src/HOL/Metis_Examples/Clausification.thy
 author blanchet Tue Jun 07 08:52:35 2011 +0200 (2011-06-07) changeset 43228 2ed2f092e990 parent 43197 c71657bbdbc0 child 47308 9caab698dbe4 permissions -rw-r--r--
obsoleted "metisFT", and added "no_types" version of Metis as fallback to Sledgehammer after noticing how useful it can be
```     1 (*  Title:      HOL/Metis_Examples/Clausification.thy
```
```     2     Author:     Jasmin Blanchette, TU Muenchen
```
```     3
```
```     4 Example that exercises Metis's Clausifier.
```
```     5 *)
```
```     6
```
```     7 header {* Example that Exercises Metis's Clausifier *}
```
```     8
```
```     9 theory Clausification
```
```    10 imports Complex_Main
```
```    11 begin
```
```    12
```
```    13 (* FIXME: shouldn't need this *)
```
```    14 declare [[unify_search_bound = 100]]
```
```    15 declare [[unify_trace_bound = 100]]
```
```    16
```
```    17
```
```    18 text {* Definitional CNF for facts *}
```
```    19
```
```    20 declare [[meson_max_clauses = 10]]
```
```    21
```
```    22 axiomatization q :: "nat \<Rightarrow> nat \<Rightarrow> bool" where
```
```    23 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)"
```
```    24
```
```    25 declare [[metis_new_skolemizer = false]]
```
```    26
```
```    27 lemma "\<exists>b. \<forall>a. (q b a \<or> q a b)"
```
```    28 by (metis qax)
```
```    29
```
```    30 lemma "\<exists>b. \<forall>a. (q b a \<or> q a b)"
```
```    31 by (metis (full_types) 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 (metis qax)
```
```    35
```
```    36 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)"
```
```    37 by (metis (full_types) qax)
```
```    38
```
```    39 declare [[metis_new_skolemizer]]
```
```    40
```
```    41 lemma "\<exists>b. \<forall>a. (q b a \<or> q a b)"
```
```    42 by (metis qax)
```
```    43
```
```    44 lemma "\<exists>b. \<forall>a. (q b a \<or> q a b)"
```
```    45 by (metis (full_types) 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 (metis qax)
```
```    49
```
```    50 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)"
```
```    51 by (metis (full_types) qax)
```
```    52
```
```    53 declare [[meson_max_clauses = 60]]
```
```    54
```
```    55 axiomatization r :: "nat \<Rightarrow> nat \<Rightarrow> bool" where
```
```    56 rax: "(r 0 0 \<and> r 0 1 \<and> r 0 2 \<and> r 0 3) \<or>
```
```    57       (r 1 0 \<and> r 1 1 \<and> r 1 2 \<and> r 1 3) \<or>
```
```    58       (r 2 0 \<and> r 2 1 \<and> r 2 2 \<and> r 2 3) \<or>
```
```    59       (r 3 0 \<and> r 3 1 \<and> r 3 2 \<and> r 3 3)"
```
```    60
```
```    61 declare [[metis_new_skolemizer = false]]
```
```    62
```
```    63 lemma "r 0 0 \<or> r 1 1 \<or> r 2 2 \<or> r 3 3"
```
```    64 by (metis rax)
```
```    65
```
```    66 lemma "r 0 0 \<or> r 1 1 \<or> r 2 2 \<or> r 3 3"
```
```    67 by (metis (full_types) rax)
```
```    68
```
```    69 declare [[metis_new_skolemizer]]
```
```    70
```
```    71 lemma "r 0 0 \<or> r 1 1 \<or> r 2 2 \<or> r 3 3"
```
```    72 by (metis rax)
```
```    73
```
```    74 lemma "r 0 0 \<or> r 1 1 \<or> r 2 2 \<or> r 3 3"
```
```    75 by (metis (full_types) rax)
```
```    76
```
```    77 lemma "(r 0 0 \<and> r 0 1 \<and> r 0 2 \<and> r 0 3) \<or>
```
```    78        (r 1 0 \<and> r 1 1 \<and> r 1 2 \<and> r 1 3) \<or>
```
```    79        (r 2 0 \<and> r 2 1 \<and> r 2 2 \<and> r 2 3) \<or>
```
```    80        (r 3 0 \<and> r 3 1 \<and> r 3 2 \<and> r 3 3)"
```
```    81 by (metis rax)
```
```    82
```
```    83 lemma "(r 0 0 \<and> r 0 1 \<and> r 0 2 \<and> r 0 3) \<or>
```
```    84        (r 1 0 \<and> r 1 1 \<and> r 1 2 \<and> r 1 3) \<or>
```
```    85        (r 2 0 \<and> r 2 1 \<and> r 2 2 \<and> r 2 3) \<or>
```
```    86        (r 3 0 \<and> r 3 1 \<and> r 3 2 \<and> r 3 3)"
```
```    87 by (metis (full_types) rax)
```
```    88
```
```    89
```
```    90 text {* Definitional CNF for goal *}
```
```    91
```
```    92 axiomatization p :: "nat \<Rightarrow> nat \<Rightarrow> bool" where
```
```    93 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)"
```
```    94
```
```    95 declare [[metis_new_skolemizer = false]]
```
```    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 (metis pax)
```
```   100
```
```   101 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>
```
```   102                    (p 0 1 \<or> \<not> x) \<and> (p 1 0 \<or> \<not> x) \<and> (p a b \<or> \<not> x)"
```
```   103 by (metis (full_types) pax)
```
```   104
```
```   105 declare [[metis_new_skolemizer]]
```
```   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 (metis pax)
```
```   110
```
```   111 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>
```
```   112                    (p 0 1 \<or> \<not> x) \<and> (p 1 0 \<or> \<not> x) \<and> (p a b \<or> \<not> x)"
```
```   113 by (metis (full_types) pax)
```
```   114
```
```   115
```
```   116 text {* New Skolemizer *}
```
```   117
```
```   118 declare [[metis_new_skolemizer]]
```
```   119
```
```   120 lemma
```
```   121   fixes x :: real
```
```   122   assumes fn_le: "!!n. f n \<le> x" and 1: "f ----> lim f"
```
```   123   shows "lim f \<le> x"
```
```   124 by (metis 1 LIMSEQ_le_const2 fn_le)
```
```   125
```
```   126 definition
```
```   127   bounded :: "'a::metric_space set \<Rightarrow> bool" where
```
```   128   "bounded S \<longleftrightarrow> (\<exists>x eee. \<forall>y\<in>S. dist x y \<le> eee)"
```
```   129
```
```   130 lemma "bounded T \<Longrightarrow> S \<subseteq> T ==> bounded S"
```
```   131 by (metis bounded_def subset_eq)
```
```   132
```
```   133 lemma
```
```   134   assumes a: "Quotient R Abs Rep"
```
```   135   shows "symp R"
```
```   136 using a unfolding Quotient_def using sympI
```
```   137 by (metis (full_types))
```
```   138
```
```   139 lemma
```
```   140   "(\<exists>x \<in> set xs. P x) \<longleftrightarrow>
```
```   141    (\<exists>ys x zs. xs = ys @ x # zs \<and> P x \<and> (\<forall>z \<in> set zs. \<not> P z))"
```
```   142 by (metis split_list_last_prop [where P = P] in_set_conv_decomp)
```
```   143
```
```   144 lemma ex_tl: "EX ys. tl ys = xs"
```
```   145 using tl.simps(2) by fast
```
```   146
```
```   147 lemma "(\<exists>ys\<Colon>nat list. tl ys = xs) \<and> (\<exists>bs\<Colon>int list. tl bs = as)"
```
```   148 by (metis ex_tl)
```
```   149
```
```   150 end
```