src/HOL/MetisExamples/TransClosure.thy
 author wenzelm Mon Mar 16 18:24:30 2009 +0100 (2009-03-16) changeset 30549 d2d7874648bd parent 28592 824f8390aaa2 child 32864 a226f29d4bdc permissions -rw-r--r--
simplified method setup;
```     1 (*  Title:      HOL/MetisTest/TransClosure.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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```     4
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```     5 Testing the metis method
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```     6 *)
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```     7
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```     8 theory TransClosure
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```     9 imports Main
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```    10 begin
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```    11
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```    12 types addr = nat
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```    13
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```    14 datatype val
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```    15   = Unit        -- "dummy result value of void expressions"
```
```    16   | Null        -- "null reference"
```
```    17   | Bool bool   -- "Boolean value"
```
```    18   | Intg int    -- "integer value"
```
```    19   | Addr addr   -- "addresses of objects in the heap"
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```    20
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```    21 consts R::"(addr \<times> addr) set"
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```    22
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```    23 consts f:: "addr \<Rightarrow> val"
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```    24
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```    25 ML {*AtpWrapper.problem_name := "TransClosure__test"*}
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```    26 lemma "\<lbrakk> f c = Intg x; \<forall> y. f b = Intg y \<longrightarrow> y \<noteq> x; (a,b) \<in> R\<^sup>*; (b,c) \<in> R\<^sup>* \<rbrakk>
```
```    27    \<Longrightarrow> \<exists> c. (b,c) \<in> R \<and> (a,c) \<in> R\<^sup>*"
```
```    28 by (metis Transitive_Closure.rtrancl_into_rtrancl converse_rtranclE trancl_reflcl)
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```    29
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```    30 lemma "\<lbrakk> f c = Intg x; \<forall> y. f b = Intg y \<longrightarrow> y \<noteq> x; (a,b) \<in> R\<^sup>*; (b,c) \<in> R\<^sup>* \<rbrakk>
```
```    31    \<Longrightarrow> \<exists> c. (b,c) \<in> R \<and> (a,c) \<in> R\<^sup>*"
```
```    32 proof (neg_clausify)
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```    33 assume 0: "f c = Intg x"
```
```    34 assume 1: "(a, b) \<in> R\<^sup>*"
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```    35 assume 2: "(b, c) \<in> R\<^sup>*"
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```    36 assume 3: "f b \<noteq> Intg x"
```
```    37 assume 4: "\<And>A. (b, A) \<notin> R \<or> (a, A) \<notin> R\<^sup>*"
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```    38 have 5: "b = c \<or> b \<in> Domain R"
```
```    39   by (metis Not_Domain_rtrancl 2)
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```    40 have 6: "\<And>X1. (a, X1) \<in> R\<^sup>* \<or> (b, X1) \<notin> R"
```
```    41   by (metis Transitive_Closure.rtrancl_into_rtrancl 1)
```
```    42 have 7: "\<And>X1. (b, X1) \<notin> R"
```
```    43   by (metis 6 4)
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```    44 have 8: "b \<notin> Domain R"
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```    45   by (metis 7 DomainE)
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```    46 have 9: "c = b"
```
```    47   by (metis 5 8)
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```    48 have 10: "f b = Intg x"
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```    49   by (metis 0 9)
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```    50 show "False"
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```    51   by (metis 10 3)
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```    52 qed
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```    53
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```    54 ML {*AtpWrapper.problem_name := "TransClosure__test_simpler"*}
```
```    55 lemma "\<lbrakk> f c = Intg x; \<forall> y. f b = Intg y \<longrightarrow> y \<noteq> x; (a,b) \<in> R\<^sup>*; (b,c) \<in> R\<^sup>* \<rbrakk>
```
```    56    \<Longrightarrow> \<exists> c. (b,c) \<in> R \<and> (a,c) \<in> R\<^sup>*"
```
```    57 apply (erule_tac x="b" in converse_rtranclE)
```
```    58 apply (metis rel_pow_0_E rel_pow_0_I)
```
```    59 apply (metis DomainE Domain_iff Transitive_Closure.rtrancl_into_rtrancl)
```
```    60 done
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```    61
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```    62 end
```