src/HOL/Lattice_Locales.thy
 author haftmann Mon Jan 30 08:20:56 2006 +0100 (2006-01-30) changeset 18851 9502ce541f01 parent 15791 446ec11266be child 21216 1c8580913738 permissions -rw-r--r--
```     1 (*  Title:      HOL/Lattices.thy
```
```     2     ID:         \$Id\$
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```     3     Author:     Tobias Nipkow
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```     4 *)
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```     5
```
```     6 header {* Lattices via Locales *}
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```     7
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```     8 theory Lattice_Locales
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```     9 imports HOL
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```    10 begin
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```    11
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```    12 subsection{* Lattices *}
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```    13
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```    14 text{* This theory of lattice locales only defines binary sup and inf
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```    15 operations. The extension to finite sets is done in theory @{text
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```    16 Finite_Set}. In the longer term it may be better to define arbitrary
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```    17 sups and infs via @{text THE}. *}
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```    18
```
```    19 locale partial_order =
```
```    20   fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "\<sqsubseteq>" 50)
```
```    21   assumes refl[iff]: "x \<sqsubseteq> x"
```
```    22   and trans: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> z"
```
```    23   and antisym: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> x \<Longrightarrow> x = y"
```
```    24
```
```    25 locale lower_semilattice = partial_order +
```
```    26   fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<sqinter>" 70)
```
```    27   assumes inf_le1: "x \<sqinter> y \<sqsubseteq> x" and inf_le2: "x \<sqinter> y \<sqsubseteq> y"
```
```    28   and inf_least: "x \<sqsubseteq> y \<Longrightarrow> x \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> y \<sqinter> z"
```
```    29
```
```    30 locale upper_semilattice = partial_order +
```
```    31   fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<squnion>" 65)
```
```    32   assumes sup_ge1: "x \<sqsubseteq> x \<squnion> y" and sup_ge2: "y \<sqsubseteq> x \<squnion> y"
```
```    33   and sup_greatest: "y \<sqsubseteq> x \<Longrightarrow> z \<sqsubseteq> x \<Longrightarrow> y \<squnion> z \<sqsubseteq> x"
```
```    34
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```    35 locale lattice = lower_semilattice + upper_semilattice
```
```    36
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```    37 lemma (in lower_semilattice) inf_commute: "(x \<sqinter> y) = (y \<sqinter> x)"
```
```    38 by(blast intro: antisym inf_le1 inf_le2 inf_least)
```
```    39
```
```    40 lemma (in upper_semilattice) sup_commute: "(x \<squnion> y) = (y \<squnion> x)"
```
```    41 by(blast intro: antisym sup_ge1 sup_ge2 sup_greatest)
```
```    42
```
```    43 lemma (in lower_semilattice) inf_assoc: "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
```
```    44 by(blast intro: antisym inf_le1 inf_le2 inf_least trans del:refl)
```
```    45
```
```    46 lemma (in upper_semilattice) sup_assoc: "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
```
```    47 by(blast intro!: antisym sup_ge1 sup_ge2 intro: sup_greatest trans del:refl)
```
```    48
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```    49 lemma (in lower_semilattice) inf_idem[simp]: "x \<sqinter> x = x"
```
```    50 by(blast intro: antisym inf_le1 inf_le2 inf_least refl)
```
```    51
```
```    52 lemma (in upper_semilattice) sup_idem[simp]: "x \<squnion> x = x"
```
```    53 by(blast intro: antisym sup_ge1 sup_ge2 sup_greatest refl)
```
```    54
```
```    55 lemma (in lower_semilattice) inf_left_idem[simp]: "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y"
```
```    56 by (simp add: inf_assoc[symmetric])
```
```    57
```
```    58 lemma (in upper_semilattice) sup_left_idem[simp]: "x \<squnion> (x \<squnion> y) = x \<squnion> y"
```
```    59 by (simp add: sup_assoc[symmetric])
```
```    60
```
```    61 lemma (in lattice) inf_sup_absorb: "x \<sqinter> (x \<squnion> y) = x"
```
```    62 by(blast intro: antisym inf_le1 inf_least sup_ge1)
```
```    63
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```    64 lemma (in lattice) sup_inf_absorb: "x \<squnion> (x \<sqinter> y) = x"
```
```    65 by(blast intro: antisym sup_ge1 sup_greatest inf_le1)
```
```    66
```
```    67 lemma (in lower_semilattice) inf_absorb: "x \<sqsubseteq> y \<Longrightarrow> x \<sqinter> y = x"
```
```    68 by(blast intro: antisym inf_le1 inf_least refl)
```
```    69
```
```    70 lemma (in upper_semilattice) sup_absorb: "x \<sqsubseteq> y \<Longrightarrow> x \<squnion> y = y"
```
```    71 by(blast intro: antisym sup_ge2 sup_greatest refl)
```
```    72
```
```    73
```
```    74 lemma (in lower_semilattice) below_inf_conv[simp]:
```
```    75  "x \<sqsubseteq> y \<sqinter> z = (x \<sqsubseteq> y \<and> x \<sqsubseteq> z)"
```
```    76 by(blast intro: antisym inf_le1 inf_le2 inf_least refl trans)
```
```    77
```
```    78 lemma (in upper_semilattice) above_sup_conv[simp]:
```
```    79  "x \<squnion> y \<sqsubseteq> z = (x \<sqsubseteq> z \<and> y \<sqsubseteq> z)"
```
```    80 by(blast intro: antisym sup_ge1 sup_ge2 sup_greatest refl trans)
```
```    81
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```    82
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```    83 text{* Towards distributivity: if you have one of them, you have them all. *}
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```    84
```
```    85 lemma (in lattice) distrib_imp1:
```
```    86 assumes D: "!!x y z. x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
```
```    87 shows "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
```
```    88 proof-
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```    89   have "x \<squnion> (y \<sqinter> z) = (x \<squnion> (x \<sqinter> z)) \<squnion> (y \<sqinter> z)" by(simp add:sup_inf_absorb)
```
```    90   also have "\<dots> = x \<squnion> (z \<sqinter> (x \<squnion> y))" by(simp add:D inf_commute sup_assoc)
```
```    91   also have "\<dots> = ((x \<squnion> y) \<sqinter> x) \<squnion> ((x \<squnion> y) \<sqinter> z)"
```
```    92     by(simp add:inf_sup_absorb inf_commute)
```
```    93   also have "\<dots> = (x \<squnion> y) \<sqinter> (x \<squnion> z)" by(simp add:D)
```
```    94   finally show ?thesis .
```
```    95 qed
```
```    96
```
```    97 lemma (in lattice) distrib_imp2:
```
```    98 assumes D: "!!x y z. x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
```
```    99 shows "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
```
```   100 proof-
```
```   101   have "x \<sqinter> (y \<squnion> z) = (x \<sqinter> (x \<squnion> z)) \<sqinter> (y \<squnion> z)" by(simp add:inf_sup_absorb)
```
```   102   also have "\<dots> = x \<sqinter> (z \<squnion> (x \<sqinter> y))" by(simp add:D sup_commute inf_assoc)
```
```   103   also have "\<dots> = ((x \<sqinter> y) \<squnion> x) \<sqinter> ((x \<sqinter> y) \<squnion> z)"
```
```   104     by(simp add:sup_inf_absorb sup_commute)
```
```   105   also have "\<dots> = (x \<sqinter> y) \<squnion> (x \<sqinter> z)" by(simp add:D)
```
```   106   finally show ?thesis .
```
```   107 qed
```
```   108
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```   109 text{* A package of rewrite rules for deciding equivalence wrt ACI: *}
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```   110
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```   111 lemma (in lower_semilattice) inf_left_commute: "x \<sqinter> (y \<sqinter> z) = y \<sqinter> (x \<sqinter> z)"
```
```   112 proof -
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```   113   have "x \<sqinter> (y \<sqinter> z) = (y \<sqinter> z) \<sqinter> x" by (simp only: inf_commute)
```
```   114   also have "... = y \<sqinter> (z \<sqinter> x)" by (simp only: inf_assoc)
```
```   115   also have "z \<sqinter> x = x \<sqinter> z" by (simp only: inf_commute)
```
```   116   finally(back_subst) show ?thesis .
```
```   117 qed
```
```   118
```
```   119 lemma (in upper_semilattice) sup_left_commute: "x \<squnion> (y \<squnion> z) = y \<squnion> (x \<squnion> z)"
```
```   120 proof -
```
```   121   have "x \<squnion> (y \<squnion> z) = (y \<squnion> z) \<squnion> x" by (simp only: sup_commute)
```
```   122   also have "... = y \<squnion> (z \<squnion> x)" by (simp only: sup_assoc)
```
```   123   also have "z \<squnion> x = x \<squnion> z" by (simp only: sup_commute)
```
```   124   finally(back_subst) show ?thesis .
```
```   125 qed
```
```   126
```
```   127 lemma (in lower_semilattice) inf_left_idem: "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y"
```
```   128 proof -
```
```   129   have "x \<sqinter> (x \<sqinter> y) = (x \<sqinter> x) \<sqinter> y" by(simp only:inf_assoc)
```
```   130   also have "\<dots> = x \<sqinter> y" by(simp)
```
```   131   finally show ?thesis .
```
```   132 qed
```
```   133
```
```   134 lemma (in upper_semilattice) sup_left_idem: "x \<squnion> (x \<squnion> y) = x \<squnion> y"
```
```   135 proof -
```
```   136   have "x \<squnion> (x \<squnion> y) = (x \<squnion> x) \<squnion> y" by(simp only:sup_assoc)
```
```   137   also have "\<dots> = x \<squnion> y" by(simp)
```
```   138   finally show ?thesis .
```
```   139 qed
```
```   140
```
```   141
```
```   142 lemmas (in lower_semilattice) inf_ACI =
```
```   143  inf_commute inf_assoc inf_left_commute inf_left_idem
```
```   144
```
```   145 lemmas (in upper_semilattice) sup_ACI =
```
```   146  sup_commute sup_assoc sup_left_commute sup_left_idem
```
```   147
```
```   148 lemmas (in lattice) ACI = inf_ACI sup_ACI
```
```   149
```
```   150
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```   151 subsection{* Distributive lattices *}
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```   152
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```   153 locale distrib_lattice = lattice +
```
```   154   assumes sup_inf_distrib1: "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
```
```   155
```
```   156 lemma (in distrib_lattice) sup_inf_distrib2:
```
```   157  "(y \<sqinter> z) \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)"
```
```   158 by(simp add:ACI sup_inf_distrib1)
```
```   159
```
```   160 lemma (in distrib_lattice) inf_sup_distrib1:
```
```   161  "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
```
```   162 by(rule distrib_imp2[OF sup_inf_distrib1])
```
```   163
```
```   164 lemma (in distrib_lattice) inf_sup_distrib2:
```
```   165  "(y \<squnion> z) \<sqinter> x = (y \<sqinter> x) \<squnion> (z \<sqinter> x)"
```
```   166 by(simp add:ACI inf_sup_distrib1)
```
```   167
```
```   168 lemmas (in distrib_lattice) distrib =
```
```   169   sup_inf_distrib1 sup_inf_distrib2 inf_sup_distrib1 inf_sup_distrib2
```
```   170
```
```   171
```
```   172 end
```