src/HOL/ex/MonoidGroup.thy
 author wenzelm Wed Jun 22 10:09:20 2016 +0200 (2016-06-22) changeset 63343 fb5d8a50c641 parent 61343 5b5656a63bd6 permissions -rw-r--r--
bundle lifting_syntax;
```     1 (*  Title:      HOL/ex/MonoidGroup.thy
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```     2     Author:     Markus Wenzel
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```     3 *)
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```     4
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```     5 section \<open>Monoids and Groups as predicates over record schemes\<close>
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```     6
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```     7 theory MonoidGroup imports Main begin
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```     8
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```     9 record 'a monoid_sig =
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```    10   times :: "'a => 'a => 'a"
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```    11   one :: 'a
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```    12
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```    13 record 'a group_sig = "'a monoid_sig" +
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```    14   inv :: "'a => 'a"
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```    15
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```    16 definition
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```    17   monoid :: "(| times :: 'a => 'a => 'a, one :: 'a, ... :: 'b |) => bool" where
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```    18   "monoid M = (\<forall>x y z.
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```    19     times M (times M x y) z = times M x (times M y z) \<and>
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```    20     times M (one M) x = x \<and> times M x (one M) = x)"
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```    21
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```    22 definition
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```    23   group :: "(| times :: 'a => 'a => 'a, one :: 'a, inv :: 'a => 'a, ... :: 'b |) => bool" where
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```    24   "group G = (monoid G \<and> (\<forall>x. times G (inv G x) x = one G))"
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```    25
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```    26 definition
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```    27   reverse :: "(| times :: 'a => 'a => 'a, one :: 'a, ... :: 'b |) =>
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```    28     (| times :: 'a => 'a => 'a, one :: 'a, ... :: 'b |)" where
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```    29   "reverse M = M (| times := \<lambda>x y. times M y x |)"
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```    30
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```    31 end
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