src/HOL/Metis_Examples/Clausify.thy
 author blanchet Thu Apr 14 11:24:05 2011 +0200 (2011-04-14) changeset 42350 128dc0fa87fc parent 42345 5ecd55993606 child 42747 f132d13fcf75 permissions -rw-r--r--
more clausification tests
```     1 (*  Title:      HOL/Metis_Examples/Clausify.thy
```
```     2     Author:     Jasmin Blanchette, TU Muenchen
```
```     3
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```     4 Testing Metis's clausifier.
```
```     5 *)
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```     6
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```     7 theory Clausify
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```     8 imports Complex_Main
```
```     9 begin
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```    10
```
```    11 (* FIXME: shouldn't need this *)
```
```    12 declare [[unify_search_bound = 100]]
```
```    13 declare [[unify_trace_bound = 100]]
```
```    14
```
```    15 text {* Definitional CNF for facts *}
```
```    16
```
```    17 declare [[meson_max_clauses = 10]]
```
```    18
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```    19 axiomatization q :: "nat \<Rightarrow> nat \<Rightarrow> bool" where
```
```    20 qax: "\<exists>b. \<forall>a. (q b a \<and> q 0 0 \<and> q 1 a \<and> q a 1) \<or> (q 0 1 \<and> q 1 0 \<and> q a b \<and> q 1 1)"
```
```    21
```
```    22 declare [[metis_new_skolemizer = false]]
```
```    23
```
```    24 lemma "\<exists>b. \<forall>a. (q b a \<or> q a b)"
```
```    25 by (metis qax)
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```    26
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```    27 lemma "\<exists>b. \<forall>a. (q b a \<or> q a b)"
```
```    28 by (metisFT qax)
```
```    29
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```    30 lemma "\<exists>b. \<forall>a. (q b a \<and> q 0 0 \<and> q 1 a \<and> q a 1) \<or> (q 0 1 \<and> q 1 0 \<and> q a b \<and> q 1 1)"
```
```    31 by (metis qax)
```
```    32
```
```    33 lemma "\<exists>b. \<forall>a. (q b a \<and> q 0 0 \<and> q 1 a \<and> q a 1) \<or> (q 0 1 \<and> q 1 0 \<and> q a b \<and> q 1 1)"
```
```    34 by (metisFT qax)
```
```    35
```
```    36 declare [[metis_new_skolemizer]]
```
```    37
```
```    38 lemma "\<exists>b. \<forall>a. (q b a \<or> q a b)"
```
```    39 by (metis qax)
```
```    40
```
```    41 lemma "\<exists>b. \<forall>a. (q b a \<or> q a b)"
```
```    42 by (metisFT qax)
```
```    43
```
```    44 lemma "\<exists>b. \<forall>a. (q b a \<and> q 0 0 \<and> q 1 a \<and> q a 1) \<or> (q 0 1 \<and> q 1 0 \<and> q a b \<and> q 1 1)"
```
```    45 by (metis qax)
```
```    46
```
```    47 lemma "\<exists>b. \<forall>a. (q b a \<and> q 0 0 \<and> q 1 a \<and> q a 1) \<or> (q 0 1 \<and> q 1 0 \<and> q a b \<and> q 1 1)"
```
```    48 by (metisFT qax)
```
```    49
```
```    50 declare [[meson_max_clauses = 60]]
```
```    51
```
```    52 axiomatization r :: "nat \<Rightarrow> nat \<Rightarrow> bool" where
```
```    53 rax: "(r 0 0 \<and> r 0 1 \<and> r 0 2 \<and> r 0 3) \<or>
```
```    54       (r 1 0 \<and> r 1 1 \<and> r 1 2 \<and> r 1 3) \<or>
```
```    55       (r 2 0 \<and> r 2 1 \<and> r 2 2 \<and> r 2 3) \<or>
```
```    56       (r 3 0 \<and> r 3 1 \<and> r 3 2 \<and> r 3 3)"
```
```    57
```
```    58 declare [[metis_new_skolemizer = false]]
```
```    59
```
```    60 lemma "r 0 0 \<or> r 1 1 \<or> r 2 2 \<or> r 3 3"
```
```    61 by (metis rax)
```
```    62
```
```    63 lemma "r 0 0 \<or> r 1 1 \<or> r 2 2 \<or> r 3 3"
```
```    64 by (metisFT rax)
```
```    65
```
```    66 declare [[metis_new_skolemizer]]
```
```    67
```
```    68 lemma "r 0 0 \<or> r 1 1 \<or> r 2 2 \<or> r 3 3"
```
```    69 by (metis rax)
```
```    70
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```    71 lemma "r 0 0 \<or> r 1 1 \<or> r 2 2 \<or> r 3 3"
```
```    72 by (metisFT rax)
```
```    73
```
```    74 lemma "(r 0 0 \<and> r 0 1 \<and> r 0 2 \<and> r 0 3) \<or>
```
```    75        (r 1 0 \<and> r 1 1 \<and> r 1 2 \<and> r 1 3) \<or>
```
```    76        (r 2 0 \<and> r 2 1 \<and> r 2 2 \<and> r 2 3) \<or>
```
```    77        (r 3 0 \<and> r 3 1 \<and> r 3 2 \<and> r 3 3)"
```
```    78 by (metis rax)
```
```    79
```
```    80 lemma "(r 0 0 \<and> r 0 1 \<and> r 0 2 \<and> r 0 3) \<or>
```
```    81        (r 1 0 \<and> r 1 1 \<and> r 1 2 \<and> r 1 3) \<or>
```
```    82        (r 2 0 \<and> r 2 1 \<and> r 2 2 \<and> r 2 3) \<or>
```
```    83        (r 3 0 \<and> r 3 1 \<and> r 3 2 \<and> r 3 3)"
```
```    84 by (metisFT rax)
```
```    85
```
```    86 text {* Definitional CNF for goal *}
```
```    87
```
```    88 axiomatization p :: "nat \<Rightarrow> nat \<Rightarrow> bool" where
```
```    89 pax: "\<exists>b. \<forall>a. (p b a \<and> p 0 0 \<and> p 1 a) \<or> (p 0 1 \<and> p 1 0 \<and> p a b)"
```
```    90
```
```    91 declare [[metis_new_skolemizer = false]]
```
```    92
```
```    93 lemma "\<exists>b. \<forall>a. \<exists>x. (p b a \<or> x) \<and> (p 0 0 \<or> x) \<and> (p 1 a \<or> x) \<and>
```
```    94                    (p 0 1 \<or> \<not> x) \<and> (p 1 0 \<or> \<not> x) \<and> (p a b \<or> \<not> x)"
```
```    95 by (metis pax)
```
```    96
```
```    97 lemma "\<exists>b. \<forall>a. \<exists>x. (p b a \<or> x) \<and> (p 0 0 \<or> x) \<and> (p 1 a \<or> x) \<and>
```
```    98                    (p 0 1 \<or> \<not> x) \<and> (p 1 0 \<or> \<not> x) \<and> (p a b \<or> \<not> x)"
```
```    99 by (metisFT pax)
```
```   100
```
```   101 declare [[metis_new_skolemizer]]
```
```   102
```
```   103 lemma "\<exists>b. \<forall>a. \<exists>x. (p b a \<or> x) \<and> (p 0 0 \<or> x) \<and> (p 1 a \<or> x) \<and>
```
```   104                    (p 0 1 \<or> \<not> x) \<and> (p 1 0 \<or> \<not> x) \<and> (p a b \<or> \<not> x)"
```
```   105 by (metis pax)
```
```   106
```
```   107 lemma "\<exists>b. \<forall>a. \<exists>x. (p b a \<or> x) \<and> (p 0 0 \<or> x) \<and> (p 1 a \<or> x) \<and>
```
```   108                    (p 0 1 \<or> \<not> x) \<and> (p 1 0 \<or> \<not> x) \<and> (p a b \<or> \<not> x)"
```
```   109 by (metisFT pax)
```
```   110
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```   111 text {* New Skolemizer *}
```
```   112
```
```   113 declare [[metis_new_skolemizer]]
```
```   114
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```   115 lemma
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```   116   fixes x :: real
```
```   117   assumes fn_le: "!!n. f n \<le> x" and 1: "f ----> lim f"
```
```   118   shows "lim f \<le> x"
```
```   119 by (metis 1 LIMSEQ_le_const2 fn_le)
```
```   120
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```   121 definition
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```   122   bounded :: "'a::metric_space set \<Rightarrow> bool" where
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```   123   "bounded S \<longleftrightarrow> (\<exists>x eee. \<forall>y\<in>S. dist x y \<le> eee)"
```
```   124
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```   125 lemma "bounded T \<Longrightarrow> S \<subseteq> T ==> bounded S"
```
```   126 by (metis bounded_def subset_eq)
```
```   127
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```   128 lemma
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```   129   assumes a: "Quotient R Abs Rep"
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```   130   shows "symp R"
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```   131 using a unfolding Quotient_def using sympI
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```   132 by metisFT
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```   133
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```   134 lemma
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```   135   "(\<exists>x \<in> set xs. P x) \<longleftrightarrow>
```
```   136    (\<exists>ys x zs. xs = ys @ x # zs \<and> P x \<and> (\<forall>z \<in> set zs. \<not> P z))"
```
```   137 by (metis split_list_last_prop [where P = P] in_set_conv_decomp)
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```   138
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```   139 lemma ex_tl: "EX ys. tl ys = xs"
```
```   140 using tl.simps(2) by fast
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```   141
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```   142 lemma "(\<exists>ys\<Colon>nat list. tl ys = xs) \<and> (\<exists>bs\<Colon>int list. tl bs = as)"
```
```   143 by (metis ex_tl)
```
```   144
```
```   145 end
```