src/HOL/ex/Set_Comprehension_Pointfree_Examples.thy
 author wenzelm Wed Jun 22 10:09:20 2016 +0200 (2016-06-22) changeset 63343 fb5d8a50c641 parent 61343 5b5656a63bd6 child 67613 ce654b0e6d69 permissions -rw-r--r--
bundle lifting_syntax;
```     1 (*  Title:      HOL/ex/Set_Comprehension_Pointfree_Examples.thy
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```     2     Author:     Lukas Bulwahn, Rafal Kolanski
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```     3     Copyright   2012 TU Muenchen
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```     4 *)
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```     5
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```     6 section \<open>Examples for the set comprehension to pointfree simproc\<close>
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```     7
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```     8 theory Set_Comprehension_Pointfree_Examples
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```     9 imports Main
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```    10 begin
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```    11
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```    12 declare [[simproc add: finite_Collect]]
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```    13
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```    14 lemma
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```    15   "finite (UNIV::'a set) ==> finite {p. EX x::'a. p = (x, x)}"
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```    16   by simp
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```    17
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```    18 lemma
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```    19   "finite A ==> finite B ==> finite {f a b| a b. a : A \<and> b : B}"
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```    20   by simp
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```    21
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```    22 lemma
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```    23   "finite B ==> finite A' ==> finite {f a b| a b. a : A \<and> a : A' \<and> b : B}"
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```    24   by simp
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```    25
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```    26 lemma
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```    27   "finite A ==> finite B ==> finite {f a b| a b. a : A \<and> b : B \<and> b : B'}"
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```    28   by simp
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```    29
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```    30 lemma
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```    31   "finite A ==> finite B ==> finite C ==> finite {f a b c| a b c. a : A \<and> b : B \<and> c : C}"
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```    32   by simp
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```    33
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```    34 lemma
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```    35   "finite A ==> finite B ==> finite C ==> finite D ==>
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```    36      finite {f a b c d| a b c d. a : A \<and> b : B \<and> c : C \<and> d : D}"
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```    37   by simp
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```    38
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```    39 lemma
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```    40   "finite A ==> finite B ==> finite C ==> finite D ==> finite E ==>
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```    41     finite {f a b c d e | a b c d e. a : A \<and> b : B \<and> c : C \<and> d : D \<and> e : E}"
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```    42   by simp
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```    43
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```    44 lemma
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```    45   "finite A ==> finite B ==> finite C ==> finite D ==> finite E \<Longrightarrow>
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```    46     finite {f a d c b e | e b c d a. b : B \<and> a : A \<and> e : E' \<and> c : C \<and> d : D \<and> e : E \<and> b : B'}"
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```    47   by simp
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```    48
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```    49 lemma
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```    50   "\<lbrakk> finite A ; finite B ; finite C ; finite D \<rbrakk>
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```    51   \<Longrightarrow> finite ({f a b c d| a b c d. a : A \<and> b : B \<and> c : C \<and> d : D})"
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```    52   by simp
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```    53
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```    54 lemma
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```    55   "finite ((\<lambda>(a,b,c,d). f a b c d) ` (A \<times> B \<times> C \<times> D))
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```    56   \<Longrightarrow> finite ({f a b c d| a b c d. a : A \<and> b : B \<and> c : C \<and> d : D})"
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```    57   by simp
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```    58
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```    59 lemma
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```    60   "finite S ==> finite {s'. EX s:S. s' = f a e s}"
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```    61   by simp
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```    62
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```    63 lemma
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```    64   "finite A ==> finite B ==> finite {f a b| a b. a : A \<and> b : B \<and> a \<notin> Z}"
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```    65   by simp
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```    66
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```    67 lemma
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```    68   "finite A ==> finite B ==> finite R ==> finite {f a b x y| a b x y. a : A \<and> b : B \<and> (x,y) \<in> R}"
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```    69 by simp
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```    70
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```    71 lemma
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```    72   "finite A ==> finite B ==> finite R ==> finite {f a b x y| a b x y. a : A \<and> (x,y) \<in> R \<and> b : B}"
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```    73 by simp
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```    74
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```    75 lemma
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```    76   "finite A ==> finite B ==> finite R ==> finite {f a (x, b) y| y b x a. a : A \<and> (x,y) \<in> R \<and> b : B}"
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```    77 by simp
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```    78
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```    79 lemma
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```    80   "finite A ==> finite AA ==> finite B ==> finite {f a b| a b. (a : A \<or> a : AA) \<and> b : B \<and> a \<notin> Z}"
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```    81 by simp
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```    82
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```    83 lemma
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```    84   "finite A1 ==> finite A2 ==> finite A3 ==> finite A4 ==> finite A5 ==> finite B ==>
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```    85      finite {f a b c | a b c. ((a : A1 \<and> a : A2) \<or> (a : A3 \<and> (a : A4 \<or> a : A5))) \<and> b : B \<and> a \<notin> Z}"
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```    86 apply simp
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```    87 oops
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```    88
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```    89 lemma "finite B ==> finite {c. EX x. x : B & c = a * x}"
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```    90 by simp
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```    91
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```    92 lemma
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```    93   "finite A ==> finite B ==> finite {f a * g b |a b. a : A & b : B}"
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```    94 by simp
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```    95
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```    96 lemma
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```    97   "finite S ==> inj (%(x, y). g x y) ==> finite {f x y| x y. g x y : S}"
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```    98   by (auto intro: finite_vimageI)
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```    99
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```   100 lemma
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```   101   "finite A ==> finite S ==> inj (%(x, y). g x y) ==> finite {f x y z | x y z. g x y : S & z : A}"
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```   102   by (auto intro: finite_vimageI)
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```   103
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```   104 lemma
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```   105   "finite S ==> finite A ==> inj (%(x, y). g x y) ==> inj (%(x, y). h x y)
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```   106     ==> finite {f a b c d | a b c d. g a c : S & h b d : A}"
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```   107   by (auto intro: finite_vimageI)
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```   108
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```   109 lemma
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```   110   assumes "finite S" shows "finite {(a,b,c,d). ([a, b], [c, d]) : S}"
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```   111 using assms by (auto intro!: finite_vimageI simp add: inj_on_def)
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```   112   (* injectivity to be automated with further rules and automation *)
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```   113
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```   114 schematic_goal (* check interaction with schematics *)
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```   115   "finite {x :: ?'A \<Rightarrow> ?'B \<Rightarrow> bool. \<exists>a b. x = Pair_Rep a b}
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```   116    = finite ((\<lambda>(b :: ?'B, a:: ?'A). Pair_Rep a b) ` (UNIV \<times> UNIV))"
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```   117   by simp
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```   118
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```   119 declare [[simproc del: finite_Collect]]
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```   120
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```   121
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```   122 section \<open>Testing simproc in code generation\<close>
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```   123
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```   124 definition union :: "nat set => nat set => nat set"
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```   125 where
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```   126   "union A B = {x. x : A \<or> x : B}"
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```   127
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```   128 definition common_subsets :: "nat set => nat set => nat set set"
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```   129 where
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```   130   "common_subsets S1 S2 = {S. S \<subseteq> S1 \<and> S \<subseteq> S2}"
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```   131
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```   132 definition products :: "nat set => nat set => nat set"
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```   133 where
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```   134   "products A B = {c. EX a b. a : A & b : B & c = a * b}"
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```   135
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```   136 export_code products in Haskell
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```   137
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```   138 export_code union common_subsets products in Haskell
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```   139
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```   140 end
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