src/HOL/Isar_Examples/Knaster_Tarski.thy
 author wenzelm Wed Dec 29 17:34:41 2010 +0100 (2010-12-29) changeset 41413 64cd30d6b0b8 parent 37671 fa53d267dab3 child 58614 7338eb25226c permissions -rw-r--r--
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```     1 (*  Title:      HOL/Isar_Examples/Knaster_Tarski.thy
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```     2     Author:     Markus Wenzel, TU Muenchen
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```     3
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```     4 Typical textbook proof example.
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```     5 *)
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```     6
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```     7 header {* Textbook-style reasoning: the Knaster-Tarski Theorem *}
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```     8
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```     9 theory Knaster_Tarski
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```    10 imports Main "~~/src/HOL/Library/Lattice_Syntax"
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```    11 begin
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```    12
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```    13
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```    14 subsection {* Prose version *}
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```    15
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```    16 text {* According to the textbook \cite[pages
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```    17   93--94]{davey-priestley}, the Knaster-Tarski fixpoint theorem is as
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```    18   follows.\footnote{We have dualized the argument, and tuned the
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```    19   notation a little bit.}
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```    20
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```    21   \textbf{The Knaster-Tarski Fixpoint Theorem.}  Let @{text L} be a
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```    22   complete lattice and @{text "f: L \<rightarrow> L"} an order-preserving map.
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```    23   Then @{text "\<Sqinter>{x \<in> L | f(x) \<le> x}"} is a fixpoint of @{text f}.
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```    24
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```    25   \textbf{Proof.} Let @{text "H = {x \<in> L | f(x) \<le> x}"} and @{text "a =
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```    26   \<Sqinter>H"}.  For all @{text "x \<in> H"} we have @{text "a \<le> x"}, so @{text
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```    27   "f(a) \<le> f(x) \<le> x"}.  Thus @{text "f(a)"} is a lower bound of @{text
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```    28   H}, whence @{text "f(a) \<le> a"}.  We now use this inequality to prove
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```    29   the reverse one (!) and thereby complete the proof that @{text a} is
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```    30   a fixpoint.  Since @{text f} is order-preserving, @{text "f(f(a)) \<le>
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```    31   f(a)"}.  This says @{text "f(a) \<in> H"}, so @{text "a \<le> f(a)"}. *}
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```    32
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```    33
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```    34 subsection {* Formal versions *}
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```    35
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```    36 text {* The Isar proof below closely follows the original
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```    37   presentation.  Virtually all of the prose narration has been
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```    38   rephrased in terms of formal Isar language elements.  Just as many
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```    39   textbook-style proofs, there is a strong bias towards forward proof,
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```    40   and several bends in the course of reasoning. *}
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```    41
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```    42 theorem Knaster_Tarski:
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```    43   fixes f :: "'a::complete_lattice \<Rightarrow> 'a"
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```    44   assumes "mono f"
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```    45   shows "\<exists>a. f a = a"
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```    46 proof
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```    47   let ?H = "{u. f u \<le> u}"
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```    48   let ?a = "\<Sqinter>?H"
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```    49   show "f ?a = ?a"
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```    50   proof -
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```    51     {
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```    52       fix x
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```    53       assume "x \<in> ?H"
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```    54       then have "?a \<le> x" by (rule Inf_lower)
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```    55       with `mono f` have "f ?a \<le> f x" ..
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```    56       also from `x \<in> ?H` have "\<dots> \<le> x" ..
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```    57       finally have "f ?a \<le> x" .
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```    58     }
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```    59     then have "f ?a \<le> ?a" by (rule Inf_greatest)
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```    60     {
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```    61       also presume "\<dots> \<le> f ?a"
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```    62       finally (order_antisym) show ?thesis .
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```    63     }
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```    64     from `mono f` and `f ?a \<le> ?a` have "f (f ?a) \<le> f ?a" ..
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```    65     then have "f ?a \<in> ?H" ..
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```    66     then show "?a \<le> f ?a" by (rule Inf_lower)
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```    67   qed
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```    68 qed
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```    69
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```    70 text {* Above we have used several advanced Isar language elements,
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```    71   such as explicit block structure and weak assumptions.  Thus we have
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```    72   mimicked the particular way of reasoning of the original text.
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```    73
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```    74   In the subsequent version the order of reasoning is changed to
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```    75   achieve structured top-down decomposition of the problem at the
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```    76   outer level, while only the inner steps of reasoning are done in a
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```    77   forward manner.  We are certainly more at ease here, requiring only
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```    78   the most basic features of the Isar language. *}
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```    79
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```    80 theorem Knaster_Tarski':
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```    81   fixes f :: "'a::complete_lattice \<Rightarrow> 'a"
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```    82   assumes "mono f"
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```    83   shows "\<exists>a. f a = a"
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```    84 proof
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```    85   let ?H = "{u. f u \<le> u}"
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```    86   let ?a = "\<Sqinter>?H"
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```    87   show "f ?a = ?a"
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```    88   proof (rule order_antisym)
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```    89     show "f ?a \<le> ?a"
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```    90     proof (rule Inf_greatest)
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```    91       fix x
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```    92       assume "x \<in> ?H"
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```    93       then have "?a \<le> x" by (rule Inf_lower)
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```    94       with `mono f` have "f ?a \<le> f x" ..
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```    95       also from `x \<in> ?H` have "\<dots> \<le> x" ..
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```    96       finally show "f ?a \<le> x" .
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```    97     qed
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```    98     show "?a \<le> f ?a"
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```    99     proof (rule Inf_lower)
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```   100       from `mono f` and `f ?a \<le> ?a` have "f (f ?a) \<le> f ?a" ..
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```   101       then show "f ?a \<in> ?H" ..
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```   102     qed
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```   103   qed
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```   104 qed
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```   105
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```   106 end
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