author | hoelzl |
Thu, 17 Jan 2013 11:59:12 +0100 | |
changeset 50936 | b28f258ebc1a |
parent 46986 | 8198cbff1771 |
child 53240 | 07593a0a27f4 |
permissions | -rw-r--r-- |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
1 |
(* Author: Steven Obua, TU Muenchen *) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
2 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
3 |
header {* Various algebraic structures combined with a lattice *} |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
4 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
5 |
theory Lattice_Algebras |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
6 |
imports Complex_Main |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
7 |
begin |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
8 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
9 |
class semilattice_inf_ab_group_add = ordered_ab_group_add + semilattice_inf |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
10 |
begin |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
11 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
12 |
lemma add_inf_distrib_left: |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
13 |
"a + inf b c = inf (a + b) (a + c)" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
14 |
apply (rule antisym) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
15 |
apply (simp_all add: le_infI) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
16 |
apply (rule add_le_imp_le_left [of "uminus a"]) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
17 |
apply (simp only: add_assoc [symmetric], simp) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
18 |
apply rule |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
19 |
apply (rule add_le_imp_le_left[of "a"], simp only: add_assoc[symmetric], simp)+ |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
20 |
done |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
21 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
22 |
lemma add_inf_distrib_right: |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
23 |
"inf a b + c = inf (a + c) (b + c)" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
24 |
proof - |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
25 |
have "c + inf a b = inf (c+a) (c+b)" by (simp add: add_inf_distrib_left) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
26 |
thus ?thesis by (simp add: add_commute) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
27 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
28 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
29 |
end |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
30 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
31 |
class semilattice_sup_ab_group_add = ordered_ab_group_add + semilattice_sup |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
32 |
begin |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
33 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
34 |
lemma add_sup_distrib_left: |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
35 |
"a + sup b c = sup (a + b) (a + c)" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
36 |
apply (rule antisym) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
37 |
apply (rule add_le_imp_le_left [of "uminus a"]) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
38 |
apply (simp only: add_assoc[symmetric], simp) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
39 |
apply rule |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
40 |
apply (rule add_le_imp_le_left [of "a"], simp only: add_assoc[symmetric], simp)+ |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
41 |
apply (rule le_supI) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
42 |
apply (simp_all) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
43 |
done |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
44 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
45 |
lemma add_sup_distrib_right: |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
46 |
"sup a b + c = sup (a+c) (b+c)" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
47 |
proof - |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
48 |
have "c + sup a b = sup (c+a) (c+b)" by (simp add: add_sup_distrib_left) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
49 |
thus ?thesis by (simp add: add_commute) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
50 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
51 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
52 |
end |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
53 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
54 |
class lattice_ab_group_add = ordered_ab_group_add + lattice |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
55 |
begin |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
56 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
57 |
subclass semilattice_inf_ab_group_add .. |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
58 |
subclass semilattice_sup_ab_group_add .. |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
59 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
60 |
lemmas add_sup_inf_distribs = add_inf_distrib_right add_inf_distrib_left add_sup_distrib_right add_sup_distrib_left |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
61 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
62 |
lemma inf_eq_neg_sup: "inf a b = - sup (-a) (-b)" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
63 |
proof (rule inf_unique) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
64 |
fix a b :: 'a |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
65 |
show "- sup (-a) (-b) \<le> a" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
66 |
by (rule add_le_imp_le_right [of _ "sup (uminus a) (uminus b)"]) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
67 |
(simp, simp add: add_sup_distrib_left) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
68 |
next |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
69 |
fix a b :: 'a |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
70 |
show "- sup (-a) (-b) \<le> b" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
71 |
by (rule add_le_imp_le_right [of _ "sup (uminus a) (uminus b)"]) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
72 |
(simp, simp add: add_sup_distrib_left) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
73 |
next |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
74 |
fix a b c :: 'a |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
75 |
assume "a \<le> b" "a \<le> c" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
76 |
then show "a \<le> - sup (-b) (-c)" by (subst neg_le_iff_le [symmetric]) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
77 |
(simp add: le_supI) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
78 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
79 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
80 |
lemma sup_eq_neg_inf: "sup a b = - inf (-a) (-b)" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
81 |
proof (rule sup_unique) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
82 |
fix a b :: 'a |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
83 |
show "a \<le> - inf (-a) (-b)" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
84 |
by (rule add_le_imp_le_right [of _ "inf (uminus a) (uminus b)"]) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
85 |
(simp, simp add: add_inf_distrib_left) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
86 |
next |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
87 |
fix a b :: 'a |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
88 |
show "b \<le> - inf (-a) (-b)" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
89 |
by (rule add_le_imp_le_right [of _ "inf (uminus a) (uminus b)"]) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
90 |
(simp, simp add: add_inf_distrib_left) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
91 |
next |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
92 |
fix a b c :: 'a |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
93 |
assume "a \<le> c" "b \<le> c" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
94 |
then show "- inf (-a) (-b) \<le> c" by (subst neg_le_iff_le [symmetric]) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
95 |
(simp add: le_infI) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
96 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
97 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
98 |
lemma neg_inf_eq_sup: "- inf a b = sup (-a) (-b)" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
99 |
by (simp add: inf_eq_neg_sup) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
100 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
101 |
lemma neg_sup_eq_inf: "- sup a b = inf (-a) (-b)" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
102 |
by (simp add: sup_eq_neg_inf) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
103 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
104 |
lemma add_eq_inf_sup: "a + b = sup a b + inf a b" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
105 |
proof - |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
106 |
have "0 = - inf 0 (a-b) + inf (a-b) 0" by (simp add: inf_commute) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
107 |
hence "0 = sup 0 (b-a) + inf (a-b) 0" by (simp add: inf_eq_neg_sup) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
108 |
hence "0 = (-a + sup a b) + (inf a b + (-b))" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
109 |
by (simp add: add_sup_distrib_left add_inf_distrib_right) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
110 |
(simp add: algebra_simps) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
111 |
thus ?thesis by (simp add: algebra_simps) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
112 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
113 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
114 |
subsection {* Positive Part, Negative Part, Absolute Value *} |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
115 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
116 |
definition |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
117 |
nprt :: "'a \<Rightarrow> 'a" where |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
118 |
"nprt x = inf x 0" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
119 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
120 |
definition |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
121 |
pprt :: "'a \<Rightarrow> 'a" where |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
122 |
"pprt x = sup x 0" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
123 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
124 |
lemma pprt_neg: "pprt (- x) = - nprt x" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
125 |
proof - |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
126 |
have "sup (- x) 0 = sup (- x) (- 0)" unfolding minus_zero .. |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
127 |
also have "\<dots> = - inf x 0" unfolding neg_inf_eq_sup .. |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
128 |
finally have "sup (- x) 0 = - inf x 0" . |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
129 |
then show ?thesis unfolding pprt_def nprt_def . |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
130 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
131 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
132 |
lemma nprt_neg: "nprt (- x) = - pprt x" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
133 |
proof - |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
134 |
from pprt_neg have "pprt (- (- x)) = - nprt (- x)" . |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
135 |
then have "pprt x = - nprt (- x)" by simp |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
136 |
then show ?thesis by simp |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
137 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
138 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
139 |
lemma prts: "a = pprt a + nprt a" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
140 |
by (simp add: pprt_def nprt_def add_eq_inf_sup[symmetric]) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
141 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
142 |
lemma zero_le_pprt[simp]: "0 \<le> pprt a" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
143 |
by (simp add: pprt_def) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
144 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
145 |
lemma nprt_le_zero[simp]: "nprt a \<le> 0" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
146 |
by (simp add: nprt_def) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
147 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
148 |
lemma le_eq_neg: "a \<le> - b \<longleftrightarrow> a + b \<le> 0" (is "?l = ?r") |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
149 |
proof - |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
150 |
have a: "?l \<longrightarrow> ?r" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
151 |
apply (auto) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
152 |
apply (rule add_le_imp_le_right[of _ "uminus b" _]) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
153 |
apply (simp add: add_assoc) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
154 |
done |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
155 |
have b: "?r \<longrightarrow> ?l" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
156 |
apply (auto) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
157 |
apply (rule add_le_imp_le_right[of _ "b" _]) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
158 |
apply (simp) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
159 |
done |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
160 |
from a b show ?thesis by blast |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
161 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
162 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
163 |
lemma pprt_0[simp]: "pprt 0 = 0" by (simp add: pprt_def) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
164 |
lemma nprt_0[simp]: "nprt 0 = 0" by (simp add: nprt_def) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
165 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35040
diff
changeset
|
166 |
lemma pprt_eq_id [simp, no_atp]: "0 \<le> x \<Longrightarrow> pprt x = x" |
46986 | 167 |
by (simp add: pprt_def sup_absorb1) |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
168 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35040
diff
changeset
|
169 |
lemma nprt_eq_id [simp, no_atp]: "x \<le> 0 \<Longrightarrow> nprt x = x" |
46986 | 170 |
by (simp add: nprt_def inf_absorb1) |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
171 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35040
diff
changeset
|
172 |
lemma pprt_eq_0 [simp, no_atp]: "x \<le> 0 \<Longrightarrow> pprt x = 0" |
46986 | 173 |
by (simp add: pprt_def sup_absorb2) |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
174 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35040
diff
changeset
|
175 |
lemma nprt_eq_0 [simp, no_atp]: "0 \<le> x \<Longrightarrow> nprt x = 0" |
46986 | 176 |
by (simp add: nprt_def inf_absorb2) |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
177 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
178 |
lemma sup_0_imp_0: "sup a (- a) = 0 \<Longrightarrow> a = 0" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
179 |
proof - |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
180 |
{ |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
181 |
fix a::'a |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
182 |
assume hyp: "sup a (-a) = 0" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
183 |
hence "sup a (-a) + a = a" by (simp) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
184 |
hence "sup (a+a) 0 = a" by (simp add: add_sup_distrib_right) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
185 |
hence "sup (a+a) 0 <= a" by (simp) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
186 |
hence "0 <= a" by (blast intro: order_trans inf_sup_ord) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
187 |
} |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
188 |
note p = this |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
189 |
assume hyp:"sup a (-a) = 0" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
190 |
hence hyp2:"sup (-a) (-(-a)) = 0" by (simp add: sup_commute) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
191 |
from p[OF hyp] p[OF hyp2] show "a = 0" by simp |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
192 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
193 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
194 |
lemma inf_0_imp_0: "inf a (-a) = 0 \<Longrightarrow> a = 0" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
195 |
apply (simp add: inf_eq_neg_sup) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
196 |
apply (simp add: sup_commute) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
197 |
apply (erule sup_0_imp_0) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
198 |
done |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
199 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35040
diff
changeset
|
200 |
lemma inf_0_eq_0 [simp, no_atp]: "inf a (- a) = 0 \<longleftrightarrow> a = 0" |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
201 |
by (rule, erule inf_0_imp_0) simp |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
202 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35040
diff
changeset
|
203 |
lemma sup_0_eq_0 [simp, no_atp]: "sup a (- a) = 0 \<longleftrightarrow> a = 0" |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
204 |
by (rule, erule sup_0_imp_0) simp |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
205 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
206 |
lemma zero_le_double_add_iff_zero_le_single_add [simp]: |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
207 |
"0 \<le> a + a \<longleftrightarrow> 0 \<le> a" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
208 |
proof |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
209 |
assume "0 <= a + a" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
210 |
hence a:"inf (a+a) 0 = 0" by (simp add: inf_commute inf_absorb1) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
211 |
have "(inf a 0)+(inf a 0) = inf (inf (a+a) 0) a" (is "?l=_") |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
212 |
by (simp add: add_sup_inf_distribs inf_aci) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
213 |
hence "?l = 0 + inf a 0" by (simp add: a, simp add: inf_commute) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
214 |
hence "inf a 0 = 0" by (simp only: add_right_cancel) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
215 |
then show "0 <= a" unfolding le_iff_inf by (simp add: inf_commute) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
216 |
next |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
217 |
assume a: "0 <= a" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
218 |
show "0 <= a + a" by (simp add: add_mono[OF a a, simplified]) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
219 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
220 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
221 |
lemma double_zero [simp]: |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
222 |
"a + a = 0 \<longleftrightarrow> a = 0" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
223 |
proof |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
224 |
assume assm: "a + a = 0" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
225 |
then have "a + a + - a = - a" by simp |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
226 |
then have "a + (a + - a) = - a" by (simp only: add_assoc) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
227 |
then have a: "- a = a" by simp |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
228 |
show "a = 0" apply (rule antisym) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
229 |
apply (unfold neg_le_iff_le [symmetric, of a]) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
230 |
unfolding a apply simp |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
231 |
unfolding zero_le_double_add_iff_zero_le_single_add [symmetric, of a] |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
232 |
unfolding assm unfolding le_less apply simp_all done |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
233 |
next |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
234 |
assume "a = 0" then show "a + a = 0" by simp |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
235 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
236 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
237 |
lemma zero_less_double_add_iff_zero_less_single_add [simp]: |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
238 |
"0 < a + a \<longleftrightarrow> 0 < a" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
239 |
proof (cases "a = 0") |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
240 |
case True then show ?thesis by auto |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
241 |
next |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
242 |
case False then show ?thesis (*FIXME tune proof*) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
243 |
unfolding less_le apply simp apply rule |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
244 |
apply clarify |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
245 |
apply rule |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
246 |
apply assumption |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
247 |
apply (rule notI) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
248 |
unfolding double_zero [symmetric, of a] apply simp |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
249 |
done |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
250 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
251 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
252 |
lemma double_add_le_zero_iff_single_add_le_zero [simp]: |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
253 |
"a + a \<le> 0 \<longleftrightarrow> a \<le> 0" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
254 |
proof - |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
255 |
have "a + a \<le> 0 \<longleftrightarrow> 0 \<le> - (a + a)" by (subst le_minus_iff, simp) |
41528 | 256 |
moreover have "\<dots> \<longleftrightarrow> a \<le> 0" by simp |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
257 |
ultimately show ?thesis by blast |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
258 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
259 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
260 |
lemma double_add_less_zero_iff_single_less_zero [simp]: |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
261 |
"a + a < 0 \<longleftrightarrow> a < 0" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
262 |
proof - |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
263 |
have "a + a < 0 \<longleftrightarrow> 0 < - (a + a)" by (subst less_minus_iff, simp) |
41528 | 264 |
moreover have "\<dots> \<longleftrightarrow> a < 0" by simp |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
265 |
ultimately show ?thesis by blast |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
266 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
267 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
268 |
declare neg_inf_eq_sup [simp] neg_sup_eq_inf [simp] |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
269 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
270 |
lemma le_minus_self_iff: "a \<le> - a \<longleftrightarrow> a \<le> 0" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
271 |
proof - |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
272 |
from add_le_cancel_left [of "uminus a" "plus a a" zero] |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
273 |
have "(a <= -a) = (a+a <= 0)" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
274 |
by (simp add: add_assoc[symmetric]) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
275 |
thus ?thesis by simp |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
276 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
277 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
278 |
lemma minus_le_self_iff: "- a \<le> a \<longleftrightarrow> 0 \<le> a" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
279 |
proof - |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
280 |
from add_le_cancel_left [of "uminus a" zero "plus a a"] |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
281 |
have "(-a <= a) = (0 <= a+a)" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
282 |
by (simp add: add_assoc[symmetric]) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
283 |
thus ?thesis by simp |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
284 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
285 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
286 |
lemma zero_le_iff_zero_nprt: "0 \<le> a \<longleftrightarrow> nprt a = 0" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
287 |
unfolding le_iff_inf by (simp add: nprt_def inf_commute) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
288 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
289 |
lemma le_zero_iff_zero_pprt: "a \<le> 0 \<longleftrightarrow> pprt a = 0" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
290 |
unfolding le_iff_sup by (simp add: pprt_def sup_commute) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
291 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
292 |
lemma le_zero_iff_pprt_id: "0 \<le> a \<longleftrightarrow> pprt a = a" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
293 |
unfolding le_iff_sup by (simp add: pprt_def sup_commute) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
294 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
295 |
lemma zero_le_iff_nprt_id: "a \<le> 0 \<longleftrightarrow> nprt a = a" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
296 |
unfolding le_iff_inf by (simp add: nprt_def inf_commute) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
297 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35040
diff
changeset
|
298 |
lemma pprt_mono [simp, no_atp]: "a \<le> b \<Longrightarrow> pprt a \<le> pprt b" |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
299 |
unfolding le_iff_sup by (simp add: pprt_def sup_aci sup_assoc [symmetric, of a]) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
300 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35040
diff
changeset
|
301 |
lemma nprt_mono [simp, no_atp]: "a \<le> b \<Longrightarrow> nprt a \<le> nprt b" |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
302 |
unfolding le_iff_inf by (simp add: nprt_def inf_aci inf_assoc [symmetric, of a]) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
303 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
304 |
end |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
305 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
306 |
lemmas add_sup_inf_distribs = add_inf_distrib_right add_inf_distrib_left add_sup_distrib_right add_sup_distrib_left |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
307 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
308 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
309 |
class lattice_ab_group_add_abs = lattice_ab_group_add + abs + |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
310 |
assumes abs_lattice: "\<bar>a\<bar> = sup a (- a)" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
311 |
begin |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
312 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
313 |
lemma abs_prts: "\<bar>a\<bar> = pprt a - nprt a" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
314 |
proof - |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
315 |
have "0 \<le> \<bar>a\<bar>" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
316 |
proof - |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
317 |
have a: "a \<le> \<bar>a\<bar>" and b: "- a \<le> \<bar>a\<bar>" by (auto simp add: abs_lattice) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
318 |
show ?thesis by (rule add_mono [OF a b, simplified]) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
319 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
320 |
then have "0 \<le> sup a (- a)" unfolding abs_lattice . |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
321 |
then have "sup (sup a (- a)) 0 = sup a (- a)" by (rule sup_absorb1) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
322 |
then show ?thesis |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
323 |
by (simp add: add_sup_inf_distribs sup_aci |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
324 |
pprt_def nprt_def diff_minus abs_lattice) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
325 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
326 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
327 |
subclass ordered_ab_group_add_abs |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
328 |
proof |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
329 |
have abs_ge_zero [simp]: "\<And>a. 0 \<le> \<bar>a\<bar>" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
330 |
proof - |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
331 |
fix a b |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
332 |
have a: "a \<le> \<bar>a\<bar>" and b: "- a \<le> \<bar>a\<bar>" by (auto simp add: abs_lattice) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
333 |
show "0 \<le> \<bar>a\<bar>" by (rule add_mono [OF a b, simplified]) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
334 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
335 |
have abs_leI: "\<And>a b. a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
336 |
by (simp add: abs_lattice le_supI) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
337 |
fix a b |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
338 |
show "0 \<le> \<bar>a\<bar>" by simp |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
339 |
show "a \<le> \<bar>a\<bar>" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
340 |
by (auto simp add: abs_lattice) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
341 |
show "\<bar>-a\<bar> = \<bar>a\<bar>" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
342 |
by (simp add: abs_lattice sup_commute) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
343 |
show "a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b" by (fact abs_leI) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
344 |
show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
345 |
proof - |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
346 |
have g:"abs a + abs b = sup (a+b) (sup (-a-b) (sup (-a+b) (a + (-b))))" (is "_=sup ?m ?n") |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
347 |
by (simp add: abs_lattice add_sup_inf_distribs sup_aci diff_minus) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
348 |
have a:"a+b <= sup ?m ?n" by (simp) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
349 |
have b:"-a-b <= ?n" by (simp) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
350 |
have c:"?n <= sup ?m ?n" by (simp) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
351 |
from b c have d: "-a-b <= sup ?m ?n" by(rule order_trans) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
352 |
have e:"-a-b = -(a+b)" by (simp add: diff_minus) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
353 |
from a d e have "abs(a+b) <= sup ?m ?n" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
354 |
by (drule_tac abs_leI, auto) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
355 |
with g[symmetric] show ?thesis by simp |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
356 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
357 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
358 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
359 |
end |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
360 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
361 |
lemma sup_eq_if: |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
362 |
fixes a :: "'a\<Colon>{lattice_ab_group_add, linorder}" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
363 |
shows "sup a (- a) = (if a < 0 then - a else a)" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
364 |
proof - |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
365 |
note add_le_cancel_right [of a a "- a", symmetric, simplified] |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
366 |
moreover note add_le_cancel_right [of "-a" a a, symmetric, simplified] |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
367 |
then show ?thesis by (auto simp: sup_max min_max.sup_absorb1 min_max.sup_absorb2) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
368 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
369 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
370 |
lemma abs_if_lattice: |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
371 |
fixes a :: "'a\<Colon>{lattice_ab_group_add_abs, linorder}" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
372 |
shows "\<bar>a\<bar> = (if a < 0 then - a else a)" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
373 |
by auto |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
374 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
375 |
lemma estimate_by_abs: |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
376 |
"a + b <= (c::'a::lattice_ab_group_add_abs) \<Longrightarrow> a <= c + abs b" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
377 |
proof - |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
378 |
assume "a+b <= c" |
37884
314a88278715
discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents:
36976
diff
changeset
|
379 |
then have "a <= c+(-b)" by (simp add: algebra_simps) |
314a88278715
discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents:
36976
diff
changeset
|
380 |
have "(-b) <= abs b" by (rule abs_ge_minus_self) |
314a88278715
discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents:
36976
diff
changeset
|
381 |
then have "c + (- b) \<le> c + \<bar>b\<bar>" by (rule add_left_mono) |
314a88278715
discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents:
36976
diff
changeset
|
382 |
with `a \<le> c + (- b)` show ?thesis by (rule order_trans) |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
383 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
384 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
385 |
class lattice_ring = ordered_ring + lattice_ab_group_add_abs |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
386 |
begin |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
387 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
388 |
subclass semilattice_inf_ab_group_add .. |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
389 |
subclass semilattice_sup_ab_group_add .. |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
390 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
391 |
end |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
392 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
393 |
lemma abs_le_mult: "abs (a * b) \<le> (abs a) * (abs (b::'a::lattice_ring))" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
394 |
proof - |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
395 |
let ?x = "pprt a * pprt b - pprt a * nprt b - nprt a * pprt b + nprt a * nprt b" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
396 |
let ?y = "pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
397 |
have a: "(abs a) * (abs b) = ?x" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
398 |
by (simp only: abs_prts[of a] abs_prts[of b] algebra_simps) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
399 |
{ |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
400 |
fix u v :: 'a |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
401 |
have bh: "\<lbrakk>u = a; v = b\<rbrakk> \<Longrightarrow> |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
402 |
u * v = pprt a * pprt b + pprt a * nprt b + |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
403 |
nprt a * pprt b + nprt a * nprt b" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
404 |
apply (subst prts[of u], subst prts[of v]) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
405 |
apply (simp add: algebra_simps) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
406 |
done |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
407 |
} |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
408 |
note b = this[OF refl[of a] refl[of b]] |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
409 |
have xy: "- ?x <= ?y" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
410 |
apply (simp) |
36976 | 411 |
apply (rule order_trans [OF add_nonpos_nonpos add_nonneg_nonneg]) |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
412 |
apply (simp_all add: mult_nonneg_nonneg mult_nonpos_nonpos) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
413 |
done |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
414 |
have yx: "?y <= ?x" |
37884
314a88278715
discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents:
36976
diff
changeset
|
415 |
apply (simp add:diff_minus) |
36976 | 416 |
apply (rule order_trans [OF add_nonpos_nonpos add_nonneg_nonneg]) |
417 |
apply (simp_all add: mult_nonneg_nonpos mult_nonpos_nonneg) |
|
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
418 |
done |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
419 |
have i1: "a*b <= abs a * abs b" by (simp only: a b yx) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
420 |
have i2: "- (abs a * abs b) <= a*b" by (simp only: a b xy) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
421 |
show ?thesis |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
422 |
apply (rule abs_leI) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
423 |
apply (simp add: i1) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
424 |
apply (simp add: i2[simplified minus_le_iff]) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
425 |
done |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
426 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
427 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
428 |
instance lattice_ring \<subseteq> ordered_ring_abs |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
429 |
proof |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
430 |
fix a b :: "'a\<Colon> lattice_ring" |
41528 | 431 |
assume a: "(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0)" |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
432 |
show "abs (a*b) = abs a * abs b" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
433 |
proof - |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
434 |
have s: "(0 <= a*b) | (a*b <= 0)" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
435 |
apply (auto) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
436 |
apply (rule_tac split_mult_pos_le) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
437 |
apply (rule_tac contrapos_np[of "a*b <= 0"]) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
438 |
apply (simp) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
439 |
apply (rule_tac split_mult_neg_le) |
41528 | 440 |
apply (insert a) |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
441 |
apply (blast) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
442 |
done |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
443 |
have mulprts: "a * b = (pprt a + nprt a) * (pprt b + nprt b)" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
444 |
by (simp add: prts[symmetric]) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
445 |
show ?thesis |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
446 |
proof cases |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
447 |
assume "0 <= a * b" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
448 |
then show ?thesis |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
449 |
apply (simp_all add: mulprts abs_prts) |
41528 | 450 |
apply (insert a) |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
451 |
apply (auto simp add: |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
452 |
algebra_simps |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
453 |
iffD1[OF zero_le_iff_zero_nprt] iffD1[OF le_zero_iff_zero_pprt] |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
454 |
iffD1[OF le_zero_iff_pprt_id] iffD1[OF zero_le_iff_nprt_id]) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
455 |
apply(drule (1) mult_nonneg_nonpos[of a b], simp) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
456 |
apply(drule (1) mult_nonneg_nonpos2[of b a], simp) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
457 |
done |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
458 |
next |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
459 |
assume "~(0 <= a*b)" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
460 |
with s have "a*b <= 0" by simp |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
461 |
then show ?thesis |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
462 |
apply (simp_all add: mulprts abs_prts) |
41528 | 463 |
apply (insert a) |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
464 |
apply (auto simp add: algebra_simps) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
465 |
apply(drule (1) mult_nonneg_nonneg[of a b],simp) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
466 |
apply(drule (1) mult_nonpos_nonpos[of a b],simp) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
467 |
done |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
468 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
469 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
470 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
471 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
472 |
lemma mult_le_prts: |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
473 |
assumes |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
474 |
"a1 <= (a::'a::lattice_ring)" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
475 |
"a <= a2" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
476 |
"b1 <= b" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
477 |
"b <= b2" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
478 |
shows |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
479 |
"a * b <= pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
480 |
proof - |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
481 |
have "a * b = (pprt a + nprt a) * (pprt b + nprt b)" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
482 |
apply (subst prts[symmetric])+ |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
483 |
apply simp |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
484 |
done |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
485 |
then have "a * b = pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
486 |
by (simp add: algebra_simps) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
487 |
moreover have "pprt a * pprt b <= pprt a2 * pprt b2" |
41528 | 488 |
by (simp_all add: assms mult_mono) |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
489 |
moreover have "pprt a * nprt b <= pprt a1 * nprt b2" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
490 |
proof - |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
491 |
have "pprt a * nprt b <= pprt a * nprt b2" |
41528 | 492 |
by (simp add: mult_left_mono assms) |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
493 |
moreover have "pprt a * nprt b2 <= pprt a1 * nprt b2" |
41528 | 494 |
by (simp add: mult_right_mono_neg assms) |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
495 |
ultimately show ?thesis |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
496 |
by simp |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
497 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
498 |
moreover have "nprt a * pprt b <= nprt a2 * pprt b1" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
499 |
proof - |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
500 |
have "nprt a * pprt b <= nprt a2 * pprt b" |
41528 | 501 |
by (simp add: mult_right_mono assms) |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
502 |
moreover have "nprt a2 * pprt b <= nprt a2 * pprt b1" |
41528 | 503 |
by (simp add: mult_left_mono_neg assms) |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
504 |
ultimately show ?thesis |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
505 |
by simp |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
506 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
507 |
moreover have "nprt a * nprt b <= nprt a1 * nprt b1" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
508 |
proof - |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
509 |
have "nprt a * nprt b <= nprt a * nprt b1" |
41528 | 510 |
by (simp add: mult_left_mono_neg assms) |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
511 |
moreover have "nprt a * nprt b1 <= nprt a1 * nprt b1" |
41528 | 512 |
by (simp add: mult_right_mono_neg assms) |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
513 |
ultimately show ?thesis |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
514 |
by simp |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
515 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
516 |
ultimately show ?thesis |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
517 |
by - (rule add_mono | simp)+ |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
518 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
519 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
520 |
lemma mult_ge_prts: |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
521 |
assumes |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
522 |
"a1 <= (a::'a::lattice_ring)" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
523 |
"a <= a2" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
524 |
"b1 <= b" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
525 |
"b <= b2" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
526 |
shows |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
527 |
"a * b >= nprt a1 * pprt b2 + nprt a2 * nprt b2 + pprt a1 * pprt b1 + pprt a2 * nprt b1" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
528 |
proof - |
41528 | 529 |
from assms have a1:"- a2 <= -a" by auto |
530 |
from assms have a2: "-a <= -a1" by auto |
|
531 |
from mult_le_prts[of "-a2" "-a" "-a1" "b1" b "b2", OF a1 a2 assms(3) assms(4), simplified nprt_neg pprt_neg] |
|
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
532 |
have le: "- (a * b) <= - nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1" by simp |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
533 |
then have "-(- nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1) <= a * b" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
534 |
by (simp only: minus_le_iff) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
535 |
then show ?thesis by simp |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
536 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
537 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
538 |
instance int :: lattice_ring |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
539 |
proof |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
540 |
fix k :: int |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
541 |
show "abs k = sup k (- k)" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
542 |
by (auto simp add: sup_int_def) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
543 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
544 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
545 |
instance real :: lattice_ring |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
546 |
proof |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
547 |
fix a :: real |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
548 |
show "abs a = sup a (- a)" |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
549 |
by (auto simp add: sup_real_def) |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
550 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
551 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
552 |
end |