon>'a \<Rightarrow> 'b\<Colon>linordered_idom) x\<bar>"
+assume 2: "\<bar>(b\<Colon>'a \<Rightarrow> 'b\<Colon>linordered_idom) (x\<Colon>'a)\<bar>
+\<le> (ca\<Colon>'b\<Colon>linordered_idom) * \<bar>(g\<Colon>'a \<Rightarrow> 'b\<Colon>linordered_idom) x\<bar>"
+assume 3: "\<not> \<bar>(a\<Colon>'a \<Rightarrow> 'b\<Colon>linordered_idom) (x\<Colon>'a)\<bar> *
+ \<bar>(b\<Colon>'a \<Rightarrow> 'b\<Colon>linordered_idom) x\<bar>
+ \<le> (c\<Colon>'b\<Colon>linordered_idom) * \<bar>(f\<Colon>'a \<Rightarrow> 'b\<Colon>linordered_idom) x\<bar> *
+ ((ca\<Colon>'b\<Colon>linordered_idom) * \<bar>(g\<Colon>'a \<Rightarrow> 'b\<Colon>linordered_idom) x\<bar>)"
+have 4: "\<bar>c\<Colon>'b\<Colon>linordered_idom\<bar> = c"
by (metis OrderedGroup.abs_of_pos 0)
-have 5: "\<And>X1\<Colon>'b\<Colon>ordered_idom. (c\<Colon>'b\<Colon>ordered_idom) * \<bar>X1\<bar> = \<bar>c * X1\<bar>"
+have 5: "\<And>X1\<Colon>'b\<Colon>linordered_idom. (c\<Colon>'b\<Colon>linordered_idom) * \<bar>X1\<bar> = \<bar>c * X1\<bar>"
by (metis Ring_and_Field.abs_mult 4)
-have 6: "(0\<Colon>'b\<Colon>ordered_idom) = (1\<Colon>'b\<Colon>ordered_idom) \<or>
-(0\<Colon>'b\<Colon>ordered_idom) < (1\<Colon>'b\<Colon>ordered_idom)"
- by (metis OrderedGroup.abs_not_less_zero Ring_and_Field.abs_one Ring_and_Field.linorder_neqE_ordered_idom)
-have 7: "(0\<Colon>'b\<Colon>ordered_idom) < (1\<Colon>'b\<Colon>ordered_idom)"
+have 6: "(0\<Colon>'b\<Colon>linordered_idom) = (1\<Colon>'b\<Colon>linordered_idom) \<or>
+(0\<Colon>'b\<Colon>linordered_idom) < (1\<Colon>'b\<Colon>linordered_idom)"
+ by (metis OrderedGroup.abs_not_less_zero Ring_and_Field.abs_one Ring_and_Field.linorder_neqE_linordered_idom)
+have 7: "(0\<Colon>'b\<Colon>linordered_idom) < (1\<Colon>'b\<Colon>linordered_idom)"
by (metis 6 Ring_and_Field.one_neq_zero)
-have 8: "\<bar>1\<Colon>'b\<Colon>ordered_idom\<bar> = (1\<Colon>'b\<Colon>ordered_idom)"
+have 8: "\<bar>1\<Colon>'b\<Colon>linordered_idom\<bar> = (1\<Colon>'b\<Colon>linordered_idom)"
by (metis OrderedGroup.abs_of_pos 7)
-have 9: "\<And>X1\<Colon>'b\<Colon>ordered_idom. (0\<Colon>'b\<Colon>ordered_idom) \<le> (c\<Colon>'b\<Colon>ordered_idom) * \<bar>X1\<bar>"
+have 9: "\<And>X1\<Colon>'b\<Colon>linordered_idom. (0\<Colon>'b\<Colon>linordered_idom) \<le> (c\<Colon>'b\<Colon>linordered_idom) * \<bar>X1\<bar>"
by (metis OrderedGroup.abs_ge_zero 5)
-have 10: "\<And>X1\<Colon>'b\<Colon>ordered_idom. X1 * (1\<Colon>'b\<Colon>ordered_idom) = X1"
+have 10: "\<And>X1\<Colon>'b\<Colon>linordered_idom. X1 * (1\<Colon>'b\<Colon>linordered_idom) = X1"
by (metis Ring_and_Field.mult_cancel_right2 mult_commute)
-have 11: "\<And>X1\<Colon>'b\<Colon>ordered_idom. \<bar>\<bar>X1\<bar>\<bar> = \<bar>X1\<bar> * \<bar>1\<Colon>'b\<Colon>ordered_idom\<bar>"
+have 11: "\<And>X1\<Colon>'b\<Colon>linordered_idom. \<bar>\<bar>X1\<bar>\<bar> = \<bar>X1\<bar> * \<bar>1\<Colon>'b\<Colon>linordered_idom\<bar>"
by (metis Ring_and_Field.abs_mult OrderedGroup.abs_idempotent 10)
-have 12: "\<And>X1\<Colon>'b\<Colon>ordered_idom. \<bar>\<bar>X1\<bar>\<bar> = \<bar>X1\<bar>"
+have 12: "\<And>X1\<Colon>'b\<Colon>linordered_idom. \<bar>\<bar>X1\<bar>\<bar> = \<bar>X1\<bar>"
by (metis 11 8 10)
-have 13: "\<And>X1\<Colon>'b\<Colon>ordered_idom. (0\<Colon>'b\<Colon>ordered_idom) \<le> \<bar>X1\<bar>"
+have 13: "\<And>X1\<Colon>'b\<Colon>linordered_idom. (0\<Colon>'b\<Colon>linordered_idom) \<le> \<bar>X1\<bar>"
by (metis OrderedGroup.abs_ge_zero 12)
-have 14: "\<not> (0\<Colon>'b\<Colon>ordered_idom)
- \<le> (c\<Colon>'b\<Colon>ordered_idom) * \<bar>(f\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) (x\<Colon>'a)\<bar> \<or>
-\<not> (0\<Colon>'b\<Colon>ordered_idom) \<le> \<bar>(b\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar> \<or>
-\<not> \<bar>b x\<bar> \<le> (ca\<Colon>'b\<Colon>ordered_idom) * \<bar>(g\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar> \<or>
-\<not> \<bar>(a\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar> \<le> c * \<bar>f x\<bar>"
+have 14: "\<not> (0\<Colon>'b\<Colon>linordered_idom)
+ \<le> (c\<Colon>'b\<Colon>linordered_idom) * \<bar>(f\<Colon>'a \<Rightarrow> 'b\<Colon>linordered_idom) (x\<Colon>'a)\<bar> \<or>
+\<not> (0\<Colon>'b\<Colon>linordered_idom) \<le> \<bar>(b\<Colon>'a \<Rightarrow> 'b\<Colon>linordered_idom) x\<bar> \<or>
+\<not> \<bar>b x\<bar> \<le> (ca\<Colon>'b\<Colon>linordered_idom) * \<bar>(g\<Colon>'a \<Rightarrow> 'b\<Colon>linordered_idom) x\<bar> \<or>
+\<not> \<bar>(a\<Colon>'a \<Rightarrow> 'b\<Colon>linordered_idom) x\<bar> \<le> c * \<bar>f x\<bar>"
by (metis 3 Ring_and_Field.mult_mono)
-have 15: "\<not> (0\<Colon>'b\<Colon>ordered_idom) \<le> \<bar>(b\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) (x\<Colon>'a)\<bar> \<or>
-\<not> \<bar>b x\<bar> \<le> (ca\<Colon>'b\<Colon>ordered_idom) * \<bar>(g\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar> \<or>
-\<not> \<bar>(a\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar>
- \<le> (c\<Colon>'b\<Colon>ordered_idom) * \<bar>(f\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar>"
+have 15: "\<not> (0\<Colon>'b\<Colon>linordered_idom) \<le> \<bar>(b\<Colon>'a \<Rightarrow> 'b\<Colon>linordered_idom) (x\<Colon>'a)\<bar> \<or>
+\<not> \<bar>b x\<bar> \<le> (ca\<Colon>'b\<Colon>linordered_idom) * \<bar>(g\<Colon>'a \<Rightarrow> 'b\<Colon>linordered_idom) x\<bar> \<or>
+\<not> \<bar>(a\<Colon>'a \<Rightarrow> 'b\<Colon>linordered_idom) x\<bar>
+ \<le> (c\<Colon>'b\<Colon>linordered_idom) * \<bar>(f\<Colon>'a \<Rightarrow> 'b\<Colon>linordered_idom) x\<bar>"
by (metis 14 9)
-have 16: "\<not> \<bar>(b\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) (x\<Colon>'a)\<bar>
- \<le> (ca\<Colon>'b\<Colon>ordered_idom) * \<bar>(g\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar> \<or>
-\<not> \<bar>(a\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar>
- \<le> (c\<Colon>'b\<Colon>ordered_idom) * \<bar>(f\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar>"
+have 16: "\<not> \<bar>(b\<Colon>'a \<Rightarrow> 'b\<Colon>linordered_idom) (x\<Colon>'a)\<bar>
+ \<le> (ca\<Colon>'b\<Colon>linordered_idom) * \<bar>(g\<Colon>'a \<Rightarrow> 'b\<Colon>linordered_idom) x\<bar> \<or>
+\<not> \<bar>(a\<Colon>'a \<Rightarrow> 'b\<Colon>linordered_idom) x\<bar>
+ \<le> (c\<Colon>'b\<Colon>linordered_idom) * \<bar>(f\<Colon>'a \<Rightarrow> 'b\<Colon>linordered_idom) x\<bar>"
by (metis 15 13)
-have 17: "\<not> \<bar>(a\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) (x\<Colon>'a)\<bar>
- \<le> (c\<Colon>'b\<Colon>ordered_idom) * \<bar>(f\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar>"
+have 17: "\<not> \<bar>(a\<Colon>'a \<Rightarrow> 'b\<Colon>linordered_idom) (x\<Colon>'a)\<bar>
+ \<le> (c\<Colon>'b\<Colon>linordered_idom) * \<bar>(f\<Colon>'a \<Rightarrow> 'b\<Colon>linordered_idom) x\<bar>"
by (metis 16 2)
show 18: "False"
by (metis 17 1)
@@ -682,7 +682,7 @@
lemma bigo_mult5: "ALL x. f x ~= 0 ==>
- O(f * g) <= (f::'a => ('b::ordered_field)) *o O(g)"
+ O(f * g) <= (f::'a => ('b::linordered_field)) *o O(g)"
proof -
assume "ALL x. f x ~= 0"
show "O(f * g) <= f *o O(g)"
@@ -712,14 +712,14 @@
declare [[ atp_problem_prefix = "BigO__bigo_mult6" ]]
lemma bigo_mult6: "ALL x. f x ~= 0 ==>
- O(f * g) = (f::'a => ('b::ordered_field)) *o O(g)"
+ O(f * g) = (f::'a => ('b::linordered_field)) *o O(g)"
by (metis bigo_mult2 bigo_mult5 order_antisym)
(*proof requires relaxing relevance: 2007-01-25*)
declare [[ atp_problem_prefix = "BigO__bigo_mult7" ]]
declare bigo_mult6 [simp]
lemma bigo_mult7: "ALL x. f x ~= 0 ==>
- O(f * g) <= O(f::'a => ('b::ordered_field)) \<otimes> O(g)"
+ O(f * g) <= O(f::'a => ('b::linordered_field)) \<otimes> O(g)"
(*sledgehammer*)
apply (subst bigo_mult6)
apply assumption
@@ -731,7 +731,7 @@
declare [[ atp_problem_prefix = "BigO__bigo_mult8" ]]
declare bigo_mult7[intro!]
lemma bigo_mult8: "ALL x. f x ~= 0 ==>
- O(f * g) = O(f::'a => ('b::ordered_field)) \<otimes> O(g)"
+ O(f * g) = O(f::'a => ('b::linordered_field)) \<otimes> O(g)"
by (metis bigo_mult bigo_mult7 order_antisym_conv)
lemma bigo_minus [intro]: "f : O(g) ==> - f : O(g)"
@@ -810,11 +810,11 @@
lemma (*bigo_const2 [intro]:*) "O(%x. c) <= O(%x. 1)"
by (metis bigo_const1 bigo_elt_subset);
-lemma bigo_const2 [intro]: "O(%x. c::'b::ordered_idom) <= O(%x. 1)";
+lemma bigo_const2 [intro]: "O(%x. c::'b::linordered_idom) <= O(%x. 1)";
(*??FAILS because the two occurrences of COMBK have different polymorphic types
proof (neg_clausify)
-assume 0: "\<not> O(COMBK (c\<Colon>'b\<Colon>ordered_idom)) \<subseteq> O(COMBK (1\<Colon>'b\<Colon>ordered_idom))"
-have 1: "COMBK (c\<Colon>'b\<Colon>ordered_idom) \<notin> O(COMBK (1\<Colon>'b\<Colon>ordered_idom))"
+assume 0: "\<not> O(COMBK (c\<Colon>'b\<Colon>linordered_idom)) \<subseteq> O(COMBK (1\<Colon>'b\<Colon>linordered_idom))"
+have 1: "COMBK (c\<Colon>'b\<Colon>linordered_idom) \<notin> O(COMBK (1\<Colon>'b\<Colon>linordered_idom))"
apply (rule notI)
apply (rule 0 [THEN notE])
apply (rule bigo_elt_subset)
@@ -830,26 +830,26 @@
done
declare [[ atp_problem_prefix = "BigO__bigo_const3" ]]
-lemma bigo_const3: "(c::'a::ordered_field) ~= 0 ==> (%x. 1) : O(%x. c)"
+lemma bigo_const3: "(c::'a::linordered_field) ~= 0 ==> (%x. 1) : O(%x. c)"
apply (simp add: bigo_def)
proof (neg_clausify)
-assume 0: "(c\<Colon>'a\<Colon>ordered_field) \<noteq> (0\<Colon>'a\<Colon>ordered_field)"
-assume 1: "\<And>A\<Colon>'a\<Colon>ordered_field. \<not> (1\<Colon>'a\<Colon>ordered_field) \<le> A * \<bar>c\<Colon>'a\<Colon>ordered_field\<bar>"
-have 2: "(0\<Colon>'a\<Colon>ordered_field) = \<bar>c\<Colon>'a\<Colon>ordered_field\<bar> \<or>
-\<not> (1\<Colon>'a\<Colon>ordered_field) \<le> (1\<Colon>'a\<Colon>ordered_field)"
+assume 0: "(c\<Colon>'a\<Colon>linordered_field) \<noteq> (0\<Colon>'a\<Colon>linordered_field)"
+assume 1: "\<And>A\<Colon>'a\<Colon>linordered_field. \<not> (1\<Colon>'a\<Colon>linordered_field) \<le> A * \<bar>c\<Colon>'a\<Colon>linordered_field\<bar>"
+have 2: "(0\<Colon>'a\<Colon>linordered_field) = \<bar>c\<Colon>'a\<Colon>linordered_field\<bar> \<or>
+\<not> (1\<Colon>'a\<Colon>linordered_field) \<le> (1\<Colon>'a\<Colon>linordered_field)"
by (metis 1 field_inverse)
-have 3: "\<bar>c\<Colon>'a\<Colon>ordered_field\<bar> = (0\<Colon>'a\<Colon>ordered_field)"
+have 3: "\<bar>c\<Colon>'a\<Colon>linordered_field\<bar> = (0\<Colon>'a\<Colon>linordered_field)"
by (metis linorder_neq_iff linorder_antisym_conv1 2)
-have 4: "(0\<Colon>'a\<Colon>ordered_field) = (c\<Colon>'a\<Colon>ordered_field)"
+have 4: "(0\<Colon>'a\<Colon>linordered_field) = (c\<Colon>'a\<Colon>linordered_field)"
by (metis 3 abs_eq_0)
show "False"
by (metis 0 4)
qed
-lemma bigo_const4: "(c::'a::ordered_field) ~= 0 ==> O(%x. 1) <= O(%x. c)"
+lemma bigo_const4: "(c::'a::linordered_field) ~= 0 ==> O(%x. 1) <= O(%x. c)"
by (rule bigo_elt_subset, rule bigo_const3, assumption)
-lemma bigo_const [simp]: "(c::'a::ordered_field) ~= 0 ==>
+lemma bigo_const [simp]: "(c::'a::linordered_field) ~= 0 ==>
O(%x. c) = O(%x. 1)"
by (rule equalityI, rule bigo_const2, rule bigo_const4, assumption)
@@ -858,9 +858,9 @@
apply (simp add: bigo_def abs_mult)
proof (neg_clausify)
fix x
-assume 0: "\<And>xa\<Colon>'b\<Colon>ordered_idom.
- \<not> \<bar>c\<Colon>'b\<Colon>ordered_idom\<bar> *
- \<bar>(f\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) ((x\<Colon>'b\<Colon>ordered_idom \<Rightarrow> 'a\<Colon>type) xa)\<bar>
+assume 0: "\<And>xa\<Colon>'b\<Colon>linordered_idom.
+ \<not> \<bar>c\<Colon>'b\<Colon>linordered_idom\<bar> *
+ \<bar>(f\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>linordered_idom) ((x\<Colon>'b\<Colon>linordered_idom \<Rightarrow> 'a\<Colon>type) xa)\<bar>
\<le> xa * \<bar>f (x xa)\<bar>"
show "False"
by (metis linorder_neq_iff linorder_antisym_conv1 0)
@@ -870,7 +870,7 @@
by (rule bigo_elt_subset, rule bigo_const_mult1)
declare [[ atp_problem_prefix = "BigO__bigo_const_mult3" ]]
-lemma bigo_const_mult3: "(c::'a::ordered_field) ~= 0 ==> f : O(%x. c * f x)"
+lemma bigo_const_mult3: "(c::'a::linordered_field) ~= 0 ==> f : O(%x. c * f x)"
apply (simp add: bigo_def)
(*sledgehammer [no luck]*);
apply (rule_tac x = "abs(inverse c)" in exI)
@@ -879,16 +879,16 @@
apply (auto );
done
-lemma bigo_const_mult4: "(c::'a::ordered_field) ~= 0 ==>
+lemma bigo_const_mult4: "(c::'a::linordered_field) ~= 0 ==>
O(f) <= O(%x. c * f x)"
by (rule bigo_elt_subset, rule bigo_const_mult3, assumption)
-lemma bigo_const_mult [simp]: "(c::'a::ordered_field) ~= 0 ==>
+lemma bigo_const_mult [simp]: "(c::'a::linordered_field) ~= 0 ==>
O(%x. c * f x) = O(f)"
by (rule equalityI, rule bigo_const_mult2, erule bigo_const_mult4)
declare [[ atp_problem_prefix = "BigO__bigo_const_mult5" ]]
-lemma bigo_const_mult5 [simp]: "(c::'a::ordered_field) ~= 0 ==>
+lemma bigo_const_mult5 [simp]: "(c::'a::linordered_field) ~= 0 ==>
(%x. c) *o O(f) = O(f)"
apply (auto del: subsetI)
apply (rule order_trans)
@@ -1057,7 +1057,7 @@
apply assumption+
done
-lemma bigo_useful_const_mult: "(c::'a::ordered_field) ~= 0 ==>
+lemma bigo_useful_const_mult: "(c::'a::linordered_field) ~= 0 ==>
(%x. c) * f =o O(h) ==> f =o O(h)"
apply (rule subsetD)
apply (subgoal_tac "(%x. 1 / c) *o O(h) <= O(h)")
@@ -1100,7 +1100,7 @@
subsection {* Less than or equal to *}
constdefs
- lesso :: "('a => 'b::ordered_idom) => ('a => 'b) => ('a => 'b)"
+ lesso :: "('a => 'b::linordered_idom) => ('a => 'b) => ('a => 'b)"
(infixl "<o" 70)
"f <o g == (%x. max (f x - g x) 0)"
@@ -1165,7 +1165,7 @@
proof (neg_clausify)
fix x
assume 0: "\<And>A. k A \<le> f A"
-have 1: "\<And>(X1\<Colon>'b\<Colon>ordered_idom) X2. \<not> max X1 X2 < X1"
+have 1: "\<And>(X1\<Colon>'b\<Colon>linordered_idom) X2. \<not> max X1 X2 < X1"
by (metis linorder_not_less le_maxI1) (*sort inserted by hand*)
assume 2: "(0\<Colon>'b) \<le> k x - g x"
have 3: "\<not> k x - g x < (0\<Colon>'b)"
@@ -1206,7 +1206,7 @@
apply (metis abs_ge_zero linorder_linear min_max.sup_absorb1 min_max.sup_commute)
done
-lemma bigo_lesso4: "f <o g =o O(k::'a=>'b::ordered_field) ==>
+lemma bigo_lesso4: "f <o g =o O(k::'a=>'b::linordered_field) ==>
g =o h +o O(k) ==> f <o h =o O(k)"
apply (unfold lesso_def)
apply (drule set_plus_imp_minus)
--- a/src/HOL/Multivariate_Analysis/Derivative.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Multivariate_Analysis/Derivative.thy Mon Feb 08 11:13:30 2010 +0100
@@ -1183,7 +1183,7 @@
fix y assume as:"y \<in> s" "0 < dist y x" "dist y x < e / (B * C * D)"
have "norm (h (f' (y - x)) (g' (y - x))) \<le> norm (f' (y - x)) * norm (g' (y - x)) * B" using B by auto
also have "\<dots> \<le> (norm (y - x) * C) * (D * norm (y - x)) * B" apply(rule mult_right_mono)
- apply(rule pordered_semiring_class.mult_mono) using B C D by (auto simp add: field_simps intro!:mult_nonneg_nonneg)
+ apply(rule mult_mono) using B C D by (auto simp add: field_simps intro!:mult_nonneg_nonneg)
also have "\<dots> = (B * C * D * norm (y - x)) * norm (y - x)" by(auto simp add:field_simps)
also have "\<dots> < e * norm (y - x)" apply(rule mult_strict_right_mono)
using as(3)[unfolded vector_dist_norm] and as(2) unfolding pos_less_divide_eq[OF bcd] by (auto simp add:field_simps)
--- a/src/HOL/Multivariate_Analysis/Determinants.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Multivariate_Analysis/Determinants.thy Mon Feb 08 11:13:30 2010 +0100
@@ -44,7 +44,7 @@
else setprod f {m..n})"
by (auto simp add: atLeastAtMostSuc_conv)
-lemma setprod_le: assumes fS: "finite S" and fg: "\<forall>x\<in>S. f x \<ge> 0 \<and> f x \<le> (g x :: 'a::ordered_idom)"
+lemma setprod_le: assumes fS: "finite S" and fg: "\<forall>x\<in>S. f x \<ge> 0 \<and> f x \<le> (g x :: 'a::linordered_idom)"
shows "setprod f S \<le> setprod g S"
using fS fg
apply(induct S)
@@ -61,7 +61,7 @@
apply simp
done
-lemma setprod_le_1: assumes fS: "finite S" and f: "\<forall>x\<in>S. f x \<ge> 0 \<and> f x \<le> (1::'a::ordered_idom)"
+lemma setprod_le_1: assumes fS: "finite S" and f: "\<forall>x\<in>S. f x \<ge> 0 \<and> f x \<le> (1::'a::linordered_idom)"
shows "setprod f S \<le> 1"
using setprod_le[OF fS f] unfolding setprod_1 .
@@ -277,7 +277,7 @@
qed
lemma det_identical_rows:
- fixes A :: "'a::ordered_idom^'n^'n"
+ fixes A :: "'a::linordered_idom^'n^'n"
assumes ij: "i \<noteq> j"
and r: "row i A = row j A"
shows "det A = 0"
@@ -295,7 +295,7 @@
qed
lemma det_identical_columns:
- fixes A :: "'a::ordered_idom^'n^'n"
+ fixes A :: "'a::linordered_idom^'n^'n"
assumes ij: "i \<noteq> j"
and r: "column i A = column j A"
shows "det A = 0"
@@ -407,7 +407,7 @@
unfolding vector_smult_lzero .
lemma det_row_operation:
- fixes A :: "'a::ordered_idom^'n^'n"
+ fixes A :: "'a::linordered_idom^'n^'n"
assumes ij: "i \<noteq> j"
shows "det (\<chi> k. if k = i then row i A + c *s row j A else row k A) = det A"
proof-
@@ -421,7 +421,7 @@
qed
lemma det_row_span:
- fixes A :: "'a:: ordered_idom^'n^'n"
+ fixes A :: "'a:: linordered_idom^'n^'n"
assumes x: "x \<in> span {row j A |j. j \<noteq> i}"
shows "det (\<chi> k. if k = i then row i A + x else row k A) = det A"
proof-
@@ -462,7 +462,7 @@
(* ------------------------------------------------------------------------- *)
lemma det_dependent_rows:
- fixes A:: "'a::ordered_idom^'n^'n"
+ fixes A:: "'a::linordered_idom^'n^'n"
assumes d: "dependent (rows A)"
shows "det A = 0"
proof-
@@ -488,7 +488,7 @@
ultimately show ?thesis by blast
qed
-lemma det_dependent_columns: assumes d: "dependent(columns (A::'a::ordered_idom^'n^'n))" shows "det A = 0"
+lemma det_dependent_columns: assumes d: "dependent(columns (A::'a::linordered_idom^'n^'n))" shows "det A = 0"
by (metis d det_dependent_rows rows_transp det_transp)
(* ------------------------------------------------------------------------- *)
@@ -608,7 +608,7 @@
qed
lemma det_mul:
- fixes A B :: "'a::ordered_idom^'n^'n"
+ fixes A B :: "'a::linordered_idom^'n^'n"
shows "det (A ** B) = det A * det B"
proof-
let ?U = "UNIV :: 'n set"
@@ -761,7 +761,7 @@
(* ------------------------------------------------------------------------- *)
lemma cramer_lemma_transp:
- fixes A:: "'a::ordered_idom^'n^'n" and x :: "'a ^'n"
+ fixes A:: "'a::linordered_idom^'n^'n" and x :: "'a ^'n"
shows "det ((\<chi> i. if i = k then setsum (\<lambda>i. x$i *s row i A) (UNIV::'n set)
else row i A)::'a^'n^'n) = x$k * det A"
(is "?lhs = ?rhs")
@@ -797,7 +797,7 @@
qed
lemma cramer_lemma:
- fixes A :: "'a::ordered_idom ^'n^'n"
+ fixes A :: "'a::linordered_idom ^'n^'n"
shows "det((\<chi> i j. if j = k then (A *v x)$i else A$i$j):: 'a^'n^'n) = x$k * det A"
proof-
let ?U = "UNIV :: 'n set"
@@ -893,7 +893,7 @@
qed
lemma det_orthogonal_matrix:
- fixes Q:: "'a::ordered_idom^'n^'n"
+ fixes Q:: "'a::linordered_idom^'n^'n"
assumes oQ: "orthogonal_matrix Q"
shows "det Q = 1 \<or> det Q = - 1"
proof-
@@ -1034,7 +1034,7 @@
definition "rotoinversion_matrix Q \<longleftrightarrow> orthogonal_matrix Q \<and> det Q = - 1"
lemma orthogonal_rotation_or_rotoinversion:
- fixes Q :: "'a::ordered_idom^'n^'n"
+ fixes Q :: "'a::linordered_idom^'n^'n"
shows " orthogonal_matrix Q \<longleftrightarrow> rotation_matrix Q \<or> rotoinversion_matrix Q"
by (metis rotoinversion_matrix_def rotation_matrix_def det_orthogonal_matrix)
(* ------------------------------------------------------------------------- *)
--- a/src/HOL/Multivariate_Analysis/Euclidean_Space.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Multivariate_Analysis/Euclidean_Space.thy Mon Feb 08 11:13:30 2010 +0100
@@ -459,7 +459,7 @@
done
lemma setsum_nonneg_eq_0_iff:
- fixes f :: "'a \<Rightarrow> 'b::pordered_ab_group_add"
+ fixes f :: "'a \<Rightarrow> 'b::ordered_ab_group_add"
shows "\<lbrakk>finite A; \<forall>x\<in>A. 0 \<le> f x\<rbrakk> \<Longrightarrow> setsum f A = 0 \<longleftrightarrow> (\<forall>x\<in>A. f x = 0)"
apply (induct set: finite, simp)
apply (simp add: add_nonneg_eq_0_iff setsum_nonneg)
@@ -836,10 +836,10 @@
lemma dot_rneg: "(x::'a::ring ^ 'n) \<bullet> (-y) = -(x \<bullet> y)" by vector
lemma dot_lzero[simp]: "0 \<bullet> x = (0::'a::{comm_monoid_add, mult_zero})" by vector
lemma dot_rzero[simp]: "x \<bullet> 0 = (0::'a::{comm_monoid_add, mult_zero})" by vector
-lemma dot_pos_le[simp]: "(0::'a\<Colon>ordered_ring_strict) <= x \<bullet> x"
+lemma dot_pos_le[simp]: "(0::'a\<Colon>linlinordered_ring_strict) <= x \<bullet> x"
by (simp add: dot_def setsum_nonneg)
-lemma setsum_squares_eq_0_iff: assumes fS: "finite F" and fp: "\<forall>x \<in> F. f x \<ge> (0 ::'a::pordered_ab_group_add)" shows "setsum f F = 0 \<longleftrightarrow> (ALL x:F. f x = 0)"
+lemma setsum_squares_eq_0_iff: assumes fS: "finite F" and fp: "\<forall>x \<in> F. f x \<ge> (0 ::'a::ordered_ab_group_add)" shows "setsum f F = 0 \<longleftrightarrow> (ALL x:F. f x = 0)"
using fS fp setsum_nonneg[OF fp]
proof (induct set: finite)
case empty thus ?case by simp
@@ -852,10 +852,10 @@
show ?case by (simp add: h)
qed
-lemma dot_eq_0: "x \<bullet> x = 0 \<longleftrightarrow> (x::'a::{ordered_ring_strict,ring_no_zero_divisors} ^ 'n) = 0"
+lemma dot_eq_0: "x \<bullet> x = 0 \<longleftrightarrow> (x::'a::{linlinordered_ring_strict,ring_no_zero_divisors} ^ 'n) = 0"
by (simp add: dot_def setsum_squares_eq_0_iff Cart_eq)
-lemma dot_pos_lt[simp]: "(0 < x \<bullet> x) \<longleftrightarrow> (x::'a::{ordered_ring_strict,ring_no_zero_divisors} ^ 'n) \<noteq> 0" using dot_eq_0[of x] dot_pos_le[of x]
+lemma dot_pos_lt[simp]: "(0 < x \<bullet> x) \<longleftrightarrow> (x::'a::{linlinordered_ring_strict,ring_no_zero_divisors} ^ 'n) \<noteq> 0" using dot_eq_0[of x] dot_pos_le[of x]
by (auto simp add: le_less)
subsection{* The collapse of the general concepts to dimension one. *}
@@ -1146,7 +1146,7 @@
shows "norm x1 \<le> b1 \<Longrightarrow> norm x2 \<le> b2 \<Longrightarrow> norm (x1 + x2) \<le> b1 + b2"
by (rule order_trans [OF norm_triangle_ineq add_mono])
-lemma ge_iff_diff_ge_0: "(a::'a::ordered_ring) \<ge> b == a - b \<ge> 0"
+lemma ge_iff_diff_ge_0: "(a::'a::linordered_ring) \<ge> b == a - b \<ge> 0"
by (simp add: ring_simps)
lemma pth_1:
@@ -3827,7 +3827,7 @@
(* FIXME : Move to some general theory ?*)
definition "pairwise R S \<longleftrightarrow> (\<forall>x \<in> S. \<forall>y\<in> S. x\<noteq>y \<longrightarrow> R x y)"
-lemma vector_sub_project_orthogonal: "(b::'a::ordered_field^'n) \<bullet> (x - ((b \<bullet> x) / (b\<bullet>b)) *s b) = 0"
+lemma vector_sub_project_orthogonal: "(b::'a::linordered_field^'n) \<bullet> (x - ((b \<bullet> x) / (b\<bullet>b)) *s b) = 0"
apply (cases "b = 0", simp)
apply (simp add: dot_rsub dot_rmult)
unfolding times_divide_eq_right[symmetric]
--- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy Mon Feb 08 11:13:30 2010 +0100
@@ -5696,27 +5696,27 @@
by (simp add: vector_smult_lneg[symmetric] vector_smult_assoc vector_sneg_minus1[symmetric])
lemma real_affinity_le:
- "0 < (m::'a::ordered_field) ==> (m * x + c \<le> y \<longleftrightarrow> x \<le> inverse(m) * y + -(c / m))"
+ "0 < (m::'a::linordered_field) ==> (m * x + c \<le> y \<longleftrightarrow> x \<le> inverse(m) * y + -(c / m))"
by (simp add: field_simps inverse_eq_divide)
lemma real_le_affinity:
- "0 < (m::'a::ordered_field) ==> (y \<le> m * x + c \<longleftrightarrow> inverse(m) * y + -(c / m) \<le> x)"
+ "0 < (m::'a::linordered_field) ==> (y \<le> m * x + c \<longleftrightarrow> inverse(m) * y + -(c / m) \<le> x)"
by (simp add: field_simps inverse_eq_divide)
lemma real_affinity_lt:
- "0 < (m::'a::ordered_field) ==> (m * x + c < y \<longleftrightarrow> x < inverse(m) * y + -(c / m))"
+ "0 < (m::'a::linordered_field) ==> (m * x + c < y \<longleftrightarrow> x < inverse(m) * y + -(c / m))"
by (simp add: field_simps inverse_eq_divide)
lemma real_lt_affinity:
- "0 < (m::'a::ordered_field) ==> (y < m * x + c \<longleftrightarrow> inverse(m) * y + -(c / m) < x)"
+ "0 < (m::'a::linordered_field) ==> (y < m * x + c \<longleftrightarrow> inverse(m) * y + -(c / m) < x)"
by (simp add: field_simps inverse_eq_divide)
lemma real_affinity_eq:
- "(m::'a::ordered_field) \<noteq> 0 ==> (m * x + c = y \<longleftrightarrow> x = inverse(m) * y + -(c / m))"
+ "(m::'a::linordered_field) \<noteq> 0 ==> (m * x + c = y \<longleftrightarrow> x = inverse(m) * y + -(c / m))"
by (simp add: field_simps inverse_eq_divide)
lemma real_eq_affinity:
- "(m::'a::ordered_field) \<noteq> 0 ==> (y = m * x + c \<longleftrightarrow> inverse(m) * y + -(c / m) = x)"
+ "(m::'a::linordered_field) \<noteq> 0 ==> (y = m * x + c \<longleftrightarrow> inverse(m) * y + -(c / m) = x)"
by (simp add: field_simps inverse_eq_divide)
lemma vector_affinity_eq:
--- a/src/HOL/NSA/HyperDef.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/NSA/HyperDef.thy Mon Feb 08 11:13:30 2010 +0100
@@ -459,7 +459,7 @@
by transfer (rule power_inverse)
lemma hyperpow_hrabs:
- "\<And>r n. abs (r::'a::{ordered_idom} star) pow n = abs (r pow n)"
+ "\<And>r n. abs (r::'a::{linordered_idom} star) pow n = abs (r pow n)"
by transfer (rule power_abs [symmetric])
lemma hyperpow_add:
@@ -475,15 +475,15 @@
by transfer simp
lemma hyperpow_gt_zero:
- "\<And>r n. (0::'a::{ordered_semidom} star) < r \<Longrightarrow> 0 < r pow n"
+ "\<And>r n. (0::'a::{linordered_semidom} star) < r \<Longrightarrow> 0 < r pow n"
by transfer (rule zero_less_power)
lemma hyperpow_ge_zero:
- "\<And>r n. (0::'a::{ordered_semidom} star) \<le> r \<Longrightarrow> 0 \<le> r pow n"
+ "\<And>r n. (0::'a::{linordered_semidom} star) \<le> r \<Longrightarrow> 0 \<le> r pow n"
by transfer (rule zero_le_power)
lemma hyperpow_le:
- "\<And>x y n. \<lbrakk>(0::'a::{ordered_semidom} star) < x; x \<le> y\<rbrakk>
+ "\<And>x y n. \<lbrakk>(0::'a::{linordered_semidom} star) < x; x \<le> y\<rbrakk>
\<Longrightarrow> x pow n \<le> y pow n"
by transfer (rule power_mono [OF _ order_less_imp_le])
@@ -492,7 +492,7 @@
by transfer (rule power_one)
lemma hrabs_hyperpow_minus_one [simp]:
- "\<And>n. abs(-1 pow n) = (1::'a::{number_ring,ordered_idom} star)"
+ "\<And>n. abs(-1 pow n) = (1::'a::{number_ring,linordered_idom} star)"
by transfer (rule abs_power_minus_one)
lemma hyperpow_mult:
@@ -501,16 +501,16 @@
by transfer (rule power_mult_distrib)
lemma hyperpow_two_le [simp]:
- "(0::'a::{monoid_mult,ordered_ring_strict} star) \<le> r pow (1 + 1)"
+ "(0::'a::{monoid_mult,linlinordered_ring_strict} star) \<le> r pow (1 + 1)"
by (auto simp add: hyperpow_two zero_le_mult_iff)
lemma hrabs_hyperpow_two [simp]:
"abs(x pow (1 + 1)) =
- (x::'a::{monoid_mult,ordered_ring_strict} star) pow (1 + 1)"
+ (x::'a::{monoid_mult,linlinordered_ring_strict} star) pow (1 + 1)"
by (simp only: abs_of_nonneg hyperpow_two_le)
lemma hyperpow_two_hrabs [simp]:
- "abs(x::'a::{ordered_idom} star) pow (1 + 1) = x pow (1 + 1)"
+ "abs(x::'a::{linordered_idom} star) pow (1 + 1) = x pow (1 + 1)"
by (simp add: hyperpow_hrabs)
text{*The precondition could be weakened to @{term "0\<le>x"}*}
@@ -519,11 +519,11 @@
by (simp add: Ring_and_Field.mult_strict_mono order_less_imp_le)
lemma hyperpow_two_gt_one:
- "\<And>r::'a::{ordered_semidom} star. 1 < r \<Longrightarrow> 1 < r pow (1 + 1)"
+ "\<And>r::'a::{linordered_semidom} star. 1 < r \<Longrightarrow> 1 < r pow (1 + 1)"
by transfer (simp add: power_gt1 del: power_Suc)
lemma hyperpow_two_ge_one:
- "\<And>r::'a::{ordered_semidom} star. 1 \<le> r \<Longrightarrow> 1 \<le> r pow (1 + 1)"
+ "\<And>r::'a::{linordered_semidom} star. 1 \<le> r \<Longrightarrow> 1 \<le> r pow (1 + 1)"
by transfer (simp add: one_le_power del: power_Suc)
lemma two_hyperpow_ge_one [simp]: "(1::hypreal) \<le> 2 pow n"
--- a/src/HOL/NSA/HyperNat.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/NSA/HyperNat.thy Mon Feb 08 11:13:30 2010 +0100
@@ -386,39 +386,39 @@
by transfer (rule of_nat_mult)
lemma of_hypnat_less_iff [simp]:
- "\<And>m n. (of_hypnat m < (of_hypnat n::'a::ordered_semidom star)) = (m < n)"
+ "\<And>m n. (of_hypnat m < (of_hypnat n::'a::linordered_semidom star)) = (m < n)"
by transfer (rule of_nat_less_iff)
lemma of_hypnat_0_less_iff [simp]:
- "\<And>n. (0 < (of_hypnat n::'a::ordered_semidom star)) = (0 < n)"
+ "\<And>n. (0 < (of_hypnat n::'a::linordered_semidom star)) = (0 < n)"
by transfer (rule of_nat_0_less_iff)
lemma of_hypnat_less_0_iff [simp]:
- "\<And>m. \<not> (of_hypnat m::'a::ordered_semidom star) < 0"
+ "\<And>m. \<not> (of_hypnat m::'a::linordered_semidom star) < 0"
by transfer (rule of_nat_less_0_iff)
lemma of_hypnat_le_iff [simp]:
- "\<And>m n. (of_hypnat m \<le> (of_hypnat n::'a::ordered_semidom star)) = (m \<le> n)"
+ "\<And>m n. (of_hypnat m \<le> (of_hypnat n::'a::linordered_semidom star)) = (m \<le> n)"
by transfer (rule of_nat_le_iff)
lemma of_hypnat_0_le_iff [simp]:
- "\<And>n. 0 \<le> (of_hypnat n::'a::ordered_semidom star)"
+ "\<And>n. 0 \<le> (of_hypnat n::'a::linordered_semidom star)"
by transfer (rule of_nat_0_le_iff)
lemma of_hypnat_le_0_iff [simp]:
- "\<And>m. ((of_hypnat m::'a::ordered_semidom star) \<le> 0) = (m = 0)"
+ "\<And>m. ((of_hypnat m::'a::linordered_semidom star) \<le> 0) = (m = 0)"
by transfer (rule of_nat_le_0_iff)
lemma of_hypnat_eq_iff [simp]:
- "\<And>m n. (of_hypnat m = (of_hypnat n::'a::ordered_semidom star)) = (m = n)"
+ "\<And>m n. (of_hypnat m = (of_hypnat n::'a::linordered_semidom star)) = (m = n)"
by transfer (rule of_nat_eq_iff)
lemma of_hypnat_eq_0_iff [simp]:
- "\<And>m. ((of_hypnat m::'a::ordered_semidom star) = 0) = (m = 0)"
+ "\<And>m. ((of_hypnat m::'a::linordered_semidom star) = 0) = (m = 0)"
by transfer (rule of_nat_eq_0_iff)
lemma HNatInfinite_of_hypnat_gt_zero:
- "N \<in> HNatInfinite \<Longrightarrow> (0::'a::ordered_semidom star) < of_hypnat N"
+ "N \<in> HNatInfinite \<Longrightarrow> (0::'a::linordered_semidom star) < of_hypnat N"
by (rule ccontr, simp add: linorder_not_less)
end
--- a/src/HOL/NSA/StarDef.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/NSA/StarDef.thy Mon Feb 08 11:13:30 2010 +0100
@@ -719,7 +719,7 @@
apply (transfer, erule (1) order_antisym)
done
-instantiation star :: (lower_semilattice) lower_semilattice
+instantiation star :: (semilattice_inf) semilattice_inf
begin
definition
@@ -730,7 +730,7 @@
end
-instantiation star :: (upper_semilattice) upper_semilattice
+instantiation star :: (semilattice_sup) semilattice_sup
begin
definition
@@ -833,27 +833,27 @@
apply (transfer, rule diff_minus)
done
-instance star :: (pordered_ab_semigroup_add) pordered_ab_semigroup_add
+instance star :: (ordered_ab_semigroup_add) ordered_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 :: (ordered_cancel_ab_semigroup_add) ordered_cancel_ab_semigroup_add ..
-instance star :: (pordered_ab_semigroup_add_imp_le) pordered_ab_semigroup_add_imp_le
+instance star :: (ordered_ab_semigroup_add_imp_le) ordered_ab_semigroup_add_imp_le
by (intro_classes, transfer, rule add_le_imp_le_left)
-instance star :: (pordered_comm_monoid_add) pordered_comm_monoid_add ..
-instance star :: (pordered_ab_group_add) pordered_ab_group_add ..
+instance star :: (ordered_comm_monoid_add) ordered_comm_monoid_add ..
+instance star :: (ordered_ab_group_add) ordered_ab_group_add ..
-instance star :: (pordered_ab_group_add_abs) pordered_ab_group_add_abs
+instance star :: (ordered_ab_group_add_abs) ordered_ab_group_add_abs
by intro_classes (transfer,
simp add: abs_ge_self abs_leI abs_triangle_ineq)+
-instance star :: (ordered_cancel_ab_semigroup_add) ordered_cancel_ab_semigroup_add ..
-instance star :: (lordered_ab_group_add_meet) lordered_ab_group_add_meet ..
-instance star :: (lordered_ab_group_add_meet) lordered_ab_group_add_meet ..
-instance star :: (lordered_ab_group_add) lordered_ab_group_add ..
+instance star :: (linordered_cancel_ab_semigroup_add) linordered_cancel_ab_semigroup_add ..
+instance star :: (semilattice_inf_ab_group_add) semilattice_inf_ab_group_add ..
+instance star :: (semilattice_inf_ab_group_add) semilattice_inf_ab_group_add ..
+instance star :: (lattice_ab_group_add) lattice_ab_group_add ..
-instance star :: (lordered_ab_group_add_abs) lordered_ab_group_add_abs
+instance star :: (lattice_ab_group_add_abs) lattice_ab_group_add_abs
by (intro_classes, transfer, rule abs_lattice)
subsection {* Ring and field classes *}
@@ -909,32 +909,32 @@
instance star :: (division_by_zero) division_by_zero
by (intro_classes, transfer, rule inverse_zero)
-instance star :: (pordered_semiring) pordered_semiring
+instance star :: (ordered_semiring) ordered_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_cancel_semiring) ordered_cancel_semiring ..
-instance star :: (ordered_semiring_strict) ordered_semiring_strict
+instance star :: (linlinordered_semiring_strict) linlinordered_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
+instance star :: (ordered_comm_semiring) ordered_comm_semiring
by (intro_classes, transfer, rule mult_mono1_class.mult_mono1)
-instance star :: (pordered_cancel_comm_semiring) pordered_cancel_comm_semiring ..
+instance star :: (ordered_cancel_comm_semiring) ordered_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_left_mono_comm)
+instance star :: (linordered_comm_semiring_strict) linordered_comm_semiring_strict
+by (intro_classes, transfer, rule mult_strict_left_mono_comm)
-instance star :: (pordered_ring) pordered_ring ..
-instance star :: (pordered_ring_abs) pordered_ring_abs
+instance star :: (ordered_ring) ordered_ring ..
+instance star :: (ordered_ring_abs) ordered_ring_abs
by intro_classes (transfer, rule abs_eq_mult)
-instance star :: (lordered_ring) lordered_ring ..
+instance star :: (lattice_ring) lattice_ring ..
instance star :: (abs_if) abs_if
by (intro_classes, transfer, rule abs_if)
@@ -942,14 +942,14 @@
instance star :: (sgn_if) sgn_if
by (intro_classes, transfer, rule sgn_if)
-instance star :: (ordered_ring_strict) ordered_ring_strict ..
-instance star :: (pordered_comm_ring) pordered_comm_ring ..
+instance star :: (linlinordered_ring_strict) linlinordered_ring_strict ..
+instance star :: (ordered_comm_ring) ordered_comm_ring ..
-instance star :: (ordered_semidom) ordered_semidom
+instance star :: (linordered_semidom) linordered_semidom
by (intro_classes, transfer, rule zero_less_one)
-instance star :: (ordered_idom) ordered_idom ..
-instance star :: (ordered_field) ordered_field ..
+instance star :: (linordered_idom) linordered_idom ..
+instance star :: (linordered_field) linordered_field ..
subsection {* Power *}
--- a/src/HOL/Nat.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Nat.thy Mon Feb 08 11:13:30 2010 +0100
@@ -741,7 +741,7 @@
done
text{*The naturals form an ordered @{text comm_semiring_1_cancel}*}
-instance nat :: ordered_semidom
+instance nat :: linordered_semidom
proof
fix i j k :: nat
show "0 < (1::nat)" by simp
@@ -1289,7 +1289,7 @@
end
-context ordered_semidom
+context linordered_semidom
begin
lemma zero_le_imp_of_nat: "0 \<le> of_nat m"
@@ -1316,7 +1316,7 @@
lemma of_nat_le_iff [simp]: "of_nat m \<le> of_nat n \<longleftrightarrow> m \<le> n"
by (simp add: not_less [symmetric] linorder_not_less [symmetric])
-text{*Every @{text ordered_semidom} has characteristic zero.*}
+text{*Every @{text linordered_semidom} has characteristic zero.*}
subclass semiring_char_0
proof qed (simp add: eq_iff order_eq_iff)
@@ -1345,7 +1345,7 @@
end
-context ordered_idom
+context linordered_idom
begin
lemma abs_of_nat [simp]: "\<bar>of_nat n\<bar> = of_nat n"
--- a/src/HOL/Nat_Numeral.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Nat_Numeral.thy Mon Feb 08 11:13:30 2010 +0100
@@ -113,7 +113,7 @@
end
-context ordered_ring_strict
+context linlinordered_ring_strict
begin
lemma sum_squares_ge_zero:
@@ -145,7 +145,7 @@
end
-context ordered_semidom
+context linordered_semidom
begin
lemma power2_le_imp_le:
@@ -162,7 +162,7 @@
end
-context ordered_idom
+context linordered_idom
begin
lemma zero_eq_power2 [simp]:
--- a/src/HOL/Nitpick.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Nitpick.thy Mon Feb 08 11:13:30 2010 +0100
@@ -224,13 +224,13 @@
"Nitpick_Nut" so that they handle the unexpanded overloaded constants
directly, but this is slightly more tricky to implement. *)
lemmas [nitpick_def] = div_int_inst.div_int div_int_inst.mod_int
- div_nat_inst.div_nat div_nat_inst.mod_nat lower_semilattice_fun_inst.inf_fun
+ div_nat_inst.div_nat div_nat_inst.mod_nat semilattice_inf_fun_inst.inf_fun
minus_fun_inst.minus_fun minus_int_inst.minus_int minus_nat_inst.minus_nat
one_int_inst.one_int one_nat_inst.one_nat ord_fun_inst.less_eq_fun
ord_int_inst.less_eq_int ord_int_inst.less_int ord_nat_inst.less_eq_nat
ord_nat_inst.less_nat plus_int_inst.plus_int plus_nat_inst.plus_nat
times_int_inst.times_int times_nat_inst.times_nat uminus_int_inst.uminus_int
- upper_semilattice_fun_inst.sup_fun zero_int_inst.zero_int
+ semilattice_sup_fun_inst.sup_fun zero_int_inst.zero_int
zero_nat_inst.zero_nat
use "Tools/Nitpick/kodkod.ML"
--- a/src/HOL/OrderedGroup.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/OrderedGroup.thy Mon Feb 08 11:13:30 2010 +0100
@@ -353,7 +353,7 @@
subsection {* (Partially) Ordered Groups *}
-class pordered_ab_semigroup_add = order + ab_semigroup_add +
+class ordered_ab_semigroup_add = order + ab_semigroup_add +
assumes add_left_mono: "a \<le> b \<Longrightarrow> c + a \<le> c + b"
begin
@@ -370,8 +370,8 @@
end
-class pordered_cancel_ab_semigroup_add =
- pordered_ab_semigroup_add + cancel_ab_semigroup_add
+class ordered_cancel_ab_semigroup_add =
+ ordered_ab_semigroup_add + cancel_ab_semigroup_add
begin
lemma add_strict_left_mono:
@@ -403,8 +403,8 @@
end
-class pordered_ab_semigroup_add_imp_le =
- pordered_cancel_ab_semigroup_add +
+class ordered_ab_semigroup_add_imp_le =
+ ordered_cancel_ab_semigroup_add +
assumes add_le_imp_le_left: "c + a \<le> c + b \<Longrightarrow> a \<le> b"
begin
@@ -464,8 +464,8 @@
subsection {* Support for reasoning about signs *}
-class pordered_comm_monoid_add =
- pordered_cancel_ab_semigroup_add + comm_monoid_add
+class ordered_comm_monoid_add =
+ ordered_cancel_ab_semigroup_add + comm_monoid_add
begin
lemma add_pos_nonneg:
@@ -550,13 +550,13 @@
end
-class pordered_ab_group_add =
- ab_group_add + pordered_ab_semigroup_add
+class ordered_ab_group_add =
+ ab_group_add + ordered_ab_semigroup_add
begin
-subclass pordered_cancel_ab_semigroup_add ..
+subclass ordered_cancel_ab_semigroup_add ..
-subclass pordered_ab_semigroup_add_imp_le
+subclass ordered_ab_semigroup_add_imp_le
proof
fix a b c :: 'a
assume "c + a \<le> c + b"
@@ -565,7 +565,7 @@
thus "a \<le> b" by simp
qed
-subclass pordered_comm_monoid_add ..
+subclass ordered_comm_monoid_add ..
lemma max_diff_distrib_left:
shows "max x y - z = max (x - z) (y - z)"
@@ -675,16 +675,16 @@
text{*Legacy - use @{text algebra_simps} *}
lemmas group_simps[noatp] = algebra_simps
-class ordered_ab_semigroup_add =
- linorder + pordered_ab_semigroup_add
+class linordered_ab_semigroup_add =
+ linorder + ordered_ab_semigroup_add
-class ordered_cancel_ab_semigroup_add =
- linorder + pordered_cancel_ab_semigroup_add
+class linordered_cancel_ab_semigroup_add =
+ linorder + ordered_cancel_ab_semigroup_add
begin
-subclass ordered_ab_semigroup_add ..
+subclass linordered_ab_semigroup_add ..
-subclass pordered_ab_semigroup_add_imp_le
+subclass ordered_ab_semigroup_add_imp_le
proof
fix a b c :: 'a
assume le: "c + a <= c + b"
@@ -705,11 +705,10 @@
end
-class ordered_ab_group_add =
- linorder + pordered_ab_group_add
+class linordered_ab_group_add = linorder + ordered_ab_group_add
begin
-subclass ordered_cancel_ab_semigroup_add ..
+subclass linordered_cancel_ab_semigroup_add ..
lemma neg_less_eq_nonneg:
"- a \<le> a \<longleftrightarrow> 0 \<le> a"
@@ -775,27 +774,27 @@
-- {* FIXME localize the following *}
lemma add_increasing:
- fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}"
+ fixes c :: "'a::{ordered_ab_semigroup_add_imp_le, comm_monoid_add}"
shows "[|0\<le>a; b\<le>c|] ==> b \<le> a + c"
by (insert add_mono [of 0 a b c], simp)
lemma add_increasing2:
- fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}"
+ fixes c :: "'a::{ordered_ab_semigroup_add_imp_le, comm_monoid_add}"
shows "[|0\<le>c; b\<le>a|] ==> b \<le> a + c"
by (simp add:add_increasing add_commute[of a])
lemma add_strict_increasing:
- fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}"
+ fixes c :: "'a::{ordered_ab_semigroup_add_imp_le, comm_monoid_add}"
shows "[|0<a; b\<le>c|] ==> b < a + c"
by (insert add_less_le_mono [of 0 a b c], simp)
lemma add_strict_increasing2:
- fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}"
+ fixes c :: "'a::{ordered_ab_semigroup_add_imp_le, comm_monoid_add}"
shows "[|0\<le>a; b<c|] ==> b < a + c"
by (insert add_le_less_mono [of 0 a b c], simp)
-class pordered_ab_group_add_abs = pordered_ab_group_add + abs +
+class ordered_ab_group_add_abs = ordered_ab_group_add + abs +
assumes abs_ge_zero [simp]: "\<bar>a\<bar> \<ge> 0"
and abs_ge_self: "a \<le> \<bar>a\<bar>"
and abs_leI: "a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b"
@@ -945,7 +944,7 @@
subsection {* Lattice Ordered (Abelian) Groups *}
-class lordered_ab_group_add_meet = pordered_ab_group_add + lower_semilattice
+class semilattice_inf_ab_group_add = ordered_ab_group_add + semilattice_inf
begin
lemma add_inf_distrib_left:
@@ -967,7 +966,7 @@
end
-class lordered_ab_group_add_join = pordered_ab_group_add + upper_semilattice
+class semilattice_sup_ab_group_add = ordered_ab_group_add + semilattice_sup
begin
lemma add_sup_distrib_left:
@@ -990,11 +989,11 @@
end
-class lordered_ab_group_add = pordered_ab_group_add + lattice
+class lattice_ab_group_add = ordered_ab_group_add + lattice
begin
-subclass lordered_ab_group_add_meet ..
-subclass lordered_ab_group_add_join ..
+subclass semilattice_inf_ab_group_add ..
+subclass semilattice_sup_ab_group_add ..
lemmas add_sup_inf_distribs = add_inf_distrib_right add_inf_distrib_left add_sup_distrib_right add_sup_distrib_left
@@ -1244,7 +1243,7 @@
lemmas add_sup_inf_distribs = add_inf_distrib_right add_inf_distrib_left add_sup_distrib_right add_sup_distrib_left
-class lordered_ab_group_add_abs = lordered_ab_group_add + abs +
+class lattice_ab_group_add_abs = lattice_ab_group_add + abs +
assumes abs_lattice: "\<bar>a\<bar> = sup a (- a)"
begin
@@ -1262,7 +1261,7 @@
pprt_def nprt_def diff_minus abs_lattice)
qed
-subclass pordered_ab_group_add_abs
+subclass ordered_ab_group_add_abs
proof
have abs_ge_zero [simp]: "\<And>a. 0 \<le> \<bar>a\<bar>"
proof -
@@ -1297,7 +1296,7 @@
end
lemma sup_eq_if:
- fixes a :: "'a\<Colon>{lordered_ab_group_add, linorder}"
+ fixes a :: "'a\<Colon>{lattice_ab_group_add, linorder}"
shows "sup a (- a) = (if a < 0 then - a else a)"
proof -
note add_le_cancel_right [of a a "- a", symmetric, simplified]
@@ -1306,7 +1305,7 @@
qed
lemma abs_if_lattice:
- fixes a :: "'a\<Colon>{lordered_ab_group_add_abs, linorder}"
+ fixes a :: "'a\<Colon>{lattice_ab_group_add_abs, linorder}"
shows "\<bar>a\<bar> = (if a < 0 then - a else a)"
by auto
@@ -1324,10 +1323,10 @@
apply (simp add: add_right_cancel[symmetric, of "x" "-z" "z", simplified])
done
-lemma less_eqI: "(x::'a::pordered_ab_group_add) - y = x' - y' \<Longrightarrow> (x < y) = (x' < y')"
+lemma less_eqI: "(x::'a::ordered_ab_group_add) - y = x' - y' \<Longrightarrow> (x < y) = (x' < y')"
by (simp add: less_iff_diff_less_0[of x y] less_iff_diff_less_0[of x' y'])
-lemma le_eqI: "(x::'a::pordered_ab_group_add) - y = x' - y' \<Longrightarrow> (y <= x) = (y' <= x')"
+lemma le_eqI: "(x::'a::ordered_ab_group_add) - y = x' - y' \<Longrightarrow> (y <= x) = (y' <= x')"
apply (simp add: le_iff_diff_le_0[of y x] le_iff_diff_le_0[of y' x'])
apply (simp add: neg_le_iff_le[symmetric, of "y-x" 0] neg_le_iff_le[symmetric, of "y'-x'" 0])
done
@@ -1340,7 +1339,7 @@
lemma le_add_right_mono:
assumes
- "a <= b + (c::'a::pordered_ab_group_add)"
+ "a <= b + (c::'a::ordered_ab_group_add)"
"c <= d"
shows "a <= b + d"
apply (rule_tac order_trans[where y = "b+c"])
@@ -1348,7 +1347,7 @@
done
lemma estimate_by_abs:
- "a + b <= (c::'a::lordered_ab_group_add_abs) \<Longrightarrow> a <= c + abs b"
+ "a + b <= (c::'a::lattice_ab_group_add_abs) \<Longrightarrow> a <= c + abs b"
proof -
assume "a+b <= c"
hence 2: "a <= c+(-b)" by (simp add: algebra_simps)
@@ -1358,16 +1357,16 @@
subsection {* Tools setup *}
-lemma add_mono_thms_ordered_semiring [noatp]:
- fixes i j k :: "'a\<Colon>pordered_ab_semigroup_add"
+lemma add_mono_thms_linordered_semiring [noatp]:
+ fixes i j k :: "'a\<Colon>ordered_ab_semigroup_add"
shows "i \<le> j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
and "i = j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
and "i \<le> j \<and> k = l \<Longrightarrow> i + k \<le> j + l"
and "i = j \<and> k = l \<Longrightarrow> i + k = j + l"
by (rule add_mono, clarify+)+
-lemma add_mono_thms_ordered_field [noatp]:
- fixes i j k :: "'a\<Colon>pordered_cancel_ab_semigroup_add"
+lemma add_mono_thms_linordered_field [noatp]:
+ fixes i j k :: "'a\<Colon>ordered_cancel_ab_semigroup_add"
shows "i < j \<and> k = l \<Longrightarrow> i + k < j + l"
and "i = j \<and> k < l \<Longrightarrow> i + k < j + l"
and "i < j \<and> k \<le> l \<Longrightarrow> i + k < j + l"
--- a/src/HOL/Orderings.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Orderings.thy Mon Feb 08 11:13:30 2010 +0100
@@ -1052,7 +1052,7 @@
subsection {* Dense orders *}
-class dense_linear_order = linorder +
+class dense_linorder = linorder +
assumes gt_ex: "\<exists>y. x < y"
and lt_ex: "\<exists>y. y < x"
and dense: "x < y \<Longrightarrow> (\<exists>z. x < z \<and> z < y)"
--- a/src/HOL/PReal.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/PReal.thy Mon Feb 08 11:13:30 2010 +0100
@@ -23,7 +23,7 @@
(\<forall>y \<in> A. ((\<forall>z. 0<z & z < y --> z \<in> A) & (\<exists>u \<in> A. y < u))))"
lemma interval_empty_iff:
- "{y. (x::'a::dense_linear_order) < y \<and> y < z} = {} \<longleftrightarrow> \<not> x < z"
+ "{y. (x::'a::dense_linorder) < y \<and> y < z} = {} \<longleftrightarrow> \<not> x < z"
by (auto dest: dense)
@@ -1155,7 +1155,7 @@
preal_add_le_cancel_right preal_add_le_cancel_left
preal_add_left_cancel_iff preal_add_right_cancel_iff
-instance preal :: ordered_cancel_ab_semigroup_add
+instance preal :: linordered_cancel_ab_semigroup_add
proof
fix a b c :: preal
show "a + b = a + c \<Longrightarrow> b = c" by (rule preal_add_left_cancel)
--- a/src/HOL/Parity.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Parity.thy Mon Feb 08 11:13:30 2010 +0100
@@ -218,7 +218,7 @@
done
lemma zero_le_even_power: "even n ==>
- 0 <= (x::'a::{ordered_ring_strict,monoid_mult}) ^ n"
+ 0 <= (x::'a::{linlinordered_ring_strict,monoid_mult}) ^ n"
apply (simp add: even_nat_equiv_def2)
apply (erule exE)
apply (erule ssubst)
@@ -227,12 +227,12 @@
done
lemma zero_le_odd_power: "odd n ==>
- (0 <= (x::'a::{ordered_idom}) ^ n) = (0 <= x)"
+ (0 <= (x::'a::{linordered_idom}) ^ n) = (0 <= x)"
apply (auto simp: odd_nat_equiv_def2 power_Suc power_add zero_le_mult_iff)
apply (metis field_power_not_zero no_zero_divirors_neq0 order_antisym_conv zero_le_square)
done
-lemma zero_le_power_eq[presburger]: "(0 <= (x::'a::{ordered_idom}) ^ n) =
+lemma zero_le_power_eq[presburger]: "(0 <= (x::'a::{linordered_idom}) ^ n) =
(even n | (odd n & 0 <= x))"
apply auto
apply (subst zero_le_odd_power [symmetric])
@@ -240,19 +240,19 @@
apply (erule zero_le_even_power)
done
-lemma zero_less_power_eq[presburger]: "(0 < (x::'a::{ordered_idom}) ^ n) =
+lemma zero_less_power_eq[presburger]: "(0 < (x::'a::{linordered_idom}) ^ n) =
(n = 0 | (even n & x ~= 0) | (odd n & 0 < x))"
unfolding order_less_le zero_le_power_eq by auto
-lemma power_less_zero_eq[presburger]: "((x::'a::{ordered_idom}) ^ n < 0) =
+lemma power_less_zero_eq[presburger]: "((x::'a::{linordered_idom}) ^ n < 0) =
(odd n & x < 0)"
apply (subst linorder_not_le [symmetric])+
apply (subst zero_le_power_eq)
apply auto
done
-lemma power_le_zero_eq[presburger]: "((x::'a::{ordered_idom}) ^ n <= 0) =
+lemma power_le_zero_eq[presburger]: "((x::'a::{linordered_idom}) ^ n <= 0) =
(n ~= 0 & ((odd n & x <= 0) | (even n & x = 0)))"
apply (subst linorder_not_less [symmetric])+
apply (subst zero_less_power_eq)
@@ -260,7 +260,7 @@
done
lemma power_even_abs: "even n ==>
- (abs (x::'a::{ordered_idom}))^n = x^n"
+ (abs (x::'a::{linordered_idom}))^n = x^n"
apply (subst power_abs [symmetric])
apply (simp add: zero_le_even_power)
done
@@ -280,7 +280,7 @@
apply simp
done
-lemma power_mono_even: fixes x y :: "'a :: {ordered_idom}"
+lemma power_mono_even: fixes x y :: "'a :: {linordered_idom}"
assumes "even n" and "\<bar>x\<bar> \<le> \<bar>y\<bar>"
shows "x^n \<le> y^n"
proof -
@@ -292,7 +292,7 @@
lemma odd_pos: "odd (n::nat) \<Longrightarrow> 0 < n" by presburger
-lemma power_mono_odd: fixes x y :: "'a :: {ordered_idom}"
+lemma power_mono_odd: fixes x y :: "'a :: {linordered_idom}"
assumes "odd n" and "x \<le> y"
shows "x^n \<le> y^n"
proof (cases "y < 0")
@@ -372,11 +372,11 @@
subsection {* An Equivalence for @{term [source] "0 \<le> a^n"} *}
lemma even_power_le_0_imp_0:
- "a ^ (2*k) \<le> (0::'a::{ordered_idom}) ==> a=0"
+ "a ^ (2*k) \<le> (0::'a::{linordered_idom}) ==> a=0"
by (induct k) (auto simp add: zero_le_mult_iff mult_le_0_iff power_Suc)
lemma zero_le_power_iff[presburger]:
- "(0 \<le> a^n) = (0 \<le> (a::'a::{ordered_idom}) | even n)"
+ "(0 \<le> a^n) = (0 \<le> (a::'a::{linordered_idom}) | even n)"
proof cases
assume even: "even n"
then obtain k where "n = 2*k"
--- a/src/HOL/Power.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Power.thy Mon Feb 08 11:13:30 2010 +0100
@@ -130,7 +130,7 @@
end
-context ordered_semidom
+context linordered_semidom
begin
lemma zero_less_power [simp]:
@@ -323,7 +323,7 @@
end
-context ordered_idom
+context linordered_idom
begin
lemma power_abs:
--- a/src/HOL/Probability/Borel.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Probability/Borel.thy Mon Feb 08 11:13:30 2010 +0100
@@ -73,7 +73,7 @@
with w have "real(Suc(natceiling(inverse(g w - f w)))) > inverse(g w - f w)"
by (metis lessI order_le_less_trans real_natceiling_ge real_of_nat_less_iff) hence "inverse(real(Suc(natceiling(inverse(g w - f w)))))
< inverse(inverse(g w - f w))"
- by (metis less_iff_diff_less_0 less_imp_inverse_less linorder_neqE_ordered_idom nz positive_imp_inverse_positive real_le_antisym real_less_def w)
+ by (metis less_iff_diff_less_0 less_imp_inverse_less linorder_neqE_linordered_idom nz positive_imp_inverse_positive real_le_antisym real_less_def w)
hence "inverse(real(Suc(natceiling(inverse(g w - f w))))) < g w - f w"
by (metis inverse_inverse_eq order_less_le_trans real_le_refl)
thus "\<exists>n. f w \<le> g w - inverse(real(Suc n))" using w
--- a/src/HOL/Quickcheck.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Quickcheck.thy Mon Feb 08 11:13:30 2010 +0100
@@ -164,7 +164,7 @@
where
"union R1 R2 = (\<lambda>s. let
(P1, s') = R1 s; (P2, s'') = R2 s'
- in (upper_semilattice_class.sup P1 P2, s''))"
+ in (semilattice_sup_class.sup P1 P2, s''))"
definition if_randompred :: "bool \<Rightarrow> unit randompred"
where
--- a/src/HOL/RComplete.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/RComplete.thy Mon Feb 08 11:13:30 2010 +0100
@@ -15,7 +15,7 @@
by simp
lemma abs_diff_less_iff:
- "(\<bar>x - a\<bar> < (r::'a::ordered_idom)) = (a - r < x \<and> x < a + r)"
+ "(\<bar>x - a\<bar> < (r::'a::linordered_idom)) = (a - r < x \<and> x < a + r)"
by auto
subsection {* Completeness of Positive Reals *}
--- a/src/HOL/Rational.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Rational.thy Mon Feb 08 11:13:30 2010 +0100
@@ -613,7 +613,7 @@
end
-instance rat :: ordered_field
+instance rat :: linordered_field
proof
fix q r s :: rat
show "q \<le> r ==> s + q \<le> s + r"
@@ -760,7 +760,7 @@
class field_char_0 = field + ring_char_0
-subclass (in ordered_field) field_char_0 ..
+subclass (in linordered_field) field_char_0 ..
context field_char_0
begin
@@ -832,7 +832,7 @@
done
lemma of_rat_less:
- "(of_rat r :: 'a::ordered_field) < of_rat s \<longleftrightarrow> r < s"
+ "(of_rat r :: 'a::linordered_field) < of_rat s \<longleftrightarrow> r < s"
proof (induct r, induct s)
fix a b c d :: int
assume not_zero: "b > 0" "d > 0"
@@ -841,14 +841,14 @@
"(of_int a :: 'a) / of_int b < of_int c / of_int d
\<longleftrightarrow> (of_int a :: 'a) * of_int d < of_int c * of_int b"
using not_zero by (simp add: pos_less_divide_eq pos_divide_less_eq)
- show "(of_rat (Fract a b) :: 'a::ordered_field) < of_rat (Fract c d)
+ show "(of_rat (Fract a b) :: 'a::linordered_field) < of_rat (Fract c d)
\<longleftrightarrow> Fract a b < Fract c d"
using not_zero `b * d > 0`
by (simp add: of_rat_rat of_int_divide_less_eq of_int_mult [symmetric] del: of_int_mult)
qed
lemma of_rat_less_eq:
- "(of_rat r :: 'a::ordered_field) \<le> of_rat s \<longleftrightarrow> r \<le> s"
+ "(of_rat r :: 'a::linordered_field) \<le> of_rat s \<longleftrightarrow> r \<le> s"
unfolding le_less by (auto simp add: of_rat_less)
lemmas of_rat_eq_0_iff [simp] = of_rat_eq_iff [of _ 0, simplified]
@@ -1083,11 +1083,11 @@
finally show ?thesis using assms by simp
qed
-lemma (in ordered_idom) sgn_greater [simp]:
+lemma (in linordered_idom) sgn_greater [simp]:
"0 < sgn a \<longleftrightarrow> 0 < a"
unfolding sgn_if by auto
-lemma (in ordered_idom) sgn_less [simp]:
+lemma (in linordered_idom) sgn_less [simp]:
"sgn a < 0 \<longleftrightarrow> a < 0"
unfolding sgn_if by auto
--- a/src/HOL/Real.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Real.thy Mon Feb 08 11:13:30 2010 +0100
@@ -3,7 +3,7 @@
begin
lemma field_le_epsilon:
- fixes x y :: "'a:: {number_ring,division_by_zero,ordered_field}"
+ fixes x y :: "'a:: {number_ring,division_by_zero,linordered_field}"
assumes e: "(!!e. 0 < e ==> x \<le> y + e)"
shows "x \<le> y"
proof (rule ccontr)
--- a/src/HOL/RealDef.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/RealDef.thy Mon Feb 08 11:13:30 2010 +0100
@@ -416,7 +416,7 @@
subsection{*The Reals Form an Ordered Field*}
-instance real :: ordered_field
+instance real :: linordered_field
proof
fix x y z :: real
show "x \<le> y ==> z + x \<le> z + y" by (rule real_add_left_mono)
@@ -426,7 +426,7 @@
by (simp only: real_sgn_def)
qed
-instance real :: lordered_ab_group_add ..
+instance real :: lattice_ab_group_add ..
text{*The function @{term real_of_preal} requires many proofs, but it seems
to be essential for proving completeness of the reals from that of the
@@ -1046,7 +1046,7 @@
lemma abs_sum_triangle_ineq: "abs ((x::real) + y + (-l + -m)) \<le> abs(x + -l) + abs(y + -m)"
by simp
-instance real :: lordered_ring
+instance real :: lattice_ring
proof
fix a::real
show "abs a = sup a (-a)"
--- a/src/HOL/Ring_and_Field.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Ring_and_Field.thy Mon Feb 08 11:13:30 2010 +0100
@@ -706,7 +706,7 @@
assumes mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
assumes mult_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c"
-class pordered_semiring = mult_mono + semiring_0 + pordered_ab_semigroup_add
+class ordered_semiring = mult_mono + semiring_0 + ordered_ab_semigroup_add
begin
lemma mult_mono:
@@ -725,12 +725,12 @@
end
-class pordered_cancel_semiring = mult_mono + pordered_ab_semigroup_add
+class ordered_cancel_semiring = mult_mono + ordered_ab_semigroup_add
+ semiring + cancel_comm_monoid_add
begin
subclass semiring_0_cancel ..
-subclass pordered_semiring ..
+subclass ordered_semiring ..
lemma mult_nonneg_nonneg: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b"
using mult_left_mono [of zero b a] by simp
@@ -750,12 +750,12 @@
end
-class ordered_semiring = semiring + comm_monoid_add + ordered_cancel_ab_semigroup_add + mult_mono
+class linordered_semiring = semiring + comm_monoid_add + linordered_cancel_ab_semigroup_add + mult_mono
begin
-subclass pordered_cancel_semiring ..
-
-subclass pordered_comm_monoid_add ..
+subclass ordered_cancel_semiring ..
+
+subclass ordered_comm_monoid_add ..
lemma mult_left_less_imp_less:
"c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
@@ -767,16 +767,16 @@
end
-class ordered_semiring_1 = ordered_semiring + semiring_1
-
-class ordered_semiring_strict = semiring + comm_monoid_add + ordered_cancel_ab_semigroup_add +
+class linlinordered_semiring_1 = linordered_semiring + semiring_1
+
+class linlinordered_semiring_strict = semiring + comm_monoid_add + linordered_cancel_ab_semigroup_add +
assumes mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
assumes mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c"
begin
subclass semiring_0_cancel ..
-subclass ordered_semiring
+subclass linordered_semiring
proof
fix a b c :: 'a
assume A: "a \<le> b" "0 \<le> c"
@@ -886,16 +886,16 @@
end
-class ordered_semiring_1_strict = ordered_semiring_strict + semiring_1
+class linlinlinordered_semiring_1_strict = linlinordered_semiring_strict + semiring_1
class mult_mono1 = times + zero + ord +
assumes mult_mono1: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
-class pordered_comm_semiring = comm_semiring_0
- + pordered_ab_semigroup_add + mult_mono1
+class ordered_comm_semiring = comm_semiring_0
+ + ordered_ab_semigroup_add + mult_mono1
begin
-subclass pordered_semiring
+subclass ordered_semiring
proof
fix a b c :: 'a
assume "a \<le> b" "0 \<le> c"
@@ -905,20 +905,20 @@
end
-class pordered_cancel_comm_semiring = comm_semiring_0_cancel
- + pordered_ab_semigroup_add + mult_mono1
+class ordered_cancel_comm_semiring = comm_semiring_0_cancel
+ + ordered_ab_semigroup_add + mult_mono1
begin
-subclass pordered_comm_semiring ..
-subclass pordered_cancel_semiring ..
+subclass ordered_comm_semiring ..
+subclass ordered_cancel_semiring ..
end
-class ordered_comm_semiring_strict = comm_semiring_0 + ordered_cancel_ab_semigroup_add +
+class linordered_comm_semiring_strict = comm_semiring_0 + linordered_cancel_ab_semigroup_add +
assumes mult_strict_left_mono_comm: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
begin
-subclass ordered_semiring_strict
+subclass linlinordered_semiring_strict
proof
fix a b c :: 'a
assume "a < b" "0 < c"
@@ -926,7 +926,7 @@
thus "a * c < b * c" by (simp only: mult_commute)
qed
-subclass pordered_cancel_comm_semiring
+subclass ordered_cancel_comm_semiring
proof
fix a b c :: 'a
assume "a \<le> b" "0 \<le> c"
@@ -937,10 +937,10 @@
end
-class pordered_ring = ring + pordered_cancel_semiring
+class ordered_ring = ring + ordered_cancel_semiring
begin
-subclass pordered_ab_group_add ..
+subclass ordered_ab_group_add ..
text{*Legacy - use @{text algebra_simps} *}
lemmas ring_simps[noatp] = algebra_simps
@@ -991,32 +991,31 @@
lemma (in sgn_if) sgn0[simp]: "sgn 0 = 0"
by(simp add:sgn_if)
-class ordered_ring = ring + ordered_semiring
- + ordered_ab_group_add + abs_if
+class linordered_ring = ring + linordered_semiring + linordered_ab_group_add + abs_if
begin
-subclass pordered_ring ..
-
-subclass pordered_ab_group_add_abs
+subclass ordered_ring ..
+
+subclass ordered_ab_group_add_abs
proof
fix a b
show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
-by (auto simp add: abs_if not_less neg_less_eq_nonneg less_eq_neg_nonpos)
- (auto simp del: minus_add_distrib simp add: minus_add_distrib [symmetric]
+ by (auto simp add: abs_if not_less neg_less_eq_nonneg less_eq_neg_nonpos)
+ (auto simp del: minus_add_distrib simp add: minus_add_distrib [symmetric]
neg_less_eq_nonneg less_eq_neg_nonpos, auto intro: add_nonneg_nonneg,
auto intro!: less_imp_le add_neg_neg)
qed (auto simp add: abs_if less_eq_neg_nonpos neg_equal_zero)
end
-(* The "strict" suffix can be seen as describing the combination of ordered_ring and no_zero_divisors.
- Basically, ordered_ring + no_zero_divisors = ordered_ring_strict.
+(* The "strict" suffix can be seen as describing the combination of linordered_ring and no_zero_divisors.
+ Basically, linordered_ring + no_zero_divisors = linlinordered_ring_strict.
*)
-class ordered_ring_strict = ring + ordered_semiring_strict
+class linlinordered_ring_strict = ring + linlinordered_semiring_strict
+ ordered_ab_group_add + abs_if
begin
-subclass ordered_ring ..
+subclass linordered_ring ..
lemma mult_strict_left_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> c * a < c * b"
using mult_strict_left_mono [of b a "- c"] by simp
@@ -1168,16 +1167,16 @@
mult_pos_pos mult_pos_neg
mult_neg_pos mult_neg_neg
-class pordered_comm_ring = comm_ring + pordered_comm_semiring
+class ordered_comm_ring = comm_ring + ordered_comm_semiring
begin
-subclass pordered_ring ..
-subclass pordered_cancel_comm_semiring ..
+subclass ordered_ring ..
+subclass ordered_cancel_comm_semiring ..
end
-class ordered_semidom = comm_semiring_1_cancel + ordered_comm_semiring_strict +
- (*previously ordered_semiring*)
+class linordered_semidom = comm_semiring_1_cancel + linordered_comm_semiring_strict +
+ (*previously linordered_semiring*)
assumes zero_less_one [simp]: "0 < 1"
begin
@@ -1202,23 +1201,23 @@
end
-class ordered_idom = comm_ring_1 +
- ordered_comm_semiring_strict + ordered_ab_group_add +
+class linordered_idom = comm_ring_1 +
+ linordered_comm_semiring_strict + ordered_ab_group_add +
abs_if + sgn_if
- (*previously ordered_ring*)
+ (*previously linordered_ring*)
begin
-subclass ordered_ring_strict ..
-subclass pordered_comm_ring ..
+subclass linlinordered_ring_strict ..
+subclass ordered_comm_ring ..
subclass idom ..
-subclass ordered_semidom
+subclass linordered_semidom
proof
have "0 \<le> 1 * 1" by (rule zero_le_square)
thus "0 < 1" by (simp add: le_less)
qed
-lemma linorder_neqE_ordered_idom:
+lemma linorder_neqE_linordered_idom:
assumes "x \<noteq> y" obtains "x < y" | "y < x"
using assms by (rule neqE)
@@ -1307,7 +1306,7 @@
end
-class ordered_field = field + ordered_idom
+class linordered_field = field + linordered_idom
text {* Simprules for comparisons where common factors can be cancelled. *}
@@ -1437,7 +1436,7 @@
subsection {* Ordered Fields *}
lemma positive_imp_inverse_positive:
-assumes a_gt_0: "0 < a" shows "0 < inverse (a::'a::ordered_field)"
+assumes a_gt_0: "0 < a" shows "0 < inverse (a::'a::linordered_field)"
proof -
have "0 < a * inverse a"
by (simp add: a_gt_0 [THEN order_less_imp_not_eq2] zero_less_one)
@@ -1446,13 +1445,13 @@
qed
lemma negative_imp_inverse_negative:
- "a < 0 ==> inverse a < (0::'a::ordered_field)"
+ "a < 0 ==> inverse a < (0::'a::linordered_field)"
by (insert positive_imp_inverse_positive [of "-a"],
simp add: nonzero_inverse_minus_eq order_less_imp_not_eq)
lemma inverse_le_imp_le:
assumes invle: "inverse a \<le> inverse b" and apos: "0 < a"
-shows "b \<le> (a::'a::ordered_field)"
+shows "b \<le> (a::'a::linordered_field)"
proof (rule classical)
assume "~ b \<le> a"
hence "a < b" by (simp add: linorder_not_le)
@@ -1466,7 +1465,7 @@
lemma inverse_positive_imp_positive:
assumes inv_gt_0: "0 < inverse a" and nz: "a \<noteq> 0"
-shows "0 < (a::'a::ordered_field)"
+shows "0 < (a::'a::linordered_field)"
proof -
have "0 < inverse (inverse a)"
using inv_gt_0 by (rule positive_imp_inverse_positive)
@@ -1475,14 +1474,14 @@
qed
lemma inverse_positive_iff_positive [simp]:
- "(0 < inverse a) = (0 < (a::'a::{ordered_field,division_by_zero}))"
+ "(0 < inverse a) = (0 < (a::'a::{linordered_field,division_by_zero}))"
apply (cases "a = 0", simp)
apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)
done
lemma inverse_negative_imp_negative:
assumes inv_less_0: "inverse a < 0" and nz: "a \<noteq> 0"
-shows "a < (0::'a::ordered_field)"
+shows "a < (0::'a::linordered_field)"
proof -
have "inverse (inverse a) < 0"
using inv_less_0 by (rule negative_imp_inverse_negative)
@@ -1490,20 +1489,20 @@
qed
lemma inverse_negative_iff_negative [simp]:
- "(inverse a < 0) = (a < (0::'a::{ordered_field,division_by_zero}))"
+ "(inverse a < 0) = (a < (0::'a::{linordered_field,division_by_zero}))"
apply (cases "a = 0", simp)
apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative)
done
lemma inverse_nonnegative_iff_nonnegative [simp]:
- "(0 \<le> inverse a) = (0 \<le> (a::'a::{ordered_field,division_by_zero}))"
+ "(0 \<le> inverse a) = (0 \<le> (a::'a::{linordered_field,division_by_zero}))"
by (simp add: linorder_not_less [symmetric])
lemma inverse_nonpositive_iff_nonpositive [simp]:
- "(inverse a \<le> 0) = (a \<le> (0::'a::{ordered_field,division_by_zero}))"
+ "(inverse a \<le> 0) = (a \<le> (0::'a::{linordered_field,division_by_zero}))"
by (simp add: linorder_not_less [symmetric])
-lemma ordered_field_no_lb: "\<forall> x. \<exists>y. y < (x::'a::ordered_field)"
+lemma linlinordered_field_no_lb: "\<forall> x. \<exists>y. y < (x::'a::linordered_field)"
proof
fix x::'a
have m1: "- (1::'a) < 0" by simp
@@ -1512,7 +1511,7 @@
thus "\<exists>y. y < x" by blast
qed
-lemma ordered_field_no_ub: "\<forall> x. \<exists>y. y > (x::'a::ordered_field)"
+lemma linlinordered_field_no_ub: "\<forall> x. \<exists>y. y > (x::'a::linordered_field)"
proof
fix x::'a
have m1: " (1::'a) > 0" by simp
@@ -1525,7 +1524,7 @@
lemma less_imp_inverse_less:
assumes less: "a < b" and apos: "0 < a"
-shows "inverse b < inverse (a::'a::ordered_field)"
+shows "inverse b < inverse (a::'a::linordered_field)"
proof (rule ccontr)
assume "~ inverse b < inverse a"
hence "inverse a \<le> inverse b" by (simp add: linorder_not_less)
@@ -1535,29 +1534,29 @@
qed
lemma inverse_less_imp_less:
- "[|inverse a < inverse b; 0 < a|] ==> b < (a::'a::ordered_field)"
+ "[|inverse a < inverse b; 0 < a|] ==> b < (a::'a::linordered_field)"
apply (simp add: order_less_le [of "inverse a"] order_less_le [of "b"])
apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq)
done
text{*Both premises are essential. Consider -1 and 1.*}
lemma inverse_less_iff_less [simp,noatp]:
- "[|0 < a; 0 < b|] ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
+ "[|0 < a; 0 < b|] ==> (inverse a < inverse b) = (b < (a::'a::linordered_field))"
by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less)
lemma le_imp_inverse_le:
- "[|a \<le> b; 0 < a|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
+ "[|a \<le> b; 0 < a|] ==> inverse b \<le> inverse (a::'a::linordered_field)"
by (force simp add: order_le_less less_imp_inverse_less)
lemma inverse_le_iff_le [simp,noatp]:
- "[|0 < a; 0 < b|] ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
+ "[|0 < a; 0 < b|] ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::linordered_field))"
by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le)
text{*These results refer to both operands being negative. The opposite-sign
case is trivial, since inverse preserves signs.*}
lemma inverse_le_imp_le_neg:
- "[|inverse a \<le> inverse b; b < 0|] ==> b \<le> (a::'a::ordered_field)"
+ "[|inverse a \<le> inverse b; b < 0|] ==> b \<le> (a::'a::linordered_field)"
apply (rule classical)
apply (subgoal_tac "a < 0")
prefer 2 apply (force simp add: linorder_not_le intro: order_less_trans)
@@ -1566,7 +1565,7 @@
done
lemma less_imp_inverse_less_neg:
- "[|a < b; b < 0|] ==> inverse b < inverse (a::'a::ordered_field)"
+ "[|a < b; b < 0|] ==> inverse b < inverse (a::'a::linordered_field)"
apply (subgoal_tac "a < 0")
prefer 2 apply (blast intro: order_less_trans)
apply (insert less_imp_inverse_less [of "-b" "-a"])
@@ -1574,7 +1573,7 @@
done
lemma inverse_less_imp_less_neg:
- "[|inverse a < inverse b; b < 0|] ==> b < (a::'a::ordered_field)"
+ "[|inverse a < inverse b; b < 0|] ==> b < (a::'a::linordered_field)"
apply (rule classical)
apply (subgoal_tac "a < 0")
prefer 2
@@ -1584,25 +1583,25 @@
done
lemma inverse_less_iff_less_neg [simp,noatp]:
- "[|a < 0; b < 0|] ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
+ "[|a < 0; b < 0|] ==> (inverse a < inverse b) = (b < (a::'a::linordered_field))"
apply (insert inverse_less_iff_less [of "-b" "-a"])
apply (simp del: inverse_less_iff_less
add: order_less_imp_not_eq nonzero_inverse_minus_eq)
done
lemma le_imp_inverse_le_neg:
- "[|a \<le> b; b < 0|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
+ "[|a \<le> b; b < 0|] ==> inverse b \<le> inverse (a::'a::linordered_field)"
by (force simp add: order_le_less less_imp_inverse_less_neg)
lemma inverse_le_iff_le_neg [simp,noatp]:
- "[|a < 0; b < 0|] ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
+ "[|a < 0; b < 0|] ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::linordered_field))"
by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg)
subsection{*Inverses and the Number One*}
lemma one_less_inverse_iff:
- "(1 < inverse x) = (0 < x & x < (1::'a::{ordered_field,division_by_zero}))"
+ "(1 < inverse x) = (0 < x & x < (1::'a::{linordered_field,division_by_zero}))"
proof cases
assume "0 < x"
with inverse_less_iff_less [OF zero_less_one, of x]
@@ -1624,22 +1623,22 @@
by (insert inverse_eq_iff_eq [of x 1], simp)
lemma one_le_inverse_iff:
- "(1 \<le> inverse x) = (0 < x & x \<le> (1::'a::{ordered_field,division_by_zero}))"
+ "(1 \<le> inverse x) = (0 < x & x \<le> (1::'a::{linordered_field,division_by_zero}))"
by (force simp add: order_le_less one_less_inverse_iff zero_less_one
eq_commute [of 1])
lemma inverse_less_1_iff:
- "(inverse x < 1) = (x \<le> 0 | 1 < (x::'a::{ordered_field,division_by_zero}))"
+ "(inverse x < 1) = (x \<le> 0 | 1 < (x::'a::{linordered_field,division_by_zero}))"
by (simp add: linorder_not_le [symmetric] one_le_inverse_iff)
lemma inverse_le_1_iff:
- "(inverse x \<le> 1) = (x \<le> 0 | 1 \<le> (x::'a::{ordered_field,division_by_zero}))"
+ "(inverse x \<le> 1) = (x \<le> 0 | 1 \<le> (x::'a::{linordered_field,division_by_zero}))"
by (simp add: linorder_not_less [symmetric] one_less_inverse_iff)
subsection{*Simplification of Inequalities Involving Literal Divisors*}
-lemma pos_le_divide_eq: "0 < (c::'a::ordered_field) ==> (a \<le> b/c) = (a*c \<le> b)"
+lemma pos_le_divide_eq: "0 < (c::'a::linordered_field) ==> (a \<le> b/c) = (a*c \<le> b)"
proof -
assume less: "0<c"
hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)"
@@ -1649,7 +1648,7 @@
finally show ?thesis .
qed
-lemma neg_le_divide_eq: "c < (0::'a::ordered_field) ==> (a \<le> b/c) = (b \<le> a*c)"
+lemma neg_le_divide_eq: "c < (0::'a::linordered_field) ==> (a \<le> b/c) = (b \<le> a*c)"
proof -
assume less: "c<0"
hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)"
@@ -1663,12 +1662,12 @@
"(a \<le> b/c) =
(if 0 < c then a*c \<le> b
else if c < 0 then b \<le> a*c
- else a \<le> (0::'a::{ordered_field,division_by_zero}))"
+ else a \<le> (0::'a::{linordered_field,division_by_zero}))"
apply (cases "c=0", simp)
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff)
done
-lemma pos_divide_le_eq: "0 < (c::'a::ordered_field) ==> (b/c \<le> a) = (b \<le> a*c)"
+lemma pos_divide_le_eq: "0 < (c::'a::linordered_field) ==> (b/c \<le> a) = (b \<le> a*c)"
proof -
assume less: "0<c"
hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)"
@@ -1678,7 +1677,7 @@
finally show ?thesis .
qed
-lemma neg_divide_le_eq: "c < (0::'a::ordered_field) ==> (b/c \<le> a) = (a*c \<le> b)"
+lemma neg_divide_le_eq: "c < (0::'a::linordered_field) ==> (b/c \<le> a) = (a*c \<le> b)"
proof -
assume less: "c<0"
hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)"
@@ -1692,13 +1691,13 @@
"(b/c \<le> a) =
(if 0 < c then b \<le> a*c
else if c < 0 then a*c \<le> b
- else 0 \<le> (a::'a::{ordered_field,division_by_zero}))"
+ else 0 \<le> (a::'a::{linordered_field,division_by_zero}))"
apply (cases "c=0", simp)
apply (force simp add: pos_divide_le_eq neg_divide_le_eq linorder_neq_iff)
done
lemma pos_less_divide_eq:
- "0 < (c::'a::ordered_field) ==> (a < b/c) = (a*c < b)"
+ "0 < (c::'a::linordered_field) ==> (a < b/c) = (a*c < b)"
proof -
assume less: "0<c"
hence "(a < b/c) = (a*c < (b/c)*c)"
@@ -1709,7 +1708,7 @@
qed
lemma neg_less_divide_eq:
- "c < (0::'a::ordered_field) ==> (a < b/c) = (b < a*c)"
+ "c < (0::'a::linordered_field) ==> (a < b/c) = (b < a*c)"
proof -
assume less: "c<0"
hence "(a < b/c) = ((b/c)*c < a*c)"
@@ -1723,13 +1722,13 @@
"(a < b/c) =
(if 0 < c then a*c < b
else if c < 0 then b < a*c
- else a < (0::'a::{ordered_field,division_by_zero}))"
+ else a < (0::'a::{linordered_field,division_by_zero}))"
apply (cases "c=0", simp)
apply (force simp add: pos_less_divide_eq neg_less_divide_eq linorder_neq_iff)
done
lemma pos_divide_less_eq:
- "0 < (c::'a::ordered_field) ==> (b/c < a) = (b < a*c)"
+ "0 < (c::'a::linordered_field) ==> (b/c < a) = (b < a*c)"
proof -
assume less: "0<c"
hence "(b/c < a) = ((b/c)*c < a*c)"
@@ -1740,7 +1739,7 @@
qed
lemma neg_divide_less_eq:
- "c < (0::'a::ordered_field) ==> (b/c < a) = (a*c < b)"
+ "c < (0::'a::linordered_field) ==> (b/c < a) = (a*c < b)"
proof -
assume less: "c<0"
hence "(b/c < a) = (a*c < (b/c)*c)"
@@ -1754,7 +1753,7 @@
"(b/c < a) =
(if 0 < c then b < a*c
else if c < 0 then a*c < b
- else 0 < (a::'a::{ordered_field,division_by_zero}))"
+ else 0 < (a::'a::{linordered_field,division_by_zero}))"
apply (cases "c=0", simp)
apply (force simp add: pos_divide_less_eq neg_divide_less_eq linorder_neq_iff)
done
@@ -1784,7 +1783,7 @@
(* Only works once linear arithmetic is installed:
text{*An example:*}
-lemma fixes a b c d e f :: "'a::ordered_field"
+lemma fixes a b c d e f :: "'a::linordered_field"
shows "\<lbrakk>a>b; c<d; e<f; 0 < u \<rbrakk> \<Longrightarrow>
((a-b)*(c-d)*(e-f))/((c-d)*(e-f)*(a-b)) <
((e-f)*(a-b)*(c-d))/((e-f)*(a-b)*(c-d)) + u"
@@ -1800,21 +1799,21 @@
subsection{*Division and Signs*}
lemma zero_less_divide_iff:
- "((0::'a::{ordered_field,division_by_zero}) < a/b) = (0 < a & 0 < b | a < 0 & b < 0)"
+ "((0::'a::{linordered_field,division_by_zero}) < a/b) = (0 < a & 0 < b | a < 0 & b < 0)"
by (simp add: divide_inverse zero_less_mult_iff)
lemma divide_less_0_iff:
- "(a/b < (0::'a::{ordered_field,division_by_zero})) =
+ "(a/b < (0::'a::{linordered_field,division_by_zero})) =
(0 < a & b < 0 | a < 0 & 0 < b)"
by (simp add: divide_inverse mult_less_0_iff)
lemma zero_le_divide_iff:
- "((0::'a::{ordered_field,division_by_zero}) \<le> a/b) =
+ "((0::'a::{linordered_field,division_by_zero}) \<le> a/b) =
(0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
by (simp add: divide_inverse zero_le_mult_iff)
lemma divide_le_0_iff:
- "(a/b \<le> (0::'a::{ordered_field,division_by_zero})) =
+ "(a/b \<le> (0::'a::{linordered_field,division_by_zero})) =
(0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
by (simp add: divide_inverse mult_le_0_iff)
@@ -1823,36 +1822,36 @@
by (simp add: divide_inverse)
lemma divide_pos_pos:
- "0 < (x::'a::ordered_field) ==> 0 < y ==> 0 < x / y"
+ "0 < (x::'a::linordered_field) ==> 0 < y ==> 0 < x / y"
by(simp add:field_simps)
lemma divide_nonneg_pos:
- "0 <= (x::'a::ordered_field) ==> 0 < y ==> 0 <= x / y"
+ "0 <= (x::'a::linordered_field) ==> 0 < y ==> 0 <= x / y"
by(simp add:field_simps)
lemma divide_neg_pos:
- "(x::'a::ordered_field) < 0 ==> 0 < y ==> x / y < 0"
+ "(x::'a::linordered_field) < 0 ==> 0 < y ==> x / y < 0"
by(simp add:field_simps)
lemma divide_nonpos_pos:
- "(x::'a::ordered_field) <= 0 ==> 0 < y ==> x / y <= 0"
+ "(x::'a::linordered_field) <= 0 ==> 0 < y ==> x / y <= 0"
by(simp add:field_simps)
lemma divide_pos_neg:
- "0 < (x::'a::ordered_field) ==> y < 0 ==> x / y < 0"
+ "0 < (x::'a::linordered_field) ==> y < 0 ==> x / y < 0"
by(simp add:field_simps)
lemma divide_nonneg_neg:
- "0 <= (x::'a::ordered_field) ==> y < 0 ==> x / y <= 0"
+ "0 <= (x::'a::linordered_field) ==> y < 0 ==> x / y <= 0"
by(simp add:field_simps)
lemma divide_neg_neg:
- "(x::'a::ordered_field) < 0 ==> y < 0 ==> 0 < x / y"
+ "(x::'a::linordered_field) < 0 ==> y < 0 ==> 0 < x / y"
by(simp add:field_simps)
lemma divide_nonpos_neg:
- "(x::'a::ordered_field) <= 0 ==> y < 0 ==> 0 <= x / y"
+ "(x::'a::linordered_field) <= 0 ==> y < 0 ==> 0 <= x / y"
by(simp add:field_simps)
@@ -1885,13 +1884,13 @@
by (simp add: eq_commute [of 1])
lemma zero_eq_1_divide_iff [simp,noatp]:
- "((0::'a::{ordered_field,division_by_zero}) = 1/a) = (a = 0)"
+ "((0::'a::{linordered_field,division_by_zero}) = 1/a) = (a = 0)"
apply (cases "a=0", simp)
apply (auto simp add: nonzero_eq_divide_eq)
done
lemma one_divide_eq_0_iff [simp,noatp]:
- "(1/a = (0::'a::{ordered_field,division_by_zero})) = (a = 0)"
+ "(1/a = (0::'a::{linordered_field,division_by_zero})) = (a = 0)"
apply (cases "a=0", simp)
apply (insert zero_neq_one [THEN not_sym])
apply (auto simp add: nonzero_divide_eq_eq)
@@ -1912,22 +1911,22 @@
subsection {* Ordering Rules for Division *}
lemma divide_strict_right_mono:
- "[|a < b; 0 < c|] ==> a / c < b / (c::'a::ordered_field)"
+ "[|a < b; 0 < c|] ==> a / c < b / (c::'a::linordered_field)"
by (simp add: order_less_imp_not_eq2 divide_inverse mult_strict_right_mono
positive_imp_inverse_positive)
lemma divide_right_mono:
- "[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/(c::'a::{ordered_field,division_by_zero})"
+ "[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/(c::'a::{linordered_field,division_by_zero})"
by (force simp add: divide_strict_right_mono order_le_less)
-lemma divide_right_mono_neg: "(a::'a::{division_by_zero,ordered_field}) <= b
+lemma divide_right_mono_neg: "(a::'a::{division_by_zero,linordered_field}) <= b
==> c <= 0 ==> b / c <= a / c"
apply (drule divide_right_mono [of _ _ "- c"])
apply auto
done
lemma divide_strict_right_mono_neg:
- "[|b < a; c < 0|] ==> a / c < b / (c::'a::ordered_field)"
+ "[|b < a; c < 0|] ==> a / c < b / (c::'a::linordered_field)"
apply (drule divide_strict_right_mono [of _ _ "-c"], simp)
apply (simp add: order_less_imp_not_eq nonzero_minus_divide_right [symmetric])
done
@@ -1935,43 +1934,43 @@
text{*The last premise ensures that @{term a} and @{term b}
have the same sign*}
lemma divide_strict_left_mono:
- "[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"
+ "[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / (b::'a::linordered_field)"
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono)
lemma divide_left_mono:
- "[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / (b::'a::ordered_field)"
+ "[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / (b::'a::linordered_field)"
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_right_mono)
-lemma divide_left_mono_neg: "(a::'a::{division_by_zero,ordered_field}) <= b
+lemma divide_left_mono_neg: "(a::'a::{division_by_zero,linordered_field}) <= b
==> c <= 0 ==> 0 < a * b ==> c / a <= c / b"
apply (drule divide_left_mono [of _ _ "- c"])
apply (auto simp add: mult_commute)
done
lemma divide_strict_left_mono_neg:
- "[|a < b; c < 0; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"
+ "[|a < b; c < 0; 0 < a*b|] ==> c / a < c / (b::'a::linordered_field)"
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono_neg)
text{*Simplify quotients that are compared with the value 1.*}
lemma le_divide_eq_1 [noatp]:
- fixes a :: "'a :: {ordered_field,division_by_zero}"
+ fixes a :: "'a :: {linordered_field,division_by_zero}"
shows "(1 \<le> b / a) = ((0 < a & a \<le> b) | (a < 0 & b \<le> a))"
by (auto simp add: le_divide_eq)
lemma divide_le_eq_1 [noatp]:
- fixes a :: "'a :: {ordered_field,division_by_zero}"
+ fixes a :: "'a :: {linordered_field,division_by_zero}"
shows "(b / a \<le> 1) = ((0 < a & b \<le> a) | (a < 0 & a \<le> b) | a=0)"
by (auto simp add: divide_le_eq)
lemma less_divide_eq_1 [noatp]:
- fixes a :: "'a :: {ordered_field,division_by_zero}"
+ fixes a :: "'a :: {linordered_field,division_by_zero}"
shows "(1 < b / a) = ((0 < a & a < b) | (a < 0 & b < a))"
by (auto simp add: less_divide_eq)
lemma divide_less_eq_1 [noatp]:
- fixes a :: "'a :: {ordered_field,division_by_zero}"
+ fixes a :: "'a :: {linordered_field,division_by_zero}"
shows "(b / a < 1) = ((0 < a & b < a) | (a < 0 & a < b) | a=0)"
by (auto simp add: divide_less_eq)
@@ -1979,83 +1978,83 @@
subsection{*Conditional Simplification Rules: No Case Splits*}
lemma le_divide_eq_1_pos [simp,noatp]:
- fixes a :: "'a :: {ordered_field,division_by_zero}"
+ fixes a :: "'a :: {linordered_field,division_by_zero}"
shows "0 < a \<Longrightarrow> (1 \<le> b/a) = (a \<le> b)"
by (auto simp add: le_divide_eq)
lemma le_divide_eq_1_neg [simp,noatp]:
- fixes a :: "'a :: {ordered_field,division_by_zero}"
+ fixes a :: "'a :: {linordered_field,division_by_zero}"
shows "a < 0 \<Longrightarrow> (1 \<le> b/a) = (b \<le> a)"
by (auto simp add: le_divide_eq)
lemma divide_le_eq_1_pos [simp,noatp]:
- fixes a :: "'a :: {ordered_field,division_by_zero}"
+ fixes a :: "'a :: {linordered_field,division_by_zero}"
shows "0 < a \<Longrightarrow> (b/a \<le> 1) = (b \<le> a)"
by (auto simp add: divide_le_eq)
lemma divide_le_eq_1_neg [simp,noatp]:
- fixes a :: "'a :: {ordered_field,division_by_zero}"
+ fixes a :: "'a :: {linordered_field,division_by_zero}"
shows "a < 0 \<Longrightarrow> (b/a \<le> 1) = (a \<le> b)"
by (auto simp add: divide_le_eq)
lemma less_divide_eq_1_pos [simp,noatp]:
- fixes a :: "'a :: {ordered_field,division_by_zero}"
+ fixes a :: "'a :: {linordered_field,division_by_zero}"
shows "0 < a \<Longrightarrow> (1 < b/a) = (a < b)"
by (auto simp add: less_divide_eq)
lemma less_divide_eq_1_neg [simp,noatp]:
- fixes a :: "'a :: {ordered_field,division_by_zero}"
+ fixes a :: "'a :: {linordered_field,division_by_zero}"
shows "a < 0 \<Longrightarrow> (1 < b/a) = (b < a)"
by (auto simp add: less_divide_eq)
lemma divide_less_eq_1_pos [simp,noatp]:
- fixes a :: "'a :: {ordered_field,division_by_zero}"
+ fixes a :: "'a :: {linordered_field,division_by_zero}"
shows "0 < a \<Longrightarrow> (b/a < 1) = (b < a)"
by (auto simp add: divide_less_eq)
lemma divide_less_eq_1_neg [simp,noatp]:
- fixes a :: "'a :: {ordered_field,division_by_zero}"
+ fixes a :: "'a :: {linordered_field,division_by_zero}"
shows "a < 0 \<Longrightarrow> b/a < 1 <-> a < b"
by (auto simp add: divide_less_eq)
lemma eq_divide_eq_1 [simp,noatp]:
- fixes a :: "'a :: {ordered_field,division_by_zero}"
+ fixes a :: "'a :: {linordered_field,division_by_zero}"
shows "(1 = b/a) = ((a \<noteq> 0 & a = b))"
by (auto simp add: eq_divide_eq)
lemma divide_eq_eq_1 [simp,noatp]:
- fixes a :: "'a :: {ordered_field,division_by_zero}"
+ fixes a :: "'a :: {linordered_field,division_by_zero}"
shows "(b/a = 1) = ((a \<noteq> 0 & a = b))"
by (auto simp add: divide_eq_eq)
subsection {* Reasoning about inequalities with division *}
-lemma mult_right_le_one_le: "0 <= (x::'a::ordered_idom) ==> 0 <= y ==> y <= 1
+lemma mult_right_le_one_le: "0 <= (x::'a::linordered_idom) ==> 0 <= y ==> y <= 1
==> x * y <= x"
by (auto simp add: mult_compare_simps)
-lemma mult_left_le_one_le: "0 <= (x::'a::ordered_idom) ==> 0 <= y ==> y <= 1
+lemma mult_left_le_one_le: "0 <= (x::'a::linordered_idom) ==> 0 <= y ==> y <= 1
==> y * x <= x"
by (auto simp add: mult_compare_simps)
-lemma mult_imp_div_pos_le: "0 < (y::'a::ordered_field) ==> x <= z * y ==>
+lemma mult_imp_div_pos_le: "0 < (y::'a::linordered_field) ==> x <= z * y ==>
x / y <= z"
by (subst pos_divide_le_eq, assumption+)
-lemma mult_imp_le_div_pos: "0 < (y::'a::ordered_field) ==> z * y <= x ==>
+lemma mult_imp_le_div_pos: "0 < (y::'a::linordered_field) ==> z * y <= x ==>
z <= x / y"
by(simp add:field_simps)
-lemma mult_imp_div_pos_less: "0 < (y::'a::ordered_field) ==> x < z * y ==>
+lemma mult_imp_div_pos_less: "0 < (y::'a::linordered_field) ==> x < z * y ==>
x / y < z"
by(simp add:field_simps)
-lemma mult_imp_less_div_pos: "0 < (y::'a::ordered_field) ==> z * y < x ==>
+lemma mult_imp_less_div_pos: "0 < (y::'a::linordered_field) ==> z * y < x ==>
z < x / y"
by(simp add:field_simps)
-lemma frac_le: "(0::'a::ordered_field) <= x ==>
+lemma frac_le: "(0::'a::linordered_field) <= x ==>
x <= y ==> 0 < w ==> w <= z ==> x / z <= y / w"
apply (rule mult_imp_div_pos_le)
apply simp
@@ -2065,7 +2064,7 @@
apply simp_all
done
-lemma frac_less: "(0::'a::ordered_field) <= x ==>
+lemma frac_less: "(0::'a::linordered_field) <= x ==>
x < y ==> 0 < w ==> w <= z ==> x / z < y / w"
apply (rule mult_imp_div_pos_less)
apply simp
@@ -2075,7 +2074,7 @@
apply simp_all
done
-lemma frac_less2: "(0::'a::ordered_field) < x ==>
+lemma frac_less2: "(0::'a::linordered_field) < x ==>
x <= y ==> 0 < w ==> w < z ==> x / z < y / w"
apply (rule mult_imp_div_pos_less)
apply simp_all
@@ -2095,7 +2094,7 @@
subsection {* Ordered Fields are Dense *}
-context ordered_semidom
+context linordered_semidom
begin
lemma less_add_one: "a < a + 1"
@@ -2110,13 +2109,13 @@
end
-lemma less_half_sum: "a < b ==> a < (a+b) / (1+1::'a::ordered_field)"
+lemma less_half_sum: "a < b ==> a < (a+b) / (1+1::'a::linordered_field)"
by (simp add: field_simps zero_less_two)
-lemma gt_half_sum: "a < b ==> (a+b)/(1+1::'a::ordered_field) < b"
+lemma gt_half_sum: "a < b ==> (a+b)/(1+1::'a::linordered_field) < b"
by (simp add: field_simps zero_less_two)
-instance ordered_field < dense_linear_order
+instance linordered_field < dense_linorder
proof
fix x y :: 'a
have "x < x + 1" by simp
@@ -2129,7 +2128,7 @@
subsection {* Absolute Value *}
-context ordered_idom
+context linordered_idom
begin
lemma mult_sgn_abs: "sgn x * abs x = x"
@@ -2137,23 +2136,23 @@
end
-lemma abs_one [simp]: "abs 1 = (1::'a::ordered_idom)"
+lemma abs_one [simp]: "abs 1 = (1::'a::linordered_idom)"
by (simp add: abs_if zero_less_one [THEN order_less_not_sym])
-class pordered_ring_abs = pordered_ring + pordered_ab_group_add_abs +
+class ordered_ring_abs = ordered_ring + ordered_ab_group_add_abs +
assumes abs_eq_mult:
"(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0) \<Longrightarrow> \<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"
-class lordered_ring = pordered_ring + lordered_ab_group_add_abs
+class lattice_ring = ordered_ring + lattice_ab_group_add_abs
begin
-subclass lordered_ab_group_add_meet ..
-subclass lordered_ab_group_add_join ..
+subclass semilattice_inf_ab_group_add ..
+subclass semilattice_sup_ab_group_add ..
end
-lemma abs_le_mult: "abs (a * b) \<le> (abs a) * (abs (b::'a::lordered_ring))"
+lemma abs_le_mult: "abs (a * b) \<le> (abs a) * (abs (b::'a::lattice_ring))"
proof -
let ?x = "pprt a * pprt b - pprt a * nprt b - nprt a * pprt b + nprt a * nprt b"
let ?y = "pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
@@ -2194,9 +2193,9 @@
done
qed
-instance lordered_ring \<subseteq> pordered_ring_abs
+instance lattice_ring \<subseteq> ordered_ring_abs
proof
- fix a b :: "'a\<Colon> lordered_ring"
+ fix a b :: "'a\<Colon> lattice_ring"
assume "(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0)"
show "abs (a*b) = abs a * abs b"
proof -
@@ -2238,10 +2237,10 @@
qed
qed
-context ordered_idom
+context linordered_idom
begin
-subclass pordered_ring_abs proof
+subclass ordered_ring_abs proof
qed (auto simp add: abs_if not_less equal_neg_zero neg_equal_zero mult_less_0_iff)
lemma abs_mult:
@@ -2255,31 +2254,31 @@
end
lemma nonzero_abs_inverse:
- "a \<noteq> 0 ==> abs (inverse (a::'a::ordered_field)) = inverse (abs a)"
+ "a \<noteq> 0 ==> abs (inverse (a::'a::linordered_field)) = inverse (abs a)"
apply (auto simp add: linorder_neq_iff abs_if nonzero_inverse_minus_eq
negative_imp_inverse_negative)
apply (blast intro: positive_imp_inverse_positive elim: order_less_asym)
done
lemma abs_inverse [simp]:
- "abs (inverse (a::'a::{ordered_field,division_by_zero})) =
+ "abs (inverse (a::'a::{linordered_field,division_by_zero})) =
inverse (abs a)"
apply (cases "a=0", simp)
apply (simp add: nonzero_abs_inverse)
done
lemma nonzero_abs_divide:
- "b \<noteq> 0 ==> abs (a / (b::'a::ordered_field)) = abs a / abs b"
+ "b \<noteq> 0 ==> abs (a / (b::'a::linordered_field)) = abs a / abs b"
by (simp add: divide_inverse abs_mult nonzero_abs_inverse)
lemma abs_divide [simp]:
- "abs (a / (b::'a::{ordered_field,division_by_zero})) = abs a / abs b"
+ "abs (a / (b::'a::{linordered_field,division_by_zero})) = abs a / abs b"
apply (cases "b=0", simp)
apply (simp add: nonzero_abs_divide)
done
lemma abs_mult_less:
- "[| abs a < c; abs b < d |] ==> abs a * abs b < c*(d::'a::ordered_idom)"
+ "[| abs a < c; abs b < d |] ==> abs a * abs b < c*(d::'a::linordered_idom)"
proof -
assume ac: "abs a < c"
hence cpos: "0<c" by (blast intro: order_le_less_trans abs_ge_zero)
@@ -2289,21 +2288,21 @@
lemmas eq_minus_self_iff[noatp] = equal_neg_zero
-lemma less_minus_self_iff: "(a < -a) = (a < (0::'a::ordered_idom))"
+lemma less_minus_self_iff: "(a < -a) = (a < (0::'a::linordered_idom))"
unfolding order_less_le less_eq_neg_nonpos equal_neg_zero ..
-lemma abs_less_iff: "(abs a < b) = (a < b & -a < (b::'a::ordered_idom))"
+lemma abs_less_iff: "(abs a < b) = (a < b & -a < (b::'a::linordered_idom))"
apply (simp add: order_less_le abs_le_iff)
apply (auto simp add: abs_if neg_less_eq_nonneg less_eq_neg_nonpos)
done
-lemma abs_mult_pos: "(0::'a::ordered_idom) <= x ==>
+lemma abs_mult_pos: "(0::'a::linordered_idom) <= x ==>
(abs y) * x = abs (y * x)"
apply (subst abs_mult)
apply simp
done
-lemma abs_div_pos: "(0::'a::{division_by_zero,ordered_field}) < y ==>
+lemma abs_div_pos: "(0::'a::{division_by_zero,linordered_field}) < y ==>
abs x / y = abs (x / y)"
apply (subst abs_divide)
apply (simp add: order_less_imp_le)
@@ -2314,7 +2313,7 @@
lemma mult_le_prts:
assumes
- "a1 <= (a::'a::lordered_ring)"
+ "a1 <= (a::'a::lattice_ring)"
"a <= a2"
"b1 <= b"
"b <= b2"
@@ -2362,7 +2361,7 @@
lemma mult_ge_prts:
assumes
- "a1 <= (a::'a::lordered_ring)"
+ "a1 <= (a::'a::lattice_ring)"
"a <= a2"
"b1 <= b"
"b <= b2"
--- a/src/HOL/Series.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Series.thy Mon Feb 08 11:13:30 2010 +0100
@@ -381,7 +381,7 @@
shows "norm x < 1 \<Longrightarrow> summable (\<lambda>n. x ^ n)"
by (rule geometric_sums [THEN sums_summable])
-lemma half: "0 < 1 / (2::'a::{number_ring,division_by_zero,ordered_field})"
+lemma half: "0 < 1 / (2::'a::{number_ring,division_by_zero,linordered_field})"
by arith
lemma power_half_series: "(\<lambda>n. (1/2::real)^Suc n) sums 1"
--- a/src/HOL/SupInf.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/SupInf.thy Mon Feb 08 11:13:30 2010 +0100
@@ -7,17 +7,17 @@
begin
lemma minus_max_eq_min:
- fixes x :: "'a::{lordered_ab_group_add, linorder}"
+ fixes x :: "'a::{lattice_ab_group_add, linorder}"
shows "- (max x y) = min (-x) (-y)"
by (metis le_imp_neg_le linorder_linear min_max.inf_absorb2 min_max.le_iff_inf min_max.le_iff_sup min_max.sup_absorb1)
lemma minus_min_eq_max:
- fixes x :: "'a::{lordered_ab_group_add, linorder}"
+ fixes x :: "'a::{lattice_ab_group_add, linorder}"
shows "- (min x y) = max (-x) (-y)"
by (metis minus_max_eq_min minus_minus)
lemma minus_Max_eq_Min [simp]:
- fixes S :: "'a::{lordered_ab_group_add, linorder} set"
+ fixes S :: "'a::{lattice_ab_group_add, linorder} set"
shows "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> - (Max S) = Min (uminus ` S)"
proof (induct S rule: finite_ne_induct)
case (singleton x)
@@ -28,7 +28,7 @@
qed
lemma minus_Min_eq_Max [simp]:
- fixes S :: "'a::{lordered_ab_group_add, linorder} set"
+ fixes S :: "'a::{lattice_ab_group_add, linorder} set"
shows "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> - (Min S) = Max (uminus ` S)"
proof (induct S rule: finite_ne_induct)
case (singleton x)
--- a/src/HOL/Tools/Nitpick/minipick.ML Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Tools/Nitpick/minipick.ML Mon Feb 08 11:13:30 2010 +0100
@@ -200,19 +200,19 @@
else
raise NOT_SUPPORTED "transitive closure for function or pair type"
| Const (@{const_name trancl}, _) => to_R_rep Ts (eta_expand Ts t 1)
- | Const (@{const_name lower_semilattice_class.inf},
+ | Const (@{const_name semilattice_inf_class.inf},
Type ("fun", [Type ("fun", [_, @{typ bool}]), _])) $ t1 $ t2 =>
Intersect (to_R_rep Ts t1, to_R_rep Ts t2)
- | Const (@{const_name lower_semilattice_class.inf}, _) $ _ =>
+ | Const (@{const_name semilattice_inf_class.inf}, _) $ _ =>
to_R_rep Ts (eta_expand Ts t 1)
- | Const (@{const_name lower_semilattice_class.inf}, _) =>
+ | Const (@{const_name semilattice_inf_class.inf}, _) =>
to_R_rep Ts (eta_expand Ts t 2)
- | Const (@{const_name upper_semilattice_class.sup},
+ | Const (@{const_name semilattice_sup_class.sup},
Type ("fun", [Type ("fun", [_, @{typ bool}]), _])) $ t1 $ t2 =>
Union (to_R_rep Ts t1, to_R_rep Ts t2)
- | Const (@{const_name upper_semilattice_class.sup}, _) $ _ =>
+ | Const (@{const_name semilattice_sup_class.sup}, _) $ _ =>
to_R_rep Ts (eta_expand Ts t 1)
- | Const (@{const_name upper_semilattice_class.sup}, _) =>
+ | Const (@{const_name semilattice_sup_class.sup}, _) =>
to_R_rep Ts (eta_expand Ts t 2)
| Const (@{const_name minus_class.minus},
Type ("fun", [Type ("fun", [_, @{typ bool}]), _])) $ t1 $ t2 =>
--- a/src/HOL/Tools/Nitpick/nitpick_hol.ML Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Tools/Nitpick/nitpick_hol.ML Mon Feb 08 11:13:30 2010 +0100
@@ -343,8 +343,8 @@
[((@{const_name of_nat}, nat_T --> int_T), 0),
((@{const_name of_nat}, @{typ "unsigned_bit word => signed_bit word"}), 0)]
val built_in_set_consts =
- [(@{const_name lower_semilattice_fun_inst.inf_fun}, 2),
- (@{const_name upper_semilattice_fun_inst.sup_fun}, 2),
+ [(@{const_name semilattice_inf_fun_inst.inf_fun}, 2),
+ (@{const_name semilattice_sup_fun_inst.sup_fun}, 2),
(@{const_name minus_fun_inst.minus_fun}, 2),
(@{const_name ord_fun_inst.less_eq_fun}, 2)]
@@ -2592,11 +2592,11 @@
if gfp then
(lbfp_prefix,
@{const "op |"},
- @{const_name upper_semilattice_fun_inst.sup_fun})
+ @{const_name semilattice_sup_fun_inst.sup_fun})
else
(ubfp_prefix,
@{const "op &"},
- @{const_name lower_semilattice_fun_inst.inf_fun})
+ @{const_name semilattice_inf_fun_inst.inf_fun})
(* unit -> term *)
fun pos () = unrolled_inductive_pred_const ext_ctxt gfp x
|> aux ss Ts js depth polar
--- a/src/HOL/Tools/Nitpick/nitpick_mono.ML Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Tools/Nitpick/nitpick_mono.ML Mon Feb 08 11:13:30 2010 +0100
@@ -667,9 +667,9 @@
in (CFun (ab_set_C, S Minus, ba_set_C), accum) end
| @{const_name trancl} => do_fragile_set_operation T accum
| @{const_name rtrancl} => (print_g "*** rtrancl"; unsolvable)
- | @{const_name lower_semilattice_fun_inst.inf_fun} =>
+ | @{const_name semilattice_inf_fun_inst.inf_fun} =>
do_robust_set_operation T accum
- | @{const_name upper_semilattice_fun_inst.sup_fun} =>
+ | @{const_name semilattice_sup_fun_inst.sup_fun} =>
do_robust_set_operation T accum
| @{const_name finite} =>
let val C1 = ctype_for (domain_type (domain_type T)) in
--- a/src/HOL/Tools/Nitpick/nitpick_nut.ML Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Tools/Nitpick/nitpick_nut.ML Mon Feb 08 11:13:30 2010 +0100
@@ -655,10 +655,10 @@
| (Const (@{const_name of_nat},
T as @{typ "unsigned_bit word => signed_bit word"}), []) =>
Cst (NatToInt, T, Any)
- | (Const (@{const_name lower_semilattice_fun_inst.inf_fun}, T),
+ | (Const (@{const_name semilattice_inf_fun_inst.inf_fun}, T),
[t1, t2]) =>
Op2 (Intersect, nth_range_type 2 T, Any, sub t1, sub t2)
- | (Const (@{const_name upper_semilattice_fun_inst.sup_fun}, T),
+ | (Const (@{const_name semilattice_sup_fun_inst.sup_fun}, T),
[t1, t2]) =>
Op2 (Union, nth_range_type 2 T, Any, sub t1, sub t2)
| (t0 as Const (@{const_name minus_fun_inst.minus_fun}, T), [t1, t2]) =>
--- a/src/HOL/Tools/int_arith.ML Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Tools/int_arith.ML Mon Feb 08 11:13:30 2010 +0100
@@ -71,8 +71,8 @@
val lhss' =
[@{cpat "(?x::?'a::ring_char_0) = (?y::?'a)"},
- @{cpat "(?x::?'a::ordered_idom) < (?y::?'a)"},
- @{cpat "(?x::?'a::ordered_idom) <= (?y::?'a)"}]
+ @{cpat "(?x::?'a::linordered_idom) < (?y::?'a)"},
+ @{cpat "(?x::?'a::linordered_idom) <= (?y::?'a)"}]
val zero_one_idom_simproc =
make_simproc {lhss = lhss' , name = "zero_one_idom_simproc",
@@ -80,9 +80,9 @@
val fast_int_arith_simproc =
Simplifier.simproc @{theory} "fast_int_arith"
- ["(m::'a::{ordered_idom,number_ring}) < n",
- "(m::'a::{ordered_idom,number_ring}) <= n",
- "(m::'a::{ordered_idom,number_ring}) = n"] (K Lin_Arith.simproc);
+ ["(m::'a::{linordered_idom,number_ring}) < n",
+ "(m::'a::{linordered_idom,number_ring}) <= n",
+ "(m::'a::{linordered_idom,number_ring}) = n"] (K Lin_Arith.simproc);
val global_setup = Simplifier.map_simpset
(fn simpset => simpset addsimprocs [fast_int_arith_simproc]);
--- a/src/HOL/Tools/lin_arith.ML Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Tools/lin_arith.ML Mon Feb 08 11:13:30 2010 +0100
@@ -236,7 +236,7 @@
end handle Rat.DIVZERO => NONE;
fun of_lin_arith_sort thy U =
- Sign.of_sort thy (U, @{sort Ring_and_Field.ordered_idom});
+ Sign.of_sort thy (U, @{sort Ring_and_Field.linordered_idom});
fun allows_lin_arith thy (discrete : string list) (U as Type (D, [])) : bool * bool =
if of_lin_arith_sort thy U then (true, member (op =) discrete D)
@@ -804,12 +804,12 @@
val init_arith_data =
Fast_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, number_of, ...} =>
- {add_mono_thms = @{thms add_mono_thms_ordered_semiring} @ @{thms add_mono_thms_ordered_field} @ add_mono_thms,
+ {add_mono_thms = @{thms add_mono_thms_linordered_semiring} @ @{thms add_mono_thms_linordered_field} @ add_mono_thms,
mult_mono_thms = @{thm mult_strict_left_mono} :: @{thm mult_left_mono} ::
@{lemma "a = b ==> c*a = c*b" by (rule arg_cong)} :: mult_mono_thms,
inj_thms = inj_thms,
lessD = lessD @ [@{thm "Suc_leI"}],
- neqE = [@{thm linorder_neqE_nat}, @{thm linorder_neqE_ordered_idom}],
+ neqE = [@{thm linorder_neqE_nat}, @{thm linorder_neqE_linordered_idom}],
simpset = HOL_basic_ss
addsimps @{thms ring_distribs}
addsimps [@{thm if_True}, @{thm if_False}]
--- a/src/HOL/Tools/numeral_simprocs.ML Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Tools/numeral_simprocs.ML Mon Feb 08 11:13:30 2010 +0100
@@ -255,20 +255,20 @@
"(l::'a::number_ring) = m * n"],
K EqCancelNumerals.proc),
("intless_cancel_numerals",
- ["(l::'a::{ordered_idom,number_ring}) + m < n",
- "(l::'a::{ordered_idom,number_ring}) < m + n",
- "(l::'a::{ordered_idom,number_ring}) - m < n",
- "(l::'a::{ordered_idom,number_ring}) < m - n",
- "(l::'a::{ordered_idom,number_ring}) * m < n",
- "(l::'a::{ordered_idom,number_ring}) < m * n"],
+ ["(l::'a::{linordered_idom,number_ring}) + m < n",
+ "(l::'a::{linordered_idom,number_ring}) < m + n",
+ "(l::'a::{linordered_idom,number_ring}) - m < n",
+ "(l::'a::{linordered_idom,number_ring}) < m - n",
+ "(l::'a::{linordered_idom,number_ring}) * m < n",
+ "(l::'a::{linordered_idom,number_ring}) < m * n"],
K LessCancelNumerals.proc),
("intle_cancel_numerals",
- ["(l::'a::{ordered_idom,number_ring}) + m <= n",
- "(l::'a::{ordered_idom,number_ring}) <= m + n",
- "(l::'a::{ordered_idom,number_ring}) - m <= n",
- "(l::'a::{ordered_idom,number_ring}) <= m - n",
- "(l::'a::{ordered_idom,number_ring}) * m <= n",
- "(l::'a::{ordered_idom,number_ring}) <= m * n"],
+ ["(l::'a::{linordered_idom,number_ring}) + m <= n",
+ "(l::'a::{linordered_idom,number_ring}) <= m + n",
+ "(l::'a::{linordered_idom,number_ring}) - m <= n",
+ "(l::'a::{linordered_idom,number_ring}) <= m - n",
+ "(l::'a::{linordered_idom,number_ring}) * m <= n",
+ "(l::'a::{linordered_idom,number_ring}) <= m * n"],
K LeCancelNumerals.proc)];
structure CombineNumeralsData =
@@ -431,12 +431,12 @@
"(l::'a::{idom,number_ring}) = m * n"],
K EqCancelNumeralFactor.proc),
("ring_less_cancel_numeral_factor",
- ["(l::'a::{ordered_idom,number_ring}) * m < n",
- "(l::'a::{ordered_idom,number_ring}) < m * n"],
+ ["(l::'a::{linordered_idom,number_ring}) * m < n",
+ "(l::'a::{linordered_idom,number_ring}) < m * n"],
K LessCancelNumeralFactor.proc),
("ring_le_cancel_numeral_factor",
- ["(l::'a::{ordered_idom,number_ring}) * m <= n",
- "(l::'a::{ordered_idom,number_ring}) <= m * n"],
+ ["(l::'a::{linordered_idom,number_ring}) * m <= n",
+ "(l::'a::{linordered_idom,number_ring}) <= m * n"],
K LeCancelNumeralFactor.proc),
("int_div_cancel_numeral_factors",
["((l::'a::{semiring_div,number_ring}) * m) div n",
@@ -581,13 +581,13 @@
["(l::'a::idom) * m = n",
"(l::'a::idom) = m * n"],
K EqCancelFactor.proc),
- ("ordered_ring_le_cancel_factor",
- ["(l::'a::ordered_ring) * m <= n",
- "(l::'a::ordered_ring) <= m * n"],
+ ("linlinordered_ring_le_cancel_factor",
+ ["(l::'a::linordered_ring) * m <= n",
+ "(l::'a::linordered_ring) <= m * n"],
K LeCancelFactor.proc),
- ("ordered_ring_less_cancel_factor",
- ["(l::'a::ordered_ring) * m < n",
- "(l::'a::ordered_ring) < m * n"],
+ ("linlinordered_ring_less_cancel_factor",
+ ["(l::'a::linordered_ring) * m < n",
+ "(l::'a::linordered_ring) < m * n"],
K LessCancelFactor.proc),
("int_div_cancel_factor",
["((l::'a::semiring_div) * m) div n", "(l::'a::semiring_div) div (m * n)"],
--- a/src/HOL/Transcendental.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/Transcendental.thy Mon Feb 08 11:13:30 2010 +0100
@@ -381,7 +381,7 @@
done
lemma real_setsum_nat_ivl_bounded2:
- fixes K :: "'a::ordered_semidom"
+ fixes K :: "'a::linordered_semidom"
assumes f: "\<And>p::nat. p < n \<Longrightarrow> f p \<le> K"
assumes K: "0 \<le> K"
shows "setsum f {0..<n-k} \<le> of_nat n * K"
--- a/src/HOL/ZF/Games.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/ZF/Games.thy Mon Feb 08 11:13:30 2010 +0100
@@ -922,7 +922,7 @@
apply (auto simp add: Pg_def quotient_def)
done
-instance Pg :: pordered_ab_group_add
+instance Pg :: ordered_ab_group_add
proof
fix a b c :: Pg
show "a - b = a + (- b)" by (simp add: Pg_diff_def)
--- a/src/HOL/ex/Numeral.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/ex/Numeral.thy Mon Feb 08 11:13:30 2010 +0100
@@ -442,12 +442,12 @@
end
subsubsection {*
- Comparisons: class @{text ordered_semidom}
+ Comparisons: class @{text linordered_semidom}
*}
text {* Could be perhaps more general than here. *}
-context ordered_semidom
+context linordered_semidom
begin
lemma of_num_pos [numeral]: "0 < of_num n"
@@ -490,7 +490,7 @@
end
-context ordered_idom
+context linordered_idom
begin
lemma minus_of_num_less_of_num_iff: "- of_num m < of_num n"
@@ -896,19 +896,19 @@
declare (in semiring_char_0) of_num_eq_one_iff [simp]
declare (in semiring_char_0) one_eq_of_num_iff [simp]
-declare (in ordered_semidom) of_num_pos [simp]
-declare (in ordered_semidom) of_num_less_eq_iff [simp]
-declare (in ordered_semidom) of_num_less_eq_one_iff [simp]
-declare (in ordered_semidom) one_less_eq_of_num_iff [simp]
-declare (in ordered_semidom) of_num_less_iff [simp]
-declare (in ordered_semidom) of_num_less_one_iff [simp]
-declare (in ordered_semidom) one_less_of_num_iff [simp]
-declare (in ordered_semidom) of_num_nonneg [simp]
-declare (in ordered_semidom) of_num_less_zero_iff [simp]
-declare (in ordered_semidom) of_num_le_zero_iff [simp]
+declare (in linordered_semidom) of_num_pos [simp]
+declare (in linordered_semidom) of_num_less_eq_iff [simp]
+declare (in linordered_semidom) of_num_less_eq_one_iff [simp]
+declare (in linordered_semidom) one_less_eq_of_num_iff [simp]
+declare (in linordered_semidom) of_num_less_iff [simp]
+declare (in linordered_semidom) of_num_less_one_iff [simp]
+declare (in linordered_semidom) one_less_of_num_iff [simp]
+declare (in linordered_semidom) of_num_nonneg [simp]
+declare (in linordered_semidom) of_num_less_zero_iff [simp]
+declare (in linordered_semidom) of_num_le_zero_iff [simp]
-declare (in ordered_idom) le_signed_numeral_special [simp]
-declare (in ordered_idom) less_signed_numeral_special [simp]
+declare (in linordered_idom) le_signed_numeral_special [simp]
+declare (in linordered_idom) less_signed_numeral_special [simp]
declare (in semiring_1_minus) Dig_of_num_minus_one [simp]
declare (in semiring_1_minus) Dig_one_minus_of_num [simp]
--- a/src/HOL/ex/RPred.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/ex/RPred.thy Mon Feb 08 11:13:30 2010 +0100
@@ -25,7 +25,7 @@
(* (infixl "\<squnion>" 80) *)
where
"supp RP1 RP2 = (\<lambda>s. let (P1, s') = RP1 s; (P2, s'') = RP2 s'
- in (upper_semilattice_class.sup P1 P2, s''))"
+ in (semilattice_sup_class.sup P1 P2, s''))"
definition if_rpred :: "bool \<Rightarrow> unit rpred"
where
--- a/src/HOL/ex/ReflectionEx.thy Mon Feb 08 11:01:47 2010 +0100
+++ b/src/HOL/ex/ReflectionEx.thy Mon Feb 08 11:13:30 2010 +0100
@@ -385,7 +385,7 @@
(* An example for equations containing type variables *)
datatype prod = Zero | One | Var nat | Mul prod prod
| Pw prod nat | PNM nat nat prod
-consts Iprod :: " prod \<Rightarrow> ('a::{ordered_idom}) list \<Rightarrow>'a"
+consts Iprod :: " prod \<Rightarrow> ('a::{linordered_idom}) list \<Rightarrow>'a"
primrec
"Iprod Zero vs = 0"
"Iprod One vs = 1"
@@ -397,7 +397,7 @@
datatype sgn = Pos prod | Neg prod | ZeroEq prod | NZeroEq prod | Tr | F
| Or sgn sgn | And sgn sgn
-consts Isgn :: " sgn \<Rightarrow> ('a::{ordered_idom}) list \<Rightarrow>bool"
+consts Isgn :: " sgn \<Rightarrow> ('a::{linordered_idom}) list \<Rightarrow>bool"
primrec
"Isgn Tr vs = True"
"Isgn F vs = False"
@@ -410,7 +410,7 @@
lemmas eqs = Isgn.simps Iprod.simps
-lemma "(x::'a::{ordered_idom})^4 * y * z * y^2 * z^23 > 0"
+lemma "(x::'a::{linordered_idom})^4 * y * z * y^2 * z^23 > 0"
apply (reify eqs)
oops