src/HOL/Library/Lattice_Algebras.thy
author haftmann
Fri, 01 Nov 2013 18:51:14 +0100
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child 54863 82acc20ded73
permissions -rw-r--r--
more simplification rules on unary and binary minus
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(* Author: Steven Obua, TU Muenchen *)
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header {* Various algebraic structures combined with a lattice *}
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theory Lattice_Algebras
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imports Complex_Main
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begin
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class semilattice_inf_ab_group_add = ordered_ab_group_add + semilattice_inf
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begin
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lemma add_inf_distrib_left: "a + inf b c = inf (a + b) (a + c)"
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  apply (rule antisym)
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  apply (simp_all add: le_infI)
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  apply (rule add_le_imp_le_left [of "uminus a"])
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  apply (simp only: add_assoc [symmetric], simp add: diff_le_eq add.commute)
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  done
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lemma add_inf_distrib_right: "inf a b + c = inf (a + c) (b + c)"
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proof -
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  have "c + inf a b = inf (c+a) (c+b)"
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    by (simp add: add_inf_distrib_left)
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  thus ?thesis by (simp add: add_commute)
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qed
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end
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class semilattice_sup_ab_group_add = ordered_ab_group_add + semilattice_sup
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begin
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lemma add_sup_distrib_left: "a + sup b c = sup (a + b) (a + c)"
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  apply (rule antisym)
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  apply (rule add_le_imp_le_left [of "uminus a"])
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  apply (simp only: add_assoc [symmetric], simp)
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  apply (simp add: le_diff_eq add.commute)
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  apply (rule le_supI)
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  apply (rule add_le_imp_le_left [of "a"], simp only: add_assoc[symmetric], simp)+
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  done
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lemma add_sup_distrib_right: "sup a b + c = sup (a+c) (b+c)"
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proof -
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  have "c + sup a b = sup (c+a) (c+b)" by (simp add: add_sup_distrib_left)
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  thus ?thesis by (simp add: add_commute)
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qed
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end
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class lattice_ab_group_add = ordered_ab_group_add + lattice
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begin
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subclass semilattice_inf_ab_group_add ..
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subclass semilattice_sup_ab_group_add ..
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lemmas add_sup_inf_distribs =
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  add_inf_distrib_right add_inf_distrib_left add_sup_distrib_right add_sup_distrib_left
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lemma inf_eq_neg_sup: "inf a b = - sup (-a) (-b)"
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proof (rule inf_unique)
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  fix a b c :: 'a
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  show "- sup (-a) (-b) \<le> a"
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    by (rule add_le_imp_le_right [of _ "sup (uminus a) (uminus b)"])
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      (simp, simp add: add_sup_distrib_left)
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  show "- sup (-a) (-b) \<le> b"
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    by (rule add_le_imp_le_right [of _ "sup (uminus a) (uminus b)"])
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      (simp, simp add: add_sup_distrib_left)
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  assume "a \<le> b" "a \<le> c"
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  then show "a \<le> - sup (-b) (-c)"
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    by (subst neg_le_iff_le [symmetric]) (simp add: le_supI)
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qed
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lemma sup_eq_neg_inf: "sup a b = - inf (-a) (-b)"
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proof (rule sup_unique)
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  fix a b c :: 'a
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  show "a \<le> - inf (-a) (-b)"
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    by (rule add_le_imp_le_right [of _ "inf (uminus a) (uminus b)"])
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      (simp, simp add: add_inf_distrib_left)
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  show "b \<le> - inf (-a) (-b)"
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    by (rule add_le_imp_le_right [of _ "inf (uminus a) (uminus b)"])
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      (simp, simp add: add_inf_distrib_left)
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  assume "a \<le> c" "b \<le> c"
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  then show "- inf (-a) (-b) \<le> c" by (subst neg_le_iff_le [symmetric]) (simp add: le_infI)
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qed
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lemma neg_inf_eq_sup: "- inf a b = sup (-a) (-b)"
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  by (simp add: inf_eq_neg_sup)
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lemma diff_inf_eq_sup: "a - inf b c = a + sup (- b) (- c)"
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  using neg_inf_eq_sup [of b c, symmetric] by simp
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lemma neg_sup_eq_inf: "- sup a b = inf (-a) (-b)"
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  by (simp add: sup_eq_neg_inf)
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lemma diff_sup_eq_inf: "a - sup b c = a + inf (- b) (- c)"
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  using neg_sup_eq_inf [of b c, symmetric] by simp
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lemma add_eq_inf_sup: "a + b = sup a b + inf a b"
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proof -
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  have "0 = - inf 0 (a-b) + inf (a-b) 0"
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    by (simp add: inf_commute)
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  hence "0 = sup 0 (b-a) + inf (a-b) 0"
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    by (simp add: inf_eq_neg_sup)
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  hence "0 = (-a + sup a b) + (inf a b + (-b))"
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    by (simp only: add_sup_distrib_left add_inf_distrib_right) simp
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  then show ?thesis by (simp add: algebra_simps)
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qed
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subsection {* Positive Part, Negative Part, Absolute Value *}
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definition nprt :: "'a \<Rightarrow> 'a"
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  where "nprt x = inf x 0"
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definition pprt :: "'a \<Rightarrow> 'a"
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  where "pprt x = sup x 0"
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lemma pprt_neg: "pprt (- x) = - nprt x"
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proof -
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  have "sup (- x) 0 = sup (- x) (- 0)" unfolding minus_zero ..
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  also have "\<dots> = - inf x 0" unfolding neg_inf_eq_sup ..
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  finally have "sup (- x) 0 = - inf x 0" .
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  then show ?thesis unfolding pprt_def nprt_def .
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qed
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lemma nprt_neg: "nprt (- x) = - pprt x"
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   125
proof -
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  from pprt_neg have "pprt (- (- x)) = - nprt (- x)" .
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  then have "pprt x = - nprt (- x)" by simp
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parents:
diff changeset
   128
  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
   129
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   130
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   131
lemma prts: "a = pprt a + nprt a"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   132
  by (simp add: pprt_def nprt_def add_eq_inf_sup[symmetric])
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   133
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   134
lemma zero_le_pprt[simp]: "0 \<le> pprt a"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   135
  by (simp add: pprt_def)
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   136
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   137
lemma nprt_le_zero[simp]: "nprt a \<le> 0"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   138
  by (simp add: nprt_def)
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   139
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   140
lemma le_eq_neg: "a \<le> - b \<longleftrightarrow> a + b \<le> 0" (is "?l = ?r")
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   141
proof
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   142
  assume ?l
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   143
  then show ?r
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   144
    apply -
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   145
    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
   146
    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
   147
    done
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   148
next
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   149
  assume ?r
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   150
  then show ?l
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   151
    apply -
35040
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 _ "b" _])
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   153
    apply simp
35040
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
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   156
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   157
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
   158
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
   159
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35040
diff changeset
   160
lemma pprt_eq_id [simp, no_atp]: "0 \<le> x \<Longrightarrow> pprt x = x"
46986
8198cbff1771 tuned proofs;
wenzelm
parents: 41528
diff changeset
   161
  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
   162
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35040
diff changeset
   163
lemma nprt_eq_id [simp, no_atp]: "x \<le> 0 \<Longrightarrow> nprt x = x"
46986
8198cbff1771 tuned proofs;
wenzelm
parents: 41528
diff changeset
   164
  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
   165
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35040
diff changeset
   166
lemma pprt_eq_0 [simp, no_atp]: "x \<le> 0 \<Longrightarrow> pprt x = 0"
46986
8198cbff1771 tuned proofs;
wenzelm
parents: 41528
diff changeset
   167
  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
   168
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35040
diff changeset
   169
lemma nprt_eq_0 [simp, no_atp]: "0 \<le> x \<Longrightarrow> nprt x = 0"
46986
8198cbff1771 tuned proofs;
wenzelm
parents: 41528
diff changeset
   170
  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
   171
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   172
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
   173
proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   174
  {
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   175
    fix a::'a
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   176
    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
   177
    hence "sup a (-a) + a = a" by (simp)
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   178
    hence "sup (a+a) 0 = a" by (simp add: add_sup_distrib_right)
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   179
    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
   180
    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
   181
  }
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   182
  note p = this
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   183
  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
   184
  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
   185
  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
   186
qed
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
lemma inf_0_imp_0: "inf a (-a) = 0 \<Longrightarrow> a = 0"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   189
  apply (simp add: inf_eq_neg_sup)
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   190
  apply (simp add: sup_commute)
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   191
  apply (erule sup_0_imp_0)
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   192
  done
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   193
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35040
diff changeset
   194
lemma inf_0_eq_0 [simp, no_atp]: "inf a (- a) = 0 \<longleftrightarrow> a = 0"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   195
  apply rule
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   196
  apply (erule inf_0_imp_0)
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   197
  apply simp
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   198
  done
35040
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 sup_0_eq_0 [simp, no_atp]: "sup a (- a) = 0 \<longleftrightarrow> a = 0"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   201
  apply rule
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   202
  apply (erule sup_0_imp_0)
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   203
  apply simp
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   204
  done
35040
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
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   221
lemma double_zero [simp]: "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
   222
proof
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   223
  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
   224
  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
   225
  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
   226
  then have a: "- a = a" by simp
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   227
  show "a = 0"
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   228
    apply (rule antisym)
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   229
    apply (unfold neg_le_iff_le [symmetric, of a])
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   230
    unfolding a
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   231
    apply simp
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   232
    unfolding zero_le_double_add_iff_zero_le_single_add [symmetric, of a]
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   233
    unfolding assm
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   234
    unfolding le_less
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   235
    apply simp_all
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   236
    done
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   237
next
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   238
  assume "a = 0"
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   239
  then show "a + 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
   240
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   241
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   242
lemma zero_less_double_add_iff_zero_less_single_add [simp]: "0 < a + a \<longleftrightarrow> 0 < a"
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   243
proof (cases "a = 0")
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   244
  case True
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   245
  then show ?thesis by auto
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   246
next
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   247
  case False
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   248
  then show ?thesis
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   249
    unfolding less_le
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   250
    apply simp
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   251
    apply rule
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   252
    apply clarify
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   253
    apply rule
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   254
    apply assumption
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   255
    apply (rule notI)
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   256
    unfolding double_zero [symmetric, of a]
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53240
diff changeset
   257
    apply blast
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   258
    done
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   259
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   260
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   261
lemma double_add_le_zero_iff_single_add_le_zero [simp]:
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   262
  "a + a \<le> 0 \<longleftrightarrow> a \<le> 0"
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   263
proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   264
  have "a + a \<le> 0 \<longleftrightarrow> 0 \<le> - (a + a)" by (subst le_minus_iff, simp)
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53240
diff changeset
   265
  moreover have "\<dots> \<longleftrightarrow> a \<le> 0"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53240
diff changeset
   266
    by (simp only: minus_add_distrib zero_le_double_add_iff_zero_le_single_add) simp
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   267
  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
   268
qed
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 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
   271
  "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
   272
proof -
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 < 0 \<longleftrightarrow> 0 < - (a + a)" by (subst less_minus_iff, simp)
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53240
diff changeset
   274
  moreover have "\<dots> \<longleftrightarrow> a < 0"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53240
diff changeset
   275
    by (simp only: minus_add_distrib zero_less_double_add_iff_zero_less_single_add) simp
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   276
  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
   277
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   278
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53240
diff changeset
   279
declare neg_inf_eq_sup [simp] neg_sup_eq_inf [simp] diff_inf_eq_sup [simp] diff_sup_eq_inf [simp]
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   280
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   281
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
   282
proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   283
  from add_le_cancel_left [of "uminus a" "plus a a" zero]
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   284
  have "(a <= -a) = (a+a <= 0)"
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   285
    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
   286
  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
   287
qed
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 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
   290
proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   291
  from add_le_cancel_left [of "uminus a" zero "plus a a"]
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   292
  have "(-a <= a) = (0 <= a+a)"
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   293
    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
   294
  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
   295
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   296
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   297
lemma zero_le_iff_zero_nprt: "0 \<le> a \<longleftrightarrow> nprt a = 0"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   298
  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
   299
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   300
lemma le_zero_iff_zero_pprt: "a \<le> 0 \<longleftrightarrow> pprt a = 0"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   301
  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
   302
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   303
lemma le_zero_iff_pprt_id: "0 \<le> a \<longleftrightarrow> pprt a = a"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   304
  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
   305
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   306
lemma zero_le_iff_nprt_id: "a \<le> 0 \<longleftrightarrow> nprt a = a"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   307
  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
   308
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35040
diff changeset
   309
lemma pprt_mono [simp, no_atp]: "a \<le> b \<Longrightarrow> pprt a \<le> pprt b"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   310
  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
   311
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35040
diff changeset
   312
lemma nprt_mono [simp, no_atp]: "a \<le> b \<Longrightarrow> nprt a \<le> nprt b"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   313
  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
   314
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   315
end
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   316
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   317
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
   318
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   319
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   320
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
   321
  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
   322
begin
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   323
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   324
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
   325
proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   326
  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
   327
  proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   328
    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
   329
    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
   330
  qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   331
  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
   332
  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
   333
  then show ?thesis
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53240
diff changeset
   334
    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
   335
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   336
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   337
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
   338
proof
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   339
  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
   340
  proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   341
    fix a b
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   342
    have a: "a \<le> \<bar>a\<bar>" and b: "- a \<le> \<bar>a\<bar>"
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   343
      by (auto simp add: abs_lattice)
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   344
    show "0 \<le> \<bar>a\<bar>"
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   345
      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
   346
  qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   347
  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
   348
    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
   349
  fix a b
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   350
  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
   351
  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
   352
    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
   353
  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
   354
    by (simp add: abs_lattice sup_commute)
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   355
  {
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   356
    assume "a \<le> b"
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   357
    then show "- a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b"
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   358
      by (rule abs_leI)
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   359
  }
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   360
  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
   361
  proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   362
    have g:"abs a + abs b = sup (a+b) (sup (-a-b) (sup (-a+b) (a + (-b))))" (is "_=sup ?m ?n")
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53240
diff changeset
   363
      by (simp add: abs_lattice add_sup_inf_distribs sup_aci ac_simps)
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   364
    have a: "a + b <= sup ?m ?n" by simp
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   365
    have b: "- a - b <= ?n" by simp
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   366
    have c: "?n <= sup ?m ?n" by simp
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   367
    from b c have d: "-a-b <= sup ?m ?n" by (rule order_trans)
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53240
diff changeset
   368
    have e:"-a-b = -(a+b)" by simp
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   369
    from a d e have "abs(a+b) <= sup ?m ?n"
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   370
      apply -
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   371
      apply (drule abs_leI)
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53240
diff changeset
   372
      apply (simp_all only: algebra_simps ac_simps minus_add)
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53240
diff changeset
   373
      apply (metis add_uminus_conv_diff d sup_commute uminus_add_conv_diff)
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   374
      done
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   375
    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
   376
  qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   377
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   378
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   379
end
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   380
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   381
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
   382
  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
   383
  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
   384
proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   385
  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
   386
  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
   387
  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
   388
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   389
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   390
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
   391
  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
   392
  shows "\<bar>a\<bar> = (if a < 0 then - a else a)"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   393
  by auto
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   394
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   395
lemma estimate_by_abs:
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   396
  "a + b <= (c::'a::lattice_ab_group_add_abs) \<Longrightarrow> a <= c + abs b"
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   397
proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   398
  assume "a+b <= c"
37884
314a88278715 discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents: 36976
diff changeset
   399
  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
   400
  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
   401
  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
   402
  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
   403
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   404
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   405
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
   406
begin
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
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
   409
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
   410
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   411
end
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   412
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   413
lemma abs_le_mult: "abs (a * b) \<le> (abs a) * (abs (b::'a::lattice_ring))"
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   414
proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   415
  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
   416
  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
   417
  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
   418
    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
   419
  {
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   420
    fix u v :: 'a
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   421
    have bh: "\<lbrakk>u = a; v = b\<rbrakk> \<Longrightarrow>
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   422
              u * v = pprt a * pprt b + pprt a * nprt b +
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   423
                      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
   424
      apply (subst prts[of u], subst prts[of v])
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   425
      apply (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
   426
      done
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
  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
   429
  have xy: "- ?x <= ?y"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53240
diff changeset
   430
    apply simp
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53240
diff changeset
   431
    apply (metis (full_types) add_increasing add_uminus_conv_diff lattice_ab_group_add_class.minus_le_self_iff minus_add_distrib mult_nonneg_nonneg 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
   432
    done
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   433
  have yx: "?y <= ?x"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53240
diff changeset
   434
    apply simp
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53240
diff changeset
   435
    apply (metis (full_types) add_nonpos_nonpos add_uminus_conv_diff lattice_ab_group_add_class.le_minus_self_iff minus_add_distrib mult_nonneg_nonpos 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
   436
    done
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   437
  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
   438
  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
   439
  show ?thesis
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   440
    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
   441
    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
   442
    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
   443
    done
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   444
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   445
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   446
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
   447
proof
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   448
  fix a b :: "'a\<Colon> lattice_ring"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 37884
diff changeset
   449
  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
   450
  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
   451
  proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   452
    have s: "(0 <= a*b) | (a*b <= 0)"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   453
      apply (auto)
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   454
      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
   455
      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
   456
      apply (simp)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   457
      apply (rule_tac split_mult_neg_le)
41528
276078f01ada eliminated global prems;
wenzelm
parents: 37884
diff changeset
   458
      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
   459
      apply (blast)
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   460
      done
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   461
    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
   462
      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
   463
    show ?thesis
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   464
    proof cases
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   465
      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
   466
      then show ?thesis
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   467
        apply (simp_all add: mulprts abs_prts)
41528
276078f01ada eliminated global prems;
wenzelm
parents: 37884
diff changeset
   468
        apply (insert a)
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   469
        apply (auto simp add:
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   470
          algebra_simps
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   471
          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
   472
          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
   473
          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
   474
          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
   475
        done
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   476
    next
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   477
      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
   478
      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
   479
      then show ?thesis
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   480
        apply (simp_all add: mulprts abs_prts)
41528
276078f01ada eliminated global prems;
wenzelm
parents: 37884
diff changeset
   481
        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
   482
        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
   483
        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
   484
        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
   485
        done
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   486
    qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   487
  qed
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
lemma mult_le_prts:
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   491
  assumes "a1 <= (a::'a::lattice_ring)"
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   492
    and "a <= a2"
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   493
    and "b1 <= b"
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   494
    and "b <= b2"
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   495
  shows "a * b <=
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   496
    pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1"
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   497
proof -
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   498
  have "a * b = (pprt a + nprt a) * (pprt b + nprt b)"
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   499
    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
   500
    apply simp
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   501
    done
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   502
  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
   503
    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
   504
  moreover have "pprt a * pprt b <= pprt a2 * pprt b2"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 37884
diff changeset
   505
    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
   506
  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
   507
  proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   508
    have "pprt a * nprt b <= pprt a * nprt b2"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 37884
diff changeset
   509
      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
   510
    moreover have "pprt a * nprt b2 <= pprt a1 * nprt b2"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 37884
diff changeset
   511
      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
   512
    ultimately show ?thesis
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   513
      by simp
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   514
  qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   515
  moreover have "nprt a * pprt b <= nprt a2 * pprt b1"
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   516
  proof -
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   517
    have "nprt a * pprt b <= nprt a2 * pprt b"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 37884
diff changeset
   518
      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
   519
    moreover have "nprt a2 * pprt b <= nprt a2 * pprt b1"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 37884
diff changeset
   520
      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
   521
    ultimately show ?thesis
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   522
      by simp
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   523
  qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   524
  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
   525
  proof -
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   526
    have "nprt a * nprt b <= nprt a * nprt b1"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 37884
diff changeset
   527
      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
   528
    moreover have "nprt a * nprt b1 <= nprt a1 * nprt b1"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 37884
diff changeset
   529
      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
   530
    ultimately show ?thesis
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   531
      by simp
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
  ultimately show ?thesis
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   534
    apply -
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   535
    apply (rule add_mono | simp)+
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   536
    done
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   537
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   538
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   539
lemma mult_ge_prts:
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   540
  assumes "a1 <= (a::'a::lattice_ring)"
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   541
    and "a <= a2"
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   542
    and "b1 <= b"
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   543
    and "b <= b2"
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   544
  shows "a * b >=
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   545
    nprt a1 * pprt b2 + nprt a2 * nprt b2 + pprt a1 * pprt b1 + pprt a2 * nprt b1"
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   546
proof -
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   547
  from assms have a1:"- a2 <= -a"
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   548
    by auto
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   549
  from assms have a2: "-a <= -a1"
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   550
    by auto
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   551
  from mult_le_prts[of "-a2" "-a" "-a1" "b1" b "b2", OF a1 a2 assms(3) assms(4), simplified nprt_neg pprt_neg]
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   552
  have le: "- (a * b) <= - nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1"
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   553
    by simp
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   554
  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
   555
    by (simp only: minus_le_iff)
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53240
diff changeset
   556
  then show ?thesis 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
   557
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   558
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   559
instance int :: lattice_ring
53240
07593a0a27f4 tuned proofs;
wenzelm
parents: 46986
diff changeset
   560
proof
35040
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   561
  fix k :: int
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   562
  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
   563
    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
   564
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   565
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   566
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
   567
proof
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   568
  fix a :: real
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   569
  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
   570
    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
   571
qed
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   572
e42e7f133d94 separate library theory for type classes combining lattices with various algebraic structures; c.f. cs. 7efe662e41b4
haftmann
parents:
diff changeset
   573
end
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53240
diff changeset
   574