src/HOL/Induct/Tree.thy
 author krauss Mon Mar 01 18:49:55 2010 +0100 (2010-03-01) changeset 35439 888993948a1d parent 35419 d78659d1723e child 39246 9e58f0499f57 permissions -rw-r--r--
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```     1 (*  Title:      HOL/Induct/Tree.thy
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```     2     Author:     Stefan Berghofer,  TU Muenchen
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```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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```     4 *)
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```     5
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```     6 header {* Infinitely branching trees *}
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```     7
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```     8 theory Tree
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```     9 imports Main
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```    10 begin
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```    11
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```    12 datatype 'a tree =
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```    13     Atom 'a
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```    14   | Branch "nat => 'a tree"
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```    15
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```    16 primrec
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```    17   map_tree :: "('a => 'b) => 'a tree => 'b tree"
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```    18 where
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```    19   "map_tree f (Atom a) = Atom (f a)"
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```    20 | "map_tree f (Branch ts) = Branch (\<lambda>x. map_tree f (ts x))"
```
```    21
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```    22 lemma tree_map_compose: "map_tree g (map_tree f t) = map_tree (g \<circ> f) t"
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```    23   by (induct t) simp_all
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```    24
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```    25 primrec
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```    26   exists_tree :: "('a => bool) => 'a tree => bool"
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```    27 where
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```    28   "exists_tree P (Atom a) = P a"
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```    29 | "exists_tree P (Branch ts) = (\<exists>x. exists_tree P (ts x))"
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```    30
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```    31 lemma exists_map:
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```    32   "(!!x. P x ==> Q (f x)) ==>
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```    33     exists_tree P ts ==> exists_tree Q (map_tree f ts)"
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```    34   by (induct ts) auto
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```    35
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```    36
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```    37 subsection{*The Brouwer ordinals, as in ZF/Induct/Brouwer.thy.*}
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```    38
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```    39 datatype brouwer = Zero | Succ "brouwer" | Lim "nat => brouwer"
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```    40
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```    41 text{*Addition of ordinals*}
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```    42 primrec
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```    43   add :: "[brouwer,brouwer] => brouwer"
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```    44 where
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```    45   "add i Zero = i"
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```    46 | "add i (Succ j) = Succ (add i j)"
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```    47 | "add i (Lim f) = Lim (%n. add i (f n))"
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```    48
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```    49 lemma add_assoc: "add (add i j) k = add i (add j k)"
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```    50   by (induct k) auto
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```    51
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```    52 text{*Multiplication of ordinals*}
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```    53 primrec
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```    54   mult :: "[brouwer,brouwer] => brouwer"
```
```    55 where
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```    56   "mult i Zero = Zero"
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```    57 | "mult i (Succ j) = add (mult i j) i"
```
```    58 | "mult i (Lim f) = Lim (%n. mult i (f n))"
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```    59
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```    60 lemma add_mult_distrib: "mult i (add j k) = add (mult i j) (mult i k)"
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```    61   by (induct k) (auto simp add: add_assoc)
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```    62
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```    63 lemma mult_assoc: "mult (mult i j) k = mult i (mult j k)"
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```    64   by (induct k) (auto simp add: add_mult_distrib)
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```    65
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```    66 text{*We could probably instantiate some axiomatic type classes and use
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```    67 the standard infix operators.*}
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```    68
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```    69 subsection{*A WF Ordering for The Brouwer ordinals (Michael Compton)*}
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```    70
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```    71 text{*To use the function package we need an ordering on the Brouwer
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```    72   ordinals.  Start with a predecessor relation and form its transitive
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```    73   closure. *}
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```    74
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```    75 definition
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```    76   brouwer_pred :: "(brouwer * brouwer) set" where
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```    77   "brouwer_pred = (\<Union>i. {(m,n). n = Succ m \<or> (EX f. n = Lim f & m = f i)})"
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```    78
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```    79 definition
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```    80   brouwer_order :: "(brouwer * brouwer) set" where
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```    81   "brouwer_order = brouwer_pred^+"
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```    82
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```    83 lemma wf_brouwer_pred: "wf brouwer_pred"
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```    84   by(unfold wf_def brouwer_pred_def, clarify, induct_tac x, blast+)
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```    85
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```    86 lemma wf_brouwer_order[simp]: "wf brouwer_order"
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```    87   by(unfold brouwer_order_def, rule wf_trancl[OF wf_brouwer_pred])
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```    88
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```    89 lemma [simp]: "(j, Succ j) : brouwer_order"
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```    90   by(auto simp add: brouwer_order_def brouwer_pred_def)
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```    91
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```    92 lemma [simp]: "(f n, Lim f) : brouwer_order"
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```    93   by(auto simp add: brouwer_order_def brouwer_pred_def)
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```    94
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```    95 text{*Example of a general function*}
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```    96
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```    97 function
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```    98   add2 :: "(brouwer*brouwer) => brouwer"
```
```    99 where
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```   100   "add2 (i, Zero) = i"
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```   101 | "add2 (i, (Succ j)) = Succ (add2 (i, j))"
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```   102 | "add2 (i, (Lim f)) = Lim (\<lambda> n. add2 (i, (f n)))"
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```   103 by pat_completeness auto
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```   104 termination by (relation "inv_image brouwer_order snd") auto
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```   105
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```   106 lemma add2_assoc: "add2 (add2 (i, j), k) = add2 (i, add2 (j, k))"
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```   107   by (induct k) auto
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```   108
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```   109 end
```