src/HOL/SizeChange/Graphs.thy
 author haftmann Fri Jan 02 08:12:46 2009 +0100 (2009-01-02) changeset 29332 edc1e2a56398 parent 28685 275122631271 child 30282 a16af775e08d permissions -rw-r--r--
named code theorem for Fract_norm
```     1 (*  Title:      HOL/Library/Graphs.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Alexander Krauss, TU Muenchen
```
```     4 *)
```
```     5
```
```     6 header {* General Graphs as Sets *}
```
```     7
```
```     8 theory Graphs
```
```     9 imports Main Misc_Tools Kleene_Algebras
```
```    10 begin
```
```    11
```
```    12 subsection {* Basic types, Size Change Graphs *}
```
```    13
```
```    14 datatype ('a, 'b) graph =
```
```    15   Graph "('a \<times> 'b \<times> 'a) set"
```
```    16
```
```    17 primrec dest_graph :: "('a, 'b) graph \<Rightarrow> ('a \<times> 'b \<times> 'a) set"
```
```    18   where "dest_graph (Graph G) = G"
```
```    19
```
```    20 lemma graph_dest_graph[simp]:
```
```    21   "Graph (dest_graph G) = G"
```
```    22   by (cases G) simp
```
```    23
```
```    24 lemma split_graph_all:
```
```    25   "(\<And>gr. PROP P gr) \<equiv> (\<And>set. PROP P (Graph set))"
```
```    26 proof
```
```    27   fix set
```
```    28   assume "\<And>gr. PROP P gr"
```
```    29   then show "PROP P (Graph set)" .
```
```    30 next
```
```    31   fix gr
```
```    32   assume "\<And>set. PROP P (Graph set)"
```
```    33   then have "PROP P (Graph (dest_graph gr))" .
```
```    34   then show "PROP P gr" by simp
```
```    35 qed
```
```    36
```
```    37 definition
```
```    38   has_edge :: "('n,'e) graph \<Rightarrow> 'n \<Rightarrow> 'e \<Rightarrow> 'n \<Rightarrow> bool"
```
```    39 ("_ \<turnstile> _ \<leadsto>\<^bsup>_\<^esup> _")
```
```    40 where
```
```    41   "has_edge G n e n' = ((n, e, n') \<in> dest_graph G)"
```
```    42
```
```    43
```
```    44 subsection {* Graph composition *}
```
```    45
```
```    46 fun grcomp :: "('n, 'e::times) graph \<Rightarrow> ('n, 'e) graph  \<Rightarrow> ('n, 'e) graph"
```
```    47 where
```
```    48   "grcomp (Graph G) (Graph H) =
```
```    49   Graph {(p,b,q) | p b q.
```
```    50   (\<exists>k e e'. (p,e,k)\<in>G \<and> (k,e',q)\<in>H \<and> b = e * e')}"
```
```    51
```
```    52
```
```    53 declare grcomp.simps[code del]
```
```    54
```
```    55
```
```    56 lemma graph_ext:
```
```    57   assumes "\<And>n e n'. has_edge G n e n' = has_edge H n e n'"
```
```    58   shows "G = H"
```
```    59   using assms
```
```    60   by (cases G, cases H) (auto simp:split_paired_all has_edge_def)
```
```    61
```
```    62
```
```    63 instantiation graph :: (type, type) comm_monoid_add
```
```    64 begin
```
```    65
```
```    66 definition
```
```    67   graph_zero_def: "0 = Graph {}"
```
```    68
```
```    69 definition
```
```    70   graph_plus_def [code del]: "G + H = Graph (dest_graph G \<union> dest_graph H)"
```
```    71
```
```    72 instance proof
```
```    73   fix x y z :: "('a,'b) graph"
```
```    74   show "x + y + z = x + (y + z)"
```
```    75    and "x + y = y + x"
```
```    76    and "0 + x = x"
```
```    77   unfolding graph_plus_def graph_zero_def by auto
```
```    78 qed
```
```    79
```
```    80 end
```
```    81
```
```    82 instantiation graph :: (type, type) "{distrib_lattice, complete_lattice}"
```
```    83 begin
```
```    84
```
```    85 definition
```
```    86   graph_leq_def [code del]: "G \<le> H \<longleftrightarrow> dest_graph G \<subseteq> dest_graph H"
```
```    87
```
```    88 definition
```
```    89   graph_less_def [code del]: "G < H \<longleftrightarrow> dest_graph G \<subset> dest_graph H"
```
```    90
```
```    91 definition
```
```    92   [code del]: "bot = Graph {}"
```
```    93
```
```    94 definition
```
```    95   [code del]: "top = Graph UNIV"
```
```    96
```
```    97 definition
```
```    98   [code del]: "inf G H = Graph (dest_graph G \<inter> dest_graph H)"
```
```    99
```
```   100 definition
```
```   101   [code del]: "sup (G \<Colon> ('a, 'b) graph) H = G + H"
```
```   102
```
```   103 definition
```
```   104   Inf_graph_def [code del]: "Inf = (\<lambda>Gs. Graph (\<Inter>(dest_graph ` Gs)))"
```
```   105
```
```   106 definition
```
```   107   Sup_graph_def [code del]: "Sup = (\<lambda>Gs. Graph (\<Union>(dest_graph ` Gs)))"
```
```   108
```
```   109 instance proof
```
```   110   fix x y z :: "('a,'b) graph"
```
```   111   fix A :: "('a, 'b) graph set"
```
```   112
```
```   113   show "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
```
```   114     unfolding graph_leq_def graph_less_def
```
```   115     by (cases x, cases y) auto
```
```   116
```
```   117   show "x \<le> x" unfolding graph_leq_def ..
```
```   118
```
```   119   { assume "x \<le> y" "y \<le> z"
```
```   120     with order_trans show "x \<le> z"
```
```   121       unfolding graph_leq_def . }
```
```   122
```
```   123   { assume "x \<le> y" "y \<le> x" thus "x = y"
```
```   124       unfolding graph_leq_def
```
```   125       by (cases x, cases y) simp }
```
```   126
```
```   127   show "inf x y \<le> x" "inf x y \<le> y"
```
```   128     unfolding inf_graph_def graph_leq_def
```
```   129     by auto
```
```   130
```
```   131   { assume "x \<le> y" "x \<le> z" thus "x \<le> inf y z"
```
```   132       unfolding inf_graph_def graph_leq_def
```
```   133       by auto }
```
```   134
```
```   135   show "x \<le> sup x y" "y \<le> sup x y"
```
```   136     unfolding sup_graph_def graph_leq_def graph_plus_def by auto
```
```   137
```
```   138   { assume "y \<le> x" "z \<le> x" thus "sup y z \<le> x"
```
```   139       unfolding sup_graph_def graph_leq_def graph_plus_def by auto }
```
```   140
```
```   141   show "bot \<le> x" unfolding graph_leq_def bot_graph_def by simp
```
```   142
```
```   143   show "x \<le> top" unfolding graph_leq_def top_graph_def by simp
```
```   144
```
```   145   show "sup x (inf y z) = inf (sup x y) (sup x z)"
```
```   146     unfolding inf_graph_def sup_graph_def graph_leq_def graph_plus_def by auto
```
```   147
```
```   148   { assume "x \<in> A" thus "Inf A \<le> x"
```
```   149       unfolding Inf_graph_def graph_leq_def by auto }
```
```   150
```
```   151   { assume "\<And>x. x \<in> A \<Longrightarrow> z \<le> x" thus "z \<le> Inf A"
```
```   152     unfolding Inf_graph_def graph_leq_def by auto }
```
```   153
```
```   154   { assume "x \<in> A" thus "x \<le> Sup A"
```
```   155       unfolding Sup_graph_def graph_leq_def by auto }
```
```   156
```
```   157   { assume "\<And>x. x \<in> A \<Longrightarrow> x \<le> z" thus "Sup A \<le> z"
```
```   158     unfolding Sup_graph_def graph_leq_def by auto }
```
```   159 qed
```
```   160
```
```   161 end
```
```   162
```
```   163 lemma in_grplus:
```
```   164   "has_edge (G + H) p b q = (has_edge G p b q \<or> has_edge H p b q)"
```
```   165   by (cases G, cases H, auto simp:has_edge_def graph_plus_def)
```
```   166
```
```   167 lemma in_grzero:
```
```   168   "has_edge 0 p b q = False"
```
```   169   by (simp add:graph_zero_def has_edge_def)
```
```   170
```
```   171
```
```   172 subsubsection {* Multiplicative Structure *}
```
```   173
```
```   174 instantiation graph :: (type, times) mult_zero
```
```   175 begin
```
```   176
```
```   177 definition
```
```   178   graph_mult_def [code del]: "G * H = grcomp G H"
```
```   179
```
```   180 instance proof
```
```   181   fix a :: "('a, 'b) graph"
```
```   182
```
```   183   show "0 * a = 0"
```
```   184     unfolding graph_mult_def graph_zero_def
```
```   185     by (cases a) (simp add:grcomp.simps)
```
```   186   show "a * 0 = 0"
```
```   187     unfolding graph_mult_def graph_zero_def
```
```   188     by (cases a) (simp add:grcomp.simps)
```
```   189 qed
```
```   190
```
```   191 end
```
```   192
```
```   193 instantiation graph :: (type, one) one
```
```   194 begin
```
```   195
```
```   196 definition
```
```   197   graph_one_def: "1 = Graph { (x, 1, x) |x. True}"
```
```   198
```
```   199 instance ..
```
```   200
```
```   201 end
```
```   202
```
```   203 lemma in_grcomp:
```
```   204   "has_edge (G * H) p b q
```
```   205   = (\<exists>k e e'. has_edge G p e k \<and> has_edge H k e' q \<and> b = e * e')"
```
```   206   by (cases G, cases H) (auto simp:graph_mult_def has_edge_def image_def)
```
```   207
```
```   208 lemma in_grunit:
```
```   209   "has_edge 1 p b q = (p = q \<and> b = 1)"
```
```   210   by (auto simp:graph_one_def has_edge_def)
```
```   211
```
```   212 instance graph :: (type, semigroup_mult) semigroup_mult
```
```   213 proof
```
```   214   fix G1 G2 G3 :: "('a,'b) graph"
```
```   215
```
```   216   show "G1 * G2 * G3 = G1 * (G2 * G3)"
```
```   217   proof (rule graph_ext, rule trans)
```
```   218     fix p J q
```
```   219     show "has_edge ((G1 * G2) * G3) p J q =
```
```   220       (\<exists>G i H j I.
```
```   221       has_edge G1 p G i
```
```   222       \<and> has_edge G2 i H j
```
```   223       \<and> has_edge G3 j I q
```
```   224       \<and> J = (G * H) * I)"
```
```   225       by (simp only:in_grcomp) blast
```
```   226     show "\<dots> = has_edge (G1 * (G2 * G3)) p J q"
```
```   227       by (simp only:in_grcomp mult_assoc) blast
```
```   228   qed
```
```   229 qed
```
```   230
```
```   231 instantiation graph :: (type, monoid_mult) "{semiring_1, idem_add, recpower, star}"
```
```   232 begin
```
```   233
```
```   234 primrec power_graph :: "('a\<Colon>type, 'b\<Colon>monoid_mult) graph \<Rightarrow> nat => ('a, 'b) graph"
```
```   235 where
```
```   236   "(A \<Colon> ('a, 'b) graph) ^ 0 = 1"
```
```   237 | "(A \<Colon> ('a, 'b) graph) ^ Suc n = A * (A ^ n)"
```
```   238
```
```   239 definition
```
```   240   graph_star_def: "star (G \<Colon> ('a, 'b) graph) = (SUP n. G ^ n)"
```
```   241
```
```   242 instance proof
```
```   243   fix a b c :: "('a, 'b) graph"
```
```   244
```
```   245   show "1 * a = a"
```
```   246     by (rule graph_ext) (auto simp:in_grcomp in_grunit)
```
```   247   show "a * 1 = a"
```
```   248     by (rule graph_ext) (auto simp:in_grcomp in_grunit)
```
```   249
```
```   250   show "(a + b) * c = a * c + b * c"
```
```   251     by (rule graph_ext, simp add:in_grcomp in_grplus) blast
```
```   252
```
```   253   show "a * (b + c) = a * b + a * c"
```
```   254     by (rule graph_ext, simp add:in_grcomp in_grplus) blast
```
```   255
```
```   256   show "(0::('a,'b) graph) \<noteq> 1" unfolding graph_zero_def graph_one_def
```
```   257     by simp
```
```   258
```
```   259   show "a + a = a" unfolding graph_plus_def by simp
```
```   260
```
```   261   show "a ^ 0 = 1" "\<And>n. a ^ (Suc n) = a * a ^ n"
```
```   262     by simp_all
```
```   263 qed
```
```   264
```
```   265 end
```
```   266
```
```   267 lemma graph_leqI:
```
```   268   assumes "\<And>n e n'. has_edge G n e n' \<Longrightarrow> has_edge H n e n'"
```
```   269   shows "G \<le> H"
```
```   270   using assms
```
```   271   unfolding graph_leq_def has_edge_def
```
```   272   by auto
```
```   273
```
```   274 lemma in_graph_plusE:
```
```   275   assumes "has_edge (G + H) n e n'"
```
```   276   assumes "has_edge G n e n' \<Longrightarrow> P"
```
```   277   assumes "has_edge H n e n' \<Longrightarrow> P"
```
```   278   shows P
```
```   279   using assms
```
```   280   by (auto simp: in_grplus)
```
```   281
```
```   282 lemma in_graph_compE:
```
```   283   assumes GH: "has_edge (G * H) n e n'"
```
```   284   obtains e1 k e2
```
```   285   where "has_edge G n e1 k" "has_edge H k e2 n'" "e = e1 * e2"
```
```   286   using GH
```
```   287   by (auto simp: in_grcomp)
```
```   288
```
```   289 lemma
```
```   290   assumes "x \<in> S k"
```
```   291   shows "x \<in> (\<Union>k. S k)"
```
```   292   using assms by blast
```
```   293
```
```   294 lemma graph_union_least:
```
```   295   assumes "\<And>n. Graph (G n) \<le> C"
```
```   296   shows "Graph (\<Union>n. G n) \<le> C"
```
```   297   using assms unfolding graph_leq_def
```
```   298   by auto
```
```   299
```
```   300 lemma Sup_graph_eq:
```
```   301   "(SUP n. Graph (G n)) = Graph (\<Union>n. G n)"
```
```   302 proof (rule order_antisym)
```
```   303   show "(SUP n. Graph (G n)) \<le> Graph (\<Union>n. G n)"
```
```   304     by  (rule SUP_leI) (auto simp add: graph_leq_def)
```
```   305
```
```   306   show "Graph (\<Union>n. G n) \<le> (SUP n. Graph (G n))"
```
```   307   by (rule graph_union_least, rule le_SUPI', rule)
```
```   308 qed
```
```   309
```
```   310 lemma has_edge_leq: "has_edge G p b q = (Graph {(p,b,q)} \<le> G)"
```
```   311   unfolding has_edge_def graph_leq_def
```
```   312   by (cases G) simp
```
```   313
```
```   314
```
```   315 lemma Sup_graph_eq2:
```
```   316   "(SUP n. G n) = Graph (\<Union>n. dest_graph (G n))"
```
```   317   using Sup_graph_eq[of "\<lambda>n. dest_graph (G n)", simplified]
```
```   318   by simp
```
```   319
```
```   320 lemma in_SUP:
```
```   321   "has_edge (SUP x. Gs x) p b q = (\<exists>x. has_edge (Gs x) p b q)"
```
```   322   unfolding Sup_graph_eq2 has_edge_leq graph_leq_def
```
```   323   by simp
```
```   324
```
```   325 instance graph :: (type, monoid_mult) kleene_by_complete_lattice
```
```   326 proof
```
```   327   fix a b c :: "('a, 'b) graph"
```
```   328
```
```   329   show "a \<le> b \<longleftrightarrow> a + b = b" unfolding graph_leq_def graph_plus_def
```
```   330     by (cases a, cases b) auto
```
```   331
```
```   332   from less_le_not_le show "a < b \<longleftrightarrow> a \<le> b \<and> \<not> b \<le> a" .
```
```   333
```
```   334   show "a * star b * c = (SUP n. a * b ^ n * c)"
```
```   335     unfolding graph_star_def
```
```   336     by (rule graph_ext) (force simp:in_SUP in_grcomp)
```
```   337 qed
```
```   338
```
```   339
```
```   340 lemma in_star:
```
```   341   "has_edge (star G) a x b = (\<exists>n. has_edge (G ^ n) a x b)"
```
```   342   by (auto simp:graph_star_def in_SUP)
```
```   343
```
```   344 lemma tcl_is_SUP:
```
```   345   "tcl (G::('a::type, 'b::monoid_mult) graph) =
```
```   346   (SUP n. G ^ (Suc n))"
```
```   347   unfolding tcl_def
```
```   348   using star_cont[of 1 G G]
```
```   349   by (simp add:power_Suc power_commutes)
```
```   350
```
```   351
```
```   352 lemma in_tcl:
```
```   353   "has_edge (tcl G) a x b = (\<exists>n>0. has_edge (G ^ n) a x b)"
```
```   354   apply (auto simp: tcl_is_SUP in_SUP simp del: power_graph.simps)
```
```   355   apply (rule_tac x = "n - 1" in exI, auto)
```
```   356   done
```
```   357
```
```   358
```
```   359 subsection {* Infinite Paths *}
```
```   360
```
```   361 types ('n, 'e) ipath = "('n \<times> 'e) sequence"
```
```   362
```
```   363 definition has_ipath :: "('n, 'e) graph \<Rightarrow> ('n, 'e) ipath \<Rightarrow> bool"
```
```   364 where
```
```   365   "has_ipath G p =
```
```   366   (\<forall>i. has_edge G (fst (p i)) (snd (p i)) (fst (p (Suc i))))"
```
```   367
```
```   368
```
```   369 subsection {* Finite Paths *}
```
```   370
```
```   371 types ('n, 'e) fpath = "('n \<times> ('e \<times> 'n) list)"
```
```   372
```
```   373 inductive  has_fpath :: "('n, 'e) graph \<Rightarrow> ('n, 'e) fpath \<Rightarrow> bool"
```
```   374   for G :: "('n, 'e) graph"
```
```   375 where
```
```   376   has_fpath_empty: "has_fpath G (n, [])"
```
```   377 | has_fpath_join: "\<lbrakk>G \<turnstile> n \<leadsto>\<^bsup>e\<^esup> n'; has_fpath G (n', es)\<rbrakk> \<Longrightarrow> has_fpath G (n, (e, n')#es)"
```
```   378
```
```   379 definition
```
```   380   "end_node p =
```
```   381   (if snd p = [] then fst p else snd (snd p ! (length (snd p) - 1)))"
```
```   382
```
```   383 definition path_nth :: "('n, 'e) fpath \<Rightarrow> nat \<Rightarrow> ('n \<times> 'e \<times> 'n)"
```
```   384 where
```
```   385   "path_nth p k = (if k = 0 then fst p else snd (snd p ! (k - 1)), snd p ! k)"
```
```   386
```
```   387 lemma endnode_nth:
```
```   388   assumes "length (snd p) = Suc k"
```
```   389   shows "end_node p = snd (snd (path_nth p k))"
```
```   390   using assms unfolding end_node_def path_nth_def
```
```   391   by auto
```
```   392
```
```   393 lemma path_nth_graph:
```
```   394   assumes "k < length (snd p)"
```
```   395   assumes "has_fpath G p"
```
```   396   shows "(\<lambda>(n,e,n'). has_edge G n e n') (path_nth p k)"
```
```   397 using assms
```
```   398 proof (induct k arbitrary: p)
```
```   399   case 0 thus ?case
```
```   400     unfolding path_nth_def by (auto elim:has_fpath.cases)
```
```   401 next
```
```   402   case (Suc k p)
```
```   403
```
```   404   from `has_fpath G p` show ?case
```
```   405   proof (rule has_fpath.cases)
```
```   406     case goal1 with Suc show ?case by simp
```
```   407   next
```
```   408     fix n e n' es
```
```   409     assume st: "p = (n, (e, n') # es)"
```
```   410        "G \<turnstile> n \<leadsto>\<^bsup>e\<^esup> n'"
```
```   411        "has_fpath G (n', es)"
```
```   412     with Suc
```
```   413     have "(\<lambda>(n, b, a). G \<turnstile> n \<leadsto>\<^bsup>b\<^esup> a) (path_nth (n', es) k)" by simp
```
```   414     with st show ?thesis by (cases k, auto simp:path_nth_def)
```
```   415   qed
```
```   416 qed
```
```   417
```
```   418 lemma path_nth_connected:
```
```   419   assumes "Suc k < length (snd p)"
```
```   420   shows "fst (path_nth p (Suc k)) = snd (snd (path_nth p k))"
```
```   421   using assms
```
```   422   unfolding path_nth_def
```
```   423   by auto
```
```   424
```
```   425 definition path_loop :: "('n, 'e) fpath \<Rightarrow> ('n, 'e) ipath" ("omega")
```
```   426 where
```
```   427   "omega p \<equiv> (\<lambda>i. (\<lambda>(n,e,n'). (n,e)) (path_nth p (i mod (length (snd p)))))"
```
```   428
```
```   429 lemma fst_p0: "fst (path_nth p 0) = fst p"
```
```   430   unfolding path_nth_def by simp
```
```   431
```
```   432 lemma path_loop_connect:
```
```   433   assumes "fst p = end_node p"
```
```   434   and "0 < length (snd p)" (is "0 < ?l")
```
```   435   shows "fst (path_nth p (Suc i mod (length (snd p))))
```
```   436   = snd (snd (path_nth p (i mod length (snd p))))"
```
```   437   (is "\<dots> = snd (snd (path_nth p ?k))")
```
```   438 proof -
```
```   439   from `0 < ?l` have "i mod ?l < ?l" (is "?k < ?l")
```
```   440     by simp
```
```   441
```
```   442   show ?thesis
```
```   443   proof (cases "Suc ?k < ?l")
```
```   444     case True
```
```   445     hence "Suc ?k \<noteq> ?l" by simp
```
```   446     with path_nth_connected[OF True]
```
```   447     show ?thesis
```
```   448       by (simp add:mod_Suc)
```
```   449   next
```
```   450     case False
```
```   451     with `?k < ?l` have wrap: "Suc ?k = ?l" by simp
```
```   452
```
```   453     hence "fst (path_nth p (Suc i mod ?l)) = fst (path_nth p 0)"
```
```   454       by (simp add: mod_Suc)
```
```   455     also from fst_p0 have "\<dots> = fst p" .
```
```   456     also have "\<dots> = end_node p" by fact
```
```   457     also have "\<dots> = snd (snd (path_nth p ?k))"
```
```   458       by (auto simp: endnode_nth wrap)
```
```   459     finally show ?thesis .
```
```   460   qed
```
```   461 qed
```
```   462
```
```   463 lemma path_loop_graph:
```
```   464   assumes "has_fpath G p"
```
```   465   and loop: "fst p = end_node p"
```
```   466   and nonempty: "0 < length (snd p)" (is "0 < ?l")
```
```   467   shows "has_ipath G (omega p)"
```
```   468 proof -
```
```   469   {
```
```   470     fix i
```
```   471     from `0 < ?l` have "i mod ?l < ?l" (is "?k < ?l")
```
```   472       by simp
```
```   473     from this and `has_fpath G p`
```
```   474     have pk_G: "(\<lambda>(n,e,n'). has_edge G n e n') (path_nth p ?k)"
```
```   475       by (rule path_nth_graph)
```
```   476
```
```   477     from path_loop_connect[OF loop nonempty] pk_G
```
```   478     have "has_edge G (fst (omega p i)) (snd (omega p i)) (fst (omega p (Suc i)))"
```
```   479       unfolding path_loop_def has_edge_def split_def
```
```   480       by simp
```
```   481   }
```
```   482   then show ?thesis by (auto simp:has_ipath_def)
```
```   483 qed
```
```   484
```
```   485 definition prod :: "('n, 'e::monoid_mult) fpath \<Rightarrow> 'e"
```
```   486 where
```
```   487   "prod p = foldr (op *) (map fst (snd p)) 1"
```
```   488
```
```   489 lemma prod_simps[simp]:
```
```   490   "prod (n, []) = 1"
```
```   491   "prod (n, (e,n')#es) = e * (prod (n',es))"
```
```   492 unfolding prod_def
```
```   493 by simp_all
```
```   494
```
```   495 lemma power_induces_path:
```
```   496   assumes a: "has_edge (A ^ k) n G m"
```
```   497   obtains p
```
```   498     where "has_fpath A p"
```
```   499       and "n = fst p" "m = end_node p"
```
```   500       and "G = prod p"
```
```   501       and "k = length (snd p)"
```
```   502   using a
```
```   503 proof (induct k arbitrary:m n G thesis)
```
```   504   case (0 m n G)
```
```   505   let ?p = "(n, [])"
```
```   506   from 0 have "has_fpath A ?p" "m = end_node ?p" "G = prod ?p"
```
```   507     by (auto simp:in_grunit end_node_def intro:has_fpath.intros)
```
```   508   thus ?case using 0 by (auto simp:end_node_def)
```
```   509 next
```
```   510   case (Suc k m n G)
```
```   511   hence "has_edge (A * A ^ k) n G m"
```
```   512     by (simp add:power_Suc power_commutes)
```
```   513   then obtain G' H j where
```
```   514     a_A: "has_edge A n G' j"
```
```   515     and H_pow: "has_edge (A ^ k) j H m"
```
```   516     and [simp]: "G = G' * H"
```
```   517     by (auto simp:in_grcomp)
```
```   518
```
```   519   from H_pow and Suc
```
```   520   obtain p
```
```   521     where p_path: "has_fpath A p"
```
```   522     and [simp]: "j = fst p" "m = end_node p" "H = prod p"
```
```   523     "k = length (snd p)"
```
```   524     by blast
```
```   525
```
```   526   let ?p' = "(n, (G', j)#snd p)"
```
```   527   from a_A and p_path
```
```   528   have "has_fpath A ?p'" "m = end_node ?p'" "G = prod ?p'"
```
```   529     by (auto simp:end_node_def nth.simps intro:has_fpath.intros split:nat.split)
```
```   530   thus ?case using Suc by auto
```
```   531 qed
```
```   532
```
```   533
```
```   534 subsection {* Sub-Paths *}
```
```   535
```
```   536 definition sub_path :: "('n, 'e) ipath \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> ('n, 'e) fpath"
```
```   537 ("(_\<langle>_,_\<rangle>)")
```
```   538 where
```
```   539   "p\<langle>i,j\<rangle> =
```
```   540   (fst (p i), map (\<lambda>k. (snd (p k), fst (p (Suc k)))) [i ..< j])"
```
```   541
```
```   542
```
```   543 lemma sub_path_is_path:
```
```   544   assumes ipath: "has_ipath G p"
```
```   545   assumes l: "i \<le> j"
```
```   546   shows "has_fpath G (p\<langle>i,j\<rangle>)"
```
```   547   using l
```
```   548 proof (induct i rule:inc_induct)
```
```   549   case base show ?case by (auto simp:sub_path_def intro:has_fpath.intros)
```
```   550 next
```
```   551   case (step i)
```
```   552   with ipath upt_rec[of i j]
```
```   553   show ?case
```
```   554     by (auto simp:sub_path_def has_ipath_def intro:has_fpath.intros)
```
```   555 qed
```
```   556
```
```   557
```
```   558 lemma sub_path_start[simp]:
```
```   559   "fst (p\<langle>i,j\<rangle>) = fst (p i)"
```
```   560   by (simp add:sub_path_def)
```
```   561
```
```   562 lemma nth_upto[simp]: "k < j - i \<Longrightarrow> [i ..< j] ! k = i + k"
```
```   563   by (induct k) auto
```
```   564
```
```   565 lemma sub_path_end[simp]:
```
```   566   "i < j \<Longrightarrow> end_node (p\<langle>i,j\<rangle>) = fst (p j)"
```
```   567   by (auto simp:sub_path_def end_node_def)
```
```   568
```
```   569 lemma foldr_map: "foldr f (map g xs) = foldr (f o g) xs"
```
```   570   by (induct xs) auto
```
```   571
```
```   572 lemma upto_append[simp]:
```
```   573   assumes "i \<le> j" "j \<le> k"
```
```   574   shows "[ i ..< j ] @ [j ..< k] = [i ..< k]"
```
```   575   using assms and upt_add_eq_append[of i j "k - j"]
```
```   576   by simp
```
```   577
```
```   578 lemma foldr_monoid: "foldr (op *) xs 1 * foldr (op *) ys 1
```
```   579   = foldr (op *) (xs @ ys) (1::'a::monoid_mult)"
```
```   580   by (induct xs) (auto simp:mult_assoc)
```
```   581
```
```   582 lemma sub_path_prod:
```
```   583   assumes "i < j"
```
```   584   assumes "j < k"
```
```   585   shows "prod (p\<langle>i,k\<rangle>) = prod (p\<langle>i,j\<rangle>) * prod (p\<langle>j,k\<rangle>)"
```
```   586   using assms
```
```   587   unfolding prod_def sub_path_def
```
```   588   by (simp add:map_compose[symmetric] comp_def)
```
```   589    (simp only:foldr_monoid map_append[symmetric] upto_append)
```
```   590
```
```   591
```
```   592 lemma path_acgpow_aux:
```
```   593   assumes "length es = l"
```
```   594   assumes "has_fpath G (n,es)"
```
```   595   shows "has_edge (G ^ l) n (prod (n,es)) (end_node (n,es))"
```
```   596 using assms
```
```   597 proof (induct l arbitrary:n es)
```
```   598   case 0 thus ?case
```
```   599     by (simp add:in_grunit end_node_def)
```
```   600 next
```
```   601   case (Suc l n es)
```
```   602   hence "es \<noteq> []" by auto
```
```   603   let ?n' = "snd (hd es)"
```
```   604   let ?es' = "tl es"
```
```   605   let ?e = "fst (hd es)"
```
```   606
```
```   607   from Suc have len: "length ?es' = l" by auto
```
```   608
```
```   609   from Suc
```
```   610   have [simp]: "end_node (n, es) = end_node (?n', ?es')"
```
```   611     by (cases es) (auto simp:end_node_def nth.simps split:nat.split)
```
```   612
```
```   613   from `has_fpath G (n,es)`
```
```   614   have "has_fpath G (?n', ?es')"
```
```   615     by (rule has_fpath.cases) (auto intro:has_fpath.intros)
```
```   616   with Suc len
```
```   617   have "has_edge (G ^ l) ?n' (prod (?n', ?es')) (end_node (?n', ?es'))"
```
```   618     by auto
```
```   619   moreover
```
```   620   from `es \<noteq> []`
```
```   621   have "prod (n, es) = ?e * (prod (?n', ?es'))"
```
```   622     by (cases es) auto
```
```   623   moreover
```
```   624   from `has_fpath G (n,es)` have c:"has_edge G n ?e ?n'"
```
```   625     by (rule has_fpath.cases) (insert `es \<noteq> []`, auto)
```
```   626
```
```   627   ultimately
```
```   628   show ?case
```
```   629      unfolding power_Suc
```
```   630      by (auto simp:in_grcomp)
```
```   631 qed
```
```   632
```
```   633
```
```   634 lemma path_acgpow:
```
```   635    "has_fpath G p
```
```   636   \<Longrightarrow> has_edge (G ^ length (snd p)) (fst p) (prod p) (end_node p)"
```
```   637 by (cases p)
```
```   638    (rule path_acgpow_aux[of "snd p" "length (snd p)" _ "fst p", simplified])
```
```   639
```
```   640
```
```   641 lemma star_paths:
```
```   642   "has_edge (star G) a x b =
```
```   643    (\<exists>p. has_fpath G p \<and> a = fst p \<and> b = end_node p \<and> x = prod p)"
```
```   644 proof
```
```   645   assume "has_edge (star G) a x b"
```
```   646   then obtain n where pow: "has_edge (G ^ n) a x b"
```
```   647     by (auto simp:in_star)
```
```   648
```
```   649   then obtain p where
```
```   650     "has_fpath G p" "a = fst p" "b = end_node p" "x = prod p"
```
```   651     by (rule power_induces_path)
```
```   652
```
```   653   thus "\<exists>p. has_fpath G p \<and> a = fst p \<and> b = end_node p \<and> x = prod p"
```
```   654     by blast
```
```   655 next
```
```   656   assume "\<exists>p. has_fpath G p \<and> a = fst p \<and> b = end_node p \<and> x = prod p"
```
```   657   then obtain p where
```
```   658     "has_fpath G p" "a = fst p" "b = end_node p" "x = prod p"
```
```   659     by blast
```
```   660
```
```   661   hence "has_edge (G ^ length (snd p)) a x b"
```
```   662     by (auto intro:path_acgpow)
```
```   663
```
```   664   thus "has_edge (star G) a x b"
```
```   665     by (auto simp:in_star)
```
```   666 qed
```
```   667
```
```   668
```
```   669 lemma plus_paths:
```
```   670   "has_edge (tcl G) a x b =
```
```   671    (\<exists>p. has_fpath G p \<and> a = fst p \<and> b = end_node p \<and> x = prod p \<and> 0 < length (snd p))"
```
```   672 proof
```
```   673   assume "has_edge (tcl G) a x b"
```
```   674
```
```   675   then obtain n where pow: "has_edge (G ^ n) a x b" and "0 < n"
```
```   676     by (auto simp:in_tcl)
```
```   677
```
```   678   from pow obtain p where
```
```   679     "has_fpath G p" "a = fst p" "b = end_node p" "x = prod p"
```
```   680     "n = length (snd p)"
```
```   681     by (rule power_induces_path)
```
```   682
```
```   683   with `0 < n`
```
```   684   show "\<exists>p. has_fpath G p \<and> a = fst p \<and> b = end_node p \<and> x = prod p \<and> 0 < length (snd p) "
```
```   685     by blast
```
```   686 next
```
```   687   assume "\<exists>p. has_fpath G p \<and> a = fst p \<and> b = end_node p \<and> x = prod p
```
```   688     \<and> 0 < length (snd p)"
```
```   689   then obtain p where
```
```   690     "has_fpath G p" "a = fst p" "b = end_node p" "x = prod p"
```
```   691     "0 < length (snd p)"
```
```   692     by blast
```
```   693
```
```   694   hence "has_edge (G ^ length (snd p)) a x b"
```
```   695     by (auto intro:path_acgpow)
```
```   696
```
```   697   with `0 < length (snd p)`
```
```   698   show "has_edge (tcl G) a x b"
```
```   699     by (auto simp:in_tcl)
```
```   700 qed
```
```   701
```
```   702
```
```   703 definition
```
```   704   "contract s p =
```
```   705   (\<lambda>i. (fst (p (s i)), prod (p\<langle>s i,s (Suc i)\<rangle>)))"
```
```   706
```
```   707 lemma ipath_contract:
```
```   708   assumes [simp]: "increasing s"
```
```   709   assumes ipath: "has_ipath G p"
```
```   710   shows "has_ipath (tcl G) (contract s p)"
```
```   711   unfolding has_ipath_def
```
```   712 proof
```
```   713   fix i
```
```   714   let ?p = "p\<langle>s i,s (Suc i)\<rangle>"
```
```   715
```
```   716   from increasing_strict
```
```   717 	have "fst (p (s (Suc i))) = end_node ?p" by simp
```
```   718   moreover
```
```   719   from increasing_strict[of s i "Suc i"] have "snd ?p \<noteq> []"
```
```   720     by (simp add:sub_path_def)
```
```   721   moreover
```
```   722   from ipath increasing_weak[of s] have "has_fpath G ?p"
```
```   723     by (rule sub_path_is_path) auto
```
```   724   ultimately
```
```   725   show "has_edge (tcl G)
```
```   726     (fst (contract s p i)) (snd (contract s p i)) (fst (contract s p (Suc i)))"
```
```   727     unfolding contract_def plus_paths
```
```   728     by (intro exI) auto
```
```   729 qed
```
```   730
```
```   731 lemma prod_unfold:
```
```   732   "i < j \<Longrightarrow> prod (p\<langle>i,j\<rangle>)
```
```   733   = snd (p i) * prod (p\<langle>Suc i, j\<rangle>)"
```
```   734   unfolding prod_def
```
```   735   by (simp add:sub_path_def upt_rec[of "i" j])
```
```   736
```
```   737
```
```   738 lemma sub_path_loop:
```
```   739   assumes "0 < k"
```
```   740   assumes k: "k = length (snd loop)"
```
```   741   assumes loop: "fst loop = end_node loop"
```
```   742   shows "(omega loop)\<langle>k * i,k * Suc i\<rangle> = loop" (is "?\<omega> = loop")
```
```   743 proof (rule prod_eqI)
```
```   744   show "fst ?\<omega> = fst loop"
```
```   745     by (auto simp:path_loop_def path_nth_def split_def k)
```
```   746
```
```   747   show "snd ?\<omega> = snd loop"
```
```   748   proof (rule nth_equalityI[rule_format])
```
```   749     show leneq: "length (snd ?\<omega>) = length (snd loop)"
```
```   750       unfolding sub_path_def k by simp
```
```   751
```
```   752     fix j assume "j < length (snd (?\<omega>))"
```
```   753     with leneq and k have "j < k" by simp
```
```   754
```
```   755     have a: "\<And>i. fst (path_nth loop (Suc i mod k))
```
```   756       = snd (snd (path_nth loop (i mod k)))"
```
```   757       unfolding k
```
```   758       apply (rule path_loop_connect[OF loop])
```
```   759       using `0 < k` and k
```
```   760       apply auto
```
```   761       done
```
```   762
```
```   763     from `j < k`
```
```   764     show "snd ?\<omega> ! j = snd loop ! j"
```
```   765       unfolding sub_path_def
```
```   766       apply (simp add:path_loop_def split_def add_ac)
```
```   767       apply (simp add:a k[symmetric])
```
```   768       apply (simp add:path_nth_def)
```
```   769       done
```
```   770   qed
```
```   771 qed
```
```   772
```
```   773 end
```