author | wenzelm |
Thu, 05 Nov 2015 10:39:49 +0100 | |
changeset 61585 | a9599d3d7610 |
parent 61546 | 53bb4172c7f7 |
child 65151 | a7394aa4d21c |
permissions | -rw-r--r-- |
61546 | 1 |
(* Author: Steven Obua, TU Muenchen *) |
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
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section \<open>Various algebraic structures combined with a lattice\<close> |
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
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parents:
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
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theory Lattice_Algebras |
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
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6 |
imports Complex_Main |
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
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7 |
begin |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
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8 |
|
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
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parents:
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class semilattice_inf_ab_group_add = ordered_ab_group_add + semilattice_inf |
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
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parents:
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10 |
begin |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
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11 |
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53240 | 12 |
lemma add_inf_distrib_left: "a + inf b c = inf (a + b) (a + c)" |
13 |
apply (rule antisym) |
|
14 |
apply (simp_all add: le_infI) |
|
15 |
apply (rule add_le_imp_le_left [of "uminus a"]) |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
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16 |
apply (simp only: add.assoc [symmetric], simp add: diff_le_eq add.commute) |
53240 | 17 |
done |
35040
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
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18 |
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53240 | 19 |
lemma add_inf_distrib_right: "inf a b + c = inf (a + c) (b + c)" |
35040
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
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20 |
proof - |
56228 | 21 |
have "c + inf a b = inf (c + a) (c + b)" |
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by (simp add: add_inf_distrib_left) |
56228 | 23 |
then show ?thesis |
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reduced name variants for assoc and commute on plus and mult
haftmann
parents:
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24 |
by (simp add: add.commute) |
35040
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
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25 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
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|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
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parents:
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27 |
end |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
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|
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
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parents:
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class semilattice_sup_ab_group_add = ordered_ab_group_add + semilattice_sup |
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
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parents:
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30 |
begin |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
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parents:
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31 |
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53240 | 32 |
lemma add_sup_distrib_left: "a + sup b c = sup (a + b) (a + c)" |
33 |
apply (rule antisym) |
|
34 |
apply (rule add_le_imp_le_left [of "uminus a"]) |
|
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reduced name variants for assoc and commute on plus and mult
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parents:
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35 |
apply (simp only: add.assoc [symmetric], simp) |
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more simplification rules on unary and binary minus
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36 |
apply (simp add: le_diff_eq add.commute) |
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more simplification rules on unary and binary minus
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parents:
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apply (rule le_supI) |
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reduced name variants for assoc and commute on plus and mult
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parents:
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apply (rule add_le_imp_le_left [of "a"], simp only: add.assoc[symmetric], simp)+ |
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done |
35040
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
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40 |
|
56228 | 41 |
lemma add_sup_distrib_right: "sup a b + c = sup (a + c) (b + c)" |
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
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42 |
proof - |
56228 | 43 |
have "c + sup a b = sup (c+a) (c+b)" |
44 |
by (simp add: add_sup_distrib_left) |
|
45 |
then show ?thesis |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
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changeset
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46 |
by (simp add: add.commute) |
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
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47 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
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48 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
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parents:
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49 |
end |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
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50 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
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51 |
class lattice_ab_group_add = ordered_ab_group_add + lattice |
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
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52 |
begin |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
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53 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
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54 |
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:
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55 |
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:
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56 |
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53240 | 57 |
lemmas add_sup_inf_distribs = |
58 |
add_inf_distrib_right add_inf_distrib_left add_sup_distrib_right add_sup_distrib_left |
|
35040
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
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59 |
|
56228 | 60 |
lemma inf_eq_neg_sup: "inf a b = - sup (- a) (- b)" |
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
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61 |
proof (rule inf_unique) |
53240 | 62 |
fix a b c :: 'a |
56228 | 63 |
show "- sup (- a) (- b) \<le> a" |
35040
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
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64 |
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
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65 |
(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
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66 |
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
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67 |
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
|
68 |
(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
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69 |
assume "a \<le> b" "a \<le> c" |
53240 | 70 |
then show "a \<le> - sup (-b) (-c)" |
71 |
by (subst neg_le_iff_le [symmetric]) (simp add: le_supI) |
|
35040
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
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72 |
qed |
53240 | 73 |
|
56228 | 74 |
lemma sup_eq_neg_inf: "sup a b = - inf (- a) (- b)" |
35040
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
75 |
proof (rule sup_unique) |
53240 | 76 |
fix a b c :: 'a |
56228 | 77 |
show "a \<le> - inf (- a) (- b)" |
35040
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
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78 |
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
|
79 |
(simp, simp add: add_inf_distrib_left) |
56228 | 80 |
show "b \<le> - inf (- a) (- b)" |
35040
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
81 |
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
|
82 |
(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
|
83 |
assume "a \<le> c" "b \<le> c" |
56228 | 84 |
then show "- inf (- a) (- b) \<le> c" |
85 |
by (subst neg_le_iff_le [symmetric]) (simp add: le_infI) |
|
35040
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
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86 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
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|
87 |
|
56228 | 88 |
lemma neg_inf_eq_sup: "- inf a b = sup (- a) (- b)" |
53240 | 89 |
by (simp add: inf_eq_neg_sup) |
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
90 |
|
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more simplification rules on unary and binary minus
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parents:
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diff
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91 |
lemma diff_inf_eq_sup: "a - inf b c = a + sup (- b) (- c)" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
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diff
changeset
|
92 |
using neg_inf_eq_sup [of b c, symmetric] by simp |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
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diff
changeset
|
93 |
|
56228 | 94 |
lemma neg_sup_eq_inf: "- sup a b = inf (- a) (- b)" |
53240 | 95 |
by (simp add: sup_eq_neg_inf) |
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
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parents:
diff
changeset
|
96 |
|
54230
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more simplification rules on unary and binary minus
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parents:
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diff
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|
97 |
lemma diff_sup_eq_inf: "a - sup b c = a + inf (- b) (- c)" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
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diff
changeset
|
98 |
using neg_sup_eq_inf [of b c, symmetric] by simp |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53240
diff
changeset
|
99 |
|
35040
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
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100 |
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
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|
101 |
proof - |
56228 | 102 |
have "0 = - inf 0 (a - b) + inf (a - b) 0" |
53240 | 103 |
by (simp add: inf_commute) |
56228 | 104 |
then have "0 = sup 0 (b - a) + inf (a - b) 0" |
53240 | 105 |
by (simp add: inf_eq_neg_sup) |
56228 | 106 |
then have "0 = (- a + sup a b) + (inf a b + (- b))" |
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more simplification rules on unary and binary minus
haftmann
parents:
53240
diff
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|
107 |
by (simp only: add_sup_distrib_left add_inf_distrib_right) simp |
56228 | 108 |
then show ?thesis |
109 |
by (simp add: algebra_simps) |
|
35040
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
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110 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
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111 |
|
53240 | 112 |
|
60500 | 113 |
subsection \<open>Positive Part, Negative Part, Absolute Value\<close> |
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
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114 |
|
53240 | 115 |
definition nprt :: "'a \<Rightarrow> 'a" |
116 |
where "nprt x = inf x 0" |
|
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
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117 |
|
53240 | 118 |
definition pprt :: "'a \<Rightarrow> 'a" |
119 |
where "pprt x = sup x 0" |
|
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
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120 |
|
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
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121 |
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:
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122 |
proof - |
56228 | 123 |
have "sup (- x) 0 = sup (- x) (- 0)" |
124 |
unfolding minus_zero .. |
|
125 |
also have "\<dots> = - inf x 0" |
|
126 |
unfolding neg_inf_eq_sup .. |
|
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
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parents:
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127 |
finally have "sup (- x) 0 = - inf x 0" . |
56228 | 128 |
then show ?thesis |
129 |
unfolding pprt_def nprt_def . |
|
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
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parents:
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130 |
qed |
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
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parents:
diff
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|
131 |
|
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
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parents:
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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
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|
133 |
proof - |
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
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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
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|
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
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|
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 |
|
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
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parents:
diff
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|
139 |
lemma prts: "a = pprt a + nprt a" |
53240 | 140 |
by (simp add: pprt_def nprt_def add_eq_inf_sup[symmetric]) |
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
141 |
|
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
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142 |
lemma zero_le_pprt[simp]: "0 \<le> pprt a" |
53240 | 143 |
by (simp add: pprt_def) |
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
144 |
|
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
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145 |
lemma nprt_le_zero[simp]: "nprt a \<le> 0" |
53240 | 146 |
by (simp add: nprt_def) |
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
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147 |
|
60698 | 148 |
lemma le_eq_neg: "a \<le> - b \<longleftrightarrow> a + b \<le> 0" |
149 |
(is "?l = ?r") |
|
53240 | 150 |
proof |
151 |
assume ?l |
|
152 |
then show ?r |
|
153 |
apply - |
|
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|
154 |
apply (rule add_le_imp_le_right[of _ "uminus b" _]) |
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|
155 |
apply (simp add: add.assoc) |
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|
156 |
done |
53240 | 157 |
next |
158 |
assume ?r |
|
159 |
then show ?l |
|
160 |
apply - |
|
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|
161 |
apply (rule add_le_imp_le_right[of _ "b" _]) |
53240 | 162 |
apply simp |
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|
163 |
done |
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parents:
diff
changeset
|
164 |
qed |
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parents:
diff
changeset
|
165 |
|
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|
166 |
lemma pprt_0[simp]: "pprt 0 = 0" by (simp add: pprt_def) |
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|
167 |
lemma nprt_0[simp]: "nprt 0 = 0" by (simp add: nprt_def) |
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|
168 |
|
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|
169 |
lemma pprt_eq_id [simp, no_atp]: "0 \<le> x \<Longrightarrow> pprt x = x" |
46986 | 170 |
by (simp add: pprt_def sup_absorb1) |
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|
171 |
|
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changeset
|
172 |
lemma nprt_eq_id [simp, no_atp]: "x \<le> 0 \<Longrightarrow> nprt x = x" |
46986 | 173 |
by (simp add: nprt_def inf_absorb1) |
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changeset
|
174 |
|
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changeset
|
175 |
lemma pprt_eq_0 [simp, no_atp]: "x \<le> 0 \<Longrightarrow> pprt x = 0" |
46986 | 176 |
by (simp add: pprt_def sup_absorb2) |
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changeset
|
177 |
|
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changeset
|
178 |
lemma nprt_eq_0 [simp, no_atp]: "0 \<le> x \<Longrightarrow> nprt x = 0" |
46986 | 179 |
by (simp add: nprt_def inf_absorb2) |
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parents:
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|
180 |
|
60698 | 181 |
lemma sup_0_imp_0: |
182 |
assumes "sup a (- a) = 0" |
|
183 |
shows "a = 0" |
|
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parents:
diff
changeset
|
184 |
proof - |
60698 | 185 |
have p: "0 \<le> a" if "sup a (- a) = 0" for a :: 'a |
186 |
proof - |
|
187 |
from that have "sup a (- a) + a = a" |
|
56228 | 188 |
by simp |
189 |
then have "sup (a + a) 0 = a" |
|
190 |
by (simp add: add_sup_distrib_right) |
|
191 |
then have "sup (a + a) 0 \<le> a" |
|
192 |
by simp |
|
60698 | 193 |
then show ?thesis |
56228 | 194 |
by (blast intro: order_trans inf_sup_ord) |
60698 | 195 |
qed |
196 |
from assms have **: "sup (-a) (-(-a)) = 0" |
|
56228 | 197 |
by (simp add: sup_commute) |
60698 | 198 |
from p[OF assms] p[OF **] show "a = 0" |
56228 | 199 |
by simp |
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changeset
|
200 |
qed |
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parents:
diff
changeset
|
201 |
|
56228 | 202 |
lemma inf_0_imp_0: "inf a (- a) = 0 \<Longrightarrow> a = 0" |
53240 | 203 |
apply (simp add: inf_eq_neg_sup) |
204 |
apply (simp add: sup_commute) |
|
205 |
apply (erule sup_0_imp_0) |
|
206 |
done |
|
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parents:
diff
changeset
|
207 |
|
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parents:
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changeset
|
208 |
lemma inf_0_eq_0 [simp, no_atp]: "inf a (- a) = 0 \<longleftrightarrow> a = 0" |
53240 | 209 |
apply rule |
210 |
apply (erule inf_0_imp_0) |
|
211 |
apply simp |
|
212 |
done |
|
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parents:
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changeset
|
213 |
|
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changeset
|
214 |
lemma sup_0_eq_0 [simp, no_atp]: "sup a (- a) = 0 \<longleftrightarrow> a = 0" |
53240 | 215 |
apply rule |
216 |
apply (erule sup_0_imp_0) |
|
217 |
apply simp |
|
218 |
done |
|
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parents:
diff
changeset
|
219 |
|
60698 | 220 |
lemma zero_le_double_add_iff_zero_le_single_add [simp]: "0 \<le> a + a \<longleftrightarrow> 0 \<le> a" |
221 |
(is "?lhs \<longleftrightarrow> ?rhs") |
|
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parents:
diff
changeset
|
222 |
proof |
60698 | 223 |
show ?rhs if ?lhs |
224 |
proof - |
|
225 |
from that have a: "inf (a + a) 0 = 0" |
|
226 |
by (simp add: inf_commute inf_absorb1) |
|
61546 | 227 |
have "inf a 0 + inf a 0 = inf (inf (a + a) 0) a" (is "?l = _") |
60698 | 228 |
by (simp add: add_sup_inf_distribs inf_aci) |
229 |
then have "?l = 0 + inf a 0" |
|
230 |
by (simp add: a, simp add: inf_commute) |
|
231 |
then have "inf a 0 = 0" |
|
232 |
by (simp only: add_right_cancel) |
|
233 |
then show ?thesis |
|
234 |
unfolding le_iff_inf by (simp add: inf_commute) |
|
235 |
qed |
|
236 |
show ?lhs if ?rhs |
|
237 |
by (simp add: add_mono[OF that that, simplified]) |
|
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parents:
diff
changeset
|
238 |
qed |
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
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parents:
diff
changeset
|
239 |
|
53240 | 240 |
lemma double_zero [simp]: "a + a = 0 \<longleftrightarrow> a = 0" |
60698 | 241 |
(is "?lhs \<longleftrightarrow> ?rhs") |
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parents:
diff
changeset
|
242 |
proof |
60698 | 243 |
show ?rhs if ?lhs |
244 |
proof - |
|
245 |
from that have "a + a + - a = - a" |
|
246 |
by simp |
|
247 |
then have "a + (a + - a) = - a" |
|
248 |
by (simp only: add.assoc) |
|
249 |
then have a: "- a = a" |
|
250 |
by simp |
|
251 |
show ?thesis |
|
252 |
apply (rule antisym) |
|
253 |
apply (unfold neg_le_iff_le [symmetric, of a]) |
|
254 |
unfolding a |
|
255 |
apply simp |
|
256 |
unfolding zero_le_double_add_iff_zero_le_single_add [symmetric, of a] |
|
257 |
unfolding that |
|
258 |
unfolding le_less |
|
259 |
apply simp_all |
|
260 |
done |
|
261 |
qed |
|
262 |
show ?lhs if ?rhs |
|
263 |
using that by simp |
|
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
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parents:
diff
changeset
|
264 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
265 |
|
53240 | 266 |
lemma zero_less_double_add_iff_zero_less_single_add [simp]: "0 < a + a \<longleftrightarrow> 0 < a" |
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changeset
|
267 |
proof (cases "a = 0") |
53240 | 268 |
case True |
269 |
then show ?thesis by auto |
|
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parents:
diff
changeset
|
270 |
next |
53240 | 271 |
case False |
272 |
then show ?thesis |
|
273 |
unfolding less_le |
|
274 |
apply simp |
|
275 |
apply rule |
|
276 |
apply clarify |
|
277 |
apply rule |
|
278 |
apply assumption |
|
279 |
apply (rule notI) |
|
280 |
unfolding double_zero [symmetric, of a] |
|
54230
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parents:
53240
diff
changeset
|
281 |
apply blast |
53240 | 282 |
done |
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
283 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
284 |
|
60698 | 285 |
lemma double_add_le_zero_iff_single_add_le_zero [simp]: "a + a \<le> 0 \<longleftrightarrow> a \<le> 0" |
35040
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
286 |
proof - |
56228 | 287 |
have "a + a \<le> 0 \<longleftrightarrow> 0 \<le> - (a + a)" |
60698 | 288 |
by (subst le_minus_iff) simp |
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haftmann
parents:
53240
diff
changeset
|
289 |
moreover have "\<dots> \<longleftrightarrow> a \<le> 0" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53240
diff
changeset
|
290 |
by (simp only: minus_add_distrib zero_le_double_add_iff_zero_le_single_add) simp |
56228 | 291 |
ultimately show ?thesis |
292 |
by blast |
|
35040
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haftmann
parents:
diff
changeset
|
293 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
294 |
|
60698 | 295 |
lemma double_add_less_zero_iff_single_less_zero [simp]: "a + a < 0 \<longleftrightarrow> a < 0" |
35040
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
296 |
proof - |
56228 | 297 |
have "a + a < 0 \<longleftrightarrow> 0 < - (a + a)" |
298 |
by (subst less_minus_iff) simp |
|
54230
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haftmann
parents:
53240
diff
changeset
|
299 |
moreover have "\<dots> \<longleftrightarrow> a < 0" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53240
diff
changeset
|
300 |
by (simp only: minus_add_distrib zero_less_double_add_iff_zero_less_single_add) simp |
56228 | 301 |
ultimately show ?thesis |
302 |
by blast |
|
35040
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
303 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
304 |
|
54230
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more simplification rules on unary and binary minus
haftmann
parents:
53240
diff
changeset
|
305 |
declare neg_inf_eq_sup [simp] neg_sup_eq_inf [simp] diff_inf_eq_sup [simp] diff_sup_eq_inf [simp] |
35040
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separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
306 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
307 |
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
|
308 |
proof - |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
309 |
from add_le_cancel_left [of "uminus a" "plus a a" zero] |
56228 | 310 |
have "a \<le> - a \<longleftrightarrow> a + a \<le> 0" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56228
diff
changeset
|
311 |
by (simp add: add.assoc[symmetric]) |
56228 | 312 |
then show ?thesis |
313 |
by simp |
|
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
314 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
315 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
316 |
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
|
317 |
proof - |
56228 | 318 |
have "- a \<le> a \<longleftrightarrow> 0 \<le> a + a" |
60698 | 319 |
using add_le_cancel_left [of "uminus a" zero "plus a a"] |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56228
diff
changeset
|
320 |
by (simp add: add.assoc[symmetric]) |
56228 | 321 |
then show ?thesis |
322 |
by simp |
|
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
323 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
324 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
325 |
lemma zero_le_iff_zero_nprt: "0 \<le> a \<longleftrightarrow> nprt a = 0" |
53240 | 326 |
unfolding le_iff_inf by (simp add: nprt_def inf_commute) |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
327 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
328 |
lemma le_zero_iff_zero_pprt: "a \<le> 0 \<longleftrightarrow> pprt a = 0" |
53240 | 329 |
unfolding le_iff_sup by (simp add: pprt_def sup_commute) |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
330 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
331 |
lemma le_zero_iff_pprt_id: "0 \<le> a \<longleftrightarrow> pprt a = a" |
53240 | 332 |
unfolding le_iff_sup by (simp add: pprt_def sup_commute) |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
333 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
334 |
lemma zero_le_iff_nprt_id: "a \<le> 0 \<longleftrightarrow> nprt a = a" |
53240 | 335 |
unfolding le_iff_inf by (simp add: nprt_def inf_commute) |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
336 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35040
diff
changeset
|
337 |
lemma pprt_mono [simp, no_atp]: "a \<le> b \<Longrightarrow> pprt a \<le> pprt b" |
53240 | 338 |
unfolding le_iff_sup by (simp add: pprt_def sup_aci sup_assoc [symmetric, of a]) |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
339 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35040
diff
changeset
|
340 |
lemma nprt_mono [simp, no_atp]: "a \<le> b \<Longrightarrow> nprt a \<le> nprt b" |
53240 | 341 |
unfolding le_iff_inf by (simp add: nprt_def inf_aci inf_assoc [symmetric, of a]) |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
342 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
343 |
end |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
344 |
|
56228 | 345 |
lemmas add_sup_inf_distribs = |
346 |
add_inf_distrib_right add_inf_distrib_left add_sup_distrib_right add_sup_distrib_left |
|
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
347 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
348 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
349 |
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
|
350 |
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
|
351 |
begin |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
352 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
353 |
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
|
354 |
proof - |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
355 |
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
|
356 |
proof - |
56228 | 357 |
have a: "a \<le> \<bar>a\<bar>" and b: "- a \<le> \<bar>a\<bar>" |
358 |
by (auto simp add: abs_lattice) |
|
359 |
show ?thesis |
|
360 |
by (rule add_mono [OF a b, simplified]) |
|
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
361 |
qed |
56228 | 362 |
then have "0 \<le> sup a (- a)" |
363 |
unfolding abs_lattice . |
|
364 |
then have "sup (sup a (- a)) 0 = sup a (- a)" |
|
365 |
by (rule sup_absorb1) |
|
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
366 |
then show ?thesis |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53240
diff
changeset
|
367 |
by (simp add: add_sup_inf_distribs ac_simps pprt_def nprt_def abs_lattice) |
35040
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 |
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
|
371 |
proof |
60698 | 372 |
have abs_ge_zero [simp]: "0 \<le> \<bar>a\<bar>" for a |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
373 |
proof - |
53240 | 374 |
have a: "a \<le> \<bar>a\<bar>" and b: "- a \<le> \<bar>a\<bar>" |
375 |
by (auto simp add: abs_lattice) |
|
376 |
show "0 \<le> \<bar>a\<bar>" |
|
377 |
by (rule add_mono [OF a b, simplified]) |
|
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
378 |
qed |
60698 | 379 |
have abs_leI: "a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b" for a b |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
380 |
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
|
381 |
fix a b |
56228 | 382 |
show "0 \<le> \<bar>a\<bar>" |
383 |
by simp |
|
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
384 |
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
|
385 |
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
|
386 |
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
|
387 |
by (simp add: abs_lattice sup_commute) |
60698 | 388 |
show "- a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b" if "a \<le> b" |
389 |
using that by (rule abs_leI) |
|
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
390 |
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
|
391 |
proof - |
56228 | 392 |
have g: "\<bar>a\<bar> + \<bar>b\<bar> = sup (a + b) (sup (- a - b) (sup (- a + b) (a + (- b))))" |
60698 | 393 |
(is "_ = sup ?m ?n") |
57862 | 394 |
by (simp add: abs_lattice add_sup_inf_distribs ac_simps) |
56228 | 395 |
have a: "a + b \<le> sup ?m ?n" |
396 |
by simp |
|
397 |
have b: "- a - b \<le> ?n" |
|
398 |
by simp |
|
399 |
have c: "?n \<le> sup ?m ?n" |
|
400 |
by simp |
|
401 |
from b c have d: "- a - b \<le> sup ?m ?n" |
|
402 |
by (rule order_trans) |
|
403 |
have e: "- a - b = - (a + b)" |
|
404 |
by simp |
|
405 |
from a d e have "\<bar>a + b\<bar> \<le> sup ?m ?n" |
|
53240 | 406 |
apply - |
407 |
apply (drule abs_leI) |
|
57862 | 408 |
apply (simp_all only: algebra_simps minus_add) |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53240
diff
changeset
|
409 |
apply (metis add_uminus_conv_diff d sup_commute uminus_add_conv_diff) |
53240 | 410 |
done |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
411 |
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
|
412 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
413 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
414 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
415 |
end |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
416 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
417 |
lemma sup_eq_if: |
60698 | 418 |
fixes a :: "'a::{lattice_ab_group_add,linorder}" |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
419 |
shows "sup a (- a) = (if a < 0 then - a else a)" |
60698 | 420 |
using add_le_cancel_right [of a a "- a", symmetric, simplified] |
421 |
and add_le_cancel_right [of "-a" a a, symmetric, simplified] |
|
422 |
by (auto simp: sup_max max.absorb1 max.absorb2) |
|
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
423 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
424 |
lemma abs_if_lattice: |
60698 | 425 |
fixes a :: "'a::{lattice_ab_group_add_abs,linorder}" |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
426 |
shows "\<bar>a\<bar> = (if a < 0 then - a else a)" |
53240 | 427 |
by auto |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
428 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
429 |
lemma estimate_by_abs: |
56228 | 430 |
fixes a b c :: "'a::lattice_ab_group_add_abs" |
60698 | 431 |
assumes "a + b \<le> c" |
432 |
shows "a \<le> c + \<bar>b\<bar>" |
|
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
433 |
proof - |
60698 | 434 |
from assms have "a \<le> c + (- b)" |
56228 | 435 |
by (simp add: algebra_simps) |
436 |
have "- b \<le> \<bar>b\<bar>" |
|
437 |
by (rule abs_ge_minus_self) |
|
438 |
then have "c + (- b) \<le> c + \<bar>b\<bar>" |
|
439 |
by (rule add_left_mono) |
|
60500 | 440 |
with \<open>a \<le> c + (- b)\<close> show ?thesis |
56228 | 441 |
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
|
442 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
443 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
444 |
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
|
445 |
begin |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
446 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
447 |
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
|
448 |
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
|
449 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
450 |
end |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
451 |
|
56228 | 452 |
lemma abs_le_mult: |
453 |
fixes a b :: "'a::lattice_ring" |
|
454 |
shows "\<bar>a * b\<bar> \<le> \<bar>a\<bar> * \<bar>b\<bar>" |
|
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
455 |
proof - |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
456 |
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
|
457 |
let ?y = "pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b" |
56228 | 458 |
have a: "\<bar>a\<bar> * \<bar>b\<bar> = ?x" |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
459 |
by (simp only: abs_prts[of a] abs_prts[of b] algebra_simps) |
60698 | 460 |
have bh: "u = a \<Longrightarrow> v = b \<Longrightarrow> |
461 |
u * v = pprt a * pprt b + pprt a * nprt b + |
|
462 |
nprt a * pprt b + nprt a * nprt b" for u v :: 'a |
|
463 |
apply (subst prts[of u], subst prts[of v]) |
|
464 |
apply (simp add: algebra_simps) |
|
465 |
done |
|
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
466 |
note b = this[OF refl[of a] refl[of b]] |
56228 | 467 |
have xy: "- ?x \<le> ?y" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53240
diff
changeset
|
468 |
apply simp |
56228 | 469 |
apply (metis (full_types) add_increasing add_uminus_conv_diff |
470 |
lattice_ab_group_add_class.minus_le_self_iff minus_add_distrib mult_nonneg_nonneg |
|
471 |
mult_nonpos_nonpos nprt_le_zero zero_le_pprt) |
|
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
472 |
done |
56228 | 473 |
have yx: "?y \<le> ?x" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53240
diff
changeset
|
474 |
apply simp |
56228 | 475 |
apply (metis (full_types) add_nonpos_nonpos add_uminus_conv_diff |
476 |
lattice_ab_group_add_class.le_minus_self_iff minus_add_distrib mult_nonneg_nonpos |
|
477 |
mult_nonpos_nonneg nprt_le_zero zero_le_pprt) |
|
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
478 |
done |
56228 | 479 |
have i1: "a * b \<le> \<bar>a\<bar> * \<bar>b\<bar>" |
480 |
by (simp only: a b yx) |
|
481 |
have i2: "- (\<bar>a\<bar> * \<bar>b\<bar>) \<le> a * b" |
|
482 |
by (simp only: a b xy) |
|
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
483 |
show ?thesis |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
484 |
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
|
485 |
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
|
486 |
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
|
487 |
done |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
488 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
489 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
490 |
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
|
491 |
proof |
56228 | 492 |
fix a b :: "'a::lattice_ring" |
41528 | 493 |
assume a: "(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0)" |
56228 | 494 |
show "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>" |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
495 |
proof - |
56228 | 496 |
have s: "(0 \<le> a * b) \<or> (a * b \<le> 0)" |
497 |
apply auto |
|
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
498 |
apply (rule_tac split_mult_pos_le) |
56228 | 499 |
apply (rule_tac contrapos_np[of "a * b \<le> 0"]) |
500 |
apply simp |
|
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
501 |
apply (rule_tac split_mult_neg_le) |
56228 | 502 |
using a |
503 |
apply blast |
|
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
504 |
done |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
505 |
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
|
506 |
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
|
507 |
show ?thesis |
56228 | 508 |
proof (cases "0 \<le> a * b") |
509 |
case True |
|
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
510 |
then show ?thesis |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
511 |
apply (simp_all add: mulprts abs_prts) |
56228 | 512 |
using a |
53240 | 513 |
apply (auto simp add: |
514 |
algebra_simps |
|
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
515 |
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
|
516 |
iffD1[OF le_zero_iff_pprt_id] iffD1[OF zero_le_iff_nprt_id]) |
56228 | 517 |
apply(drule (1) mult_nonneg_nonpos[of a b], simp) |
518 |
apply(drule (1) mult_nonneg_nonpos2[of b a], simp) |
|
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
519 |
done |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
520 |
next |
56228 | 521 |
case False |
522 |
with s have "a * b \<le> 0" |
|
523 |
by simp |
|
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
524 |
then show ?thesis |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
525 |
apply (simp_all add: mulprts abs_prts) |
41528 | 526 |
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
|
527 |
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
|
528 |
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
|
529 |
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
|
530 |
done |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
531 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
532 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
533 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
534 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
535 |
lemma mult_le_prts: |
56228 | 536 |
fixes a b :: "'a::lattice_ring" |
537 |
assumes "a1 \<le> a" |
|
538 |
and "a \<le> a2" |
|
539 |
and "b1 \<le> b" |
|
540 |
and "b \<le> b2" |
|
541 |
shows "a * b \<le> |
|
53240 | 542 |
pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1" |
543 |
proof - |
|
544 |
have "a * b = (pprt a + nprt a) * (pprt b + nprt b)" |
|
60698 | 545 |
by (subst prts[symmetric])+ simp |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
546 |
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
|
547 |
by (simp add: algebra_simps) |
56228 | 548 |
moreover have "pprt a * pprt b \<le> pprt a2 * pprt b2" |
41528 | 549 |
by (simp_all add: assms mult_mono) |
56228 | 550 |
moreover have "pprt a * nprt b \<le> pprt a1 * nprt b2" |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
551 |
proof - |
56228 | 552 |
have "pprt a * nprt b \<le> pprt a * nprt b2" |
41528 | 553 |
by (simp add: mult_left_mono assms) |
56228 | 554 |
moreover have "pprt a * nprt b2 \<le> pprt a1 * nprt b2" |
41528 | 555 |
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
|
556 |
ultimately show ?thesis |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
557 |
by simp |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
558 |
qed |
56228 | 559 |
moreover have "nprt a * pprt b \<le> nprt a2 * pprt b1" |
53240 | 560 |
proof - |
56228 | 561 |
have "nprt a * pprt b \<le> nprt a2 * pprt b" |
41528 | 562 |
by (simp add: mult_right_mono assms) |
56228 | 563 |
moreover have "nprt a2 * pprt b \<le> nprt a2 * pprt b1" |
41528 | 564 |
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
|
565 |
ultimately show ?thesis |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
566 |
by simp |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
567 |
qed |
56228 | 568 |
moreover have "nprt a * nprt b \<le> nprt a1 * nprt b1" |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
569 |
proof - |
56228 | 570 |
have "nprt a * nprt b \<le> nprt a * nprt b1" |
41528 | 571 |
by (simp add: mult_left_mono_neg assms) |
56228 | 572 |
moreover have "nprt a * nprt b1 \<le> nprt a1 * nprt b1" |
41528 | 573 |
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
|
574 |
ultimately show ?thesis |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
575 |
by simp |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
576 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
577 |
ultimately show ?thesis |
60698 | 578 |
by - (rule add_mono | simp)+ |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
579 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
580 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
581 |
lemma mult_ge_prts: |
56228 | 582 |
fixes a b :: "'a::lattice_ring" |
583 |
assumes "a1 \<le> a" |
|
584 |
and "a \<le> a2" |
|
585 |
and "b1 \<le> b" |
|
586 |
and "b \<le> b2" |
|
587 |
shows "a * b \<ge> |
|
53240 | 588 |
nprt a1 * pprt b2 + nprt a2 * nprt b2 + pprt a1 * pprt b1 + pprt a2 * nprt b1" |
589 |
proof - |
|
56228 | 590 |
from assms have a1: "- a2 \<le> -a" |
53240 | 591 |
by auto |
56228 | 592 |
from assms have a2: "- a \<le> -a1" |
53240 | 593 |
by auto |
56228 | 594 |
from mult_le_prts[of "- a2" "- a" "- a1" "b1" b "b2", |
595 |
OF a1 a2 assms(3) assms(4), simplified nprt_neg pprt_neg] |
|
60698 | 596 |
have le: "- (a * b) \<le> |
597 |
- nprt a1 * pprt b2 + - nprt a2 * nprt b2 + |
|
56228 | 598 |
- pprt a1 * pprt b1 + - pprt a2 * nprt b1" |
53240 | 599 |
by simp |
56228 | 600 |
then have "- (- nprt a1 * pprt b2 + - nprt a2 * nprt b2 + |
601 |
- pprt a1 * pprt b1 + - pprt a2 * nprt b1) \<le> a * b" |
|
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
602 |
by (simp only: minus_le_iff) |
56228 | 603 |
then show ?thesis |
604 |
by (simp add: algebra_simps) |
|
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
605 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
606 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
607 |
instance int :: lattice_ring |
53240 | 608 |
proof |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
609 |
fix k :: int |
56228 | 610 |
show "\<bar>k\<bar> = sup k (- k)" |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
611 |
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
|
612 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
613 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
614 |
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
|
615 |
proof |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
616 |
fix a :: real |
56228 | 617 |
show "\<bar>a\<bar> = sup a (- a)" |
35040
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
618 |
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
|
619 |
qed |
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
620 |
|
e42e7f133d94
separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff
changeset
|
621 |
end |