src/HOL/Library/Graphs.thy
 author haftmann Fri Aug 24 14:14:20 2007 +0200 (2007-08-24) changeset 24423 ae9cd0e92423 parent 24345 86a3557a9ebb permissions -rw-r--r--
overloaded definitions accompanied by explicit constants
```     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 SCT_Misc 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 fun 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 instance graph :: (type, type) "{comm_monoid_add}"
```
```    64   graph_zero_def: "0 == Graph {}"
```
```    65   graph_plus_def: "G + H == Graph (dest_graph G \<union> dest_graph H)"
```
```    66 proof
```
```    67   fix x y z :: "('a,'b) graph"
```
```    68
```
```    69   show "x + y + z = x + (y + z)"
```
```    70    and "x + y = y + x"
```
```    71    and "0 + x = x"
```
```    72   unfolding graph_plus_def graph_zero_def
```
```    73   by auto
```
```    74 qed
```
```    75
```
```    76 lemmas [code func del] = graph_plus_def
```
```    77
```
```    78 instance graph :: (type, type) "{distrib_lattice, complete_lattice}"
```
```    79   graph_leq_def: "G \<le> H \<equiv> dest_graph G \<subseteq> dest_graph H"
```
```    80   graph_less_def: "G < H \<equiv> dest_graph G \<subset> dest_graph H"
```
```    81   "inf G H \<equiv> Graph (dest_graph G \<inter> dest_graph H)"
```
```    82   "sup G H \<equiv> G + H"
```
```    83   Inf_graph_def: "Inf \<equiv> \<lambda>Gs. Graph (\<Inter>(dest_graph ` Gs))"
```
```    84   Sup_graph_def: "Sup \<equiv> \<lambda>Gs. Graph (\<Union>(dest_graph ` Gs))"
```
```    85 proof
```
```    86   fix x y z :: "('a,'b) graph"
```
```    87   fix A :: "('a, 'b) graph set"
```
```    88
```
```    89   show "(x < y) = (x \<le> y \<and> x \<noteq> y)"
```
```    90     unfolding graph_leq_def graph_less_def
```
```    91     by (cases x, cases y) auto
```
```    92
```
```    93   show "x \<le> x" unfolding graph_leq_def ..
```
```    94
```
```    95   { assume "x \<le> y" "y \<le> z"
```
```    96     with order_trans show "x \<le> z"
```
```    97       unfolding graph_leq_def . }
```
```    98
```
```    99   { assume "x \<le> y" "y \<le> x" thus "x = y"
```
```   100       unfolding graph_leq_def
```
```   101       by (cases x, cases y) simp }
```
```   102
```
```   103   show "inf x y \<le> x" "inf x y \<le> y"
```
```   104     unfolding inf_graph_def graph_leq_def
```
```   105     by auto
```
```   106
```
```   107   { assume "x \<le> y" "x \<le> z" thus "x \<le> inf y z"
```
```   108       unfolding inf_graph_def graph_leq_def
```
```   109       by auto }
```
```   110
```
```   111   show "x \<le> sup x y" "y \<le> sup x y"
```
```   112     unfolding sup_graph_def graph_leq_def graph_plus_def by auto
```
```   113
```
```   114   { assume "y \<le> x" "z \<le> x" thus "sup y z \<le> x"
```
```   115       unfolding sup_graph_def graph_leq_def graph_plus_def by auto }
```
```   116
```
```   117   show "sup x (inf y z) = inf (sup x y) (sup x z)"
```
```   118     unfolding inf_graph_def sup_graph_def graph_leq_def graph_plus_def by auto
```
```   119
```
```   120   { assume "x \<in> A" thus "Inf A \<le> x"
```
```   121       unfolding Inf_graph_def graph_leq_def by auto }
```
```   122
```
```   123   { assume "\<And>x. x \<in> A \<Longrightarrow> z \<le> x" thus "z \<le> Inf A"
```
```   124     unfolding Inf_graph_def graph_leq_def by auto }
```
```   125
```
```   126   { assume "x \<in> A" thus "x \<le> Sup A"
```
```   127       unfolding Sup_graph_def graph_leq_def by auto }
```
```   128
```
```   129   { assume "\<And>x. x \<in> A \<Longrightarrow> x \<le> z" thus "Sup A \<le> z"
```
```   130     unfolding Sup_graph_def graph_leq_def by auto }
```
```   131 qed
```
```   132
```
```   133 lemmas [code func del] = graph_leq_def graph_less_def
```
```   134   inf_graph_def sup_graph_def Inf_graph_def Sup_graph_def
```
```   135
```
```   136 lemma in_grplus:
```
```   137   "has_edge (G + H) p b q = (has_edge G p b q \<or> has_edge H p b q)"
```
```   138   by (cases G, cases H, auto simp:has_edge_def graph_plus_def)
```
```   139
```
```   140 lemma in_grzero:
```
```   141   "has_edge 0 p b q = False"
```
```   142   by (simp add:graph_zero_def has_edge_def)
```
```   143
```
```   144
```
```   145 subsubsection {* Multiplicative Structure *}
```
```   146
```
```   147 instance graph :: (type, times) mult_zero
```
```   148   graph_mult_def: "G * H == grcomp G H"
```
```   149 proof
```
```   150   fix a :: "('a, 'b) graph"
```
```   151
```
```   152   show "0 * a = 0"
```
```   153     unfolding graph_mult_def graph_zero_def
```
```   154     by (cases a) (simp add:grcomp.simps)
```
```   155   show "a * 0 = 0"
```
```   156     unfolding graph_mult_def graph_zero_def
```
```   157     by (cases a) (simp add:grcomp.simps)
```
```   158 qed
```
```   159
```
```   160 lemmas [code func del] = graph_mult_def
```
```   161
```
```   162 instance graph :: (type, one) one
```
```   163   graph_one_def: "1 == Graph { (x, 1, x) |x. True}" ..
```
```   164
```
```   165 lemma in_grcomp:
```
```   166   "has_edge (G * H) p b q
```
```   167   = (\<exists>k e e'. has_edge G p e k \<and> has_edge H k e' q \<and> b = e * e')"
```
```   168   by (cases G, cases H) (auto simp:graph_mult_def has_edge_def image_def)
```
```   169
```
```   170 lemma in_grunit:
```
```   171   "has_edge 1 p b q = (p = q \<and> b = 1)"
```
```   172   by (auto simp:graph_one_def has_edge_def)
```
```   173
```
```   174 instance graph :: (type, semigroup_mult) semigroup_mult
```
```   175 proof
```
```   176   fix G1 G2 G3 :: "('a,'b) graph"
```
```   177
```
```   178   show "G1 * G2 * G3 = G1 * (G2 * G3)"
```
```   179   proof (rule graph_ext, rule trans)
```
```   180     fix p J q
```
```   181     show "has_edge ((G1 * G2) * G3) p J q =
```
```   182       (\<exists>G i H j I.
```
```   183       has_edge G1 p G i
```
```   184       \<and> has_edge G2 i H j
```
```   185       \<and> has_edge G3 j I q
```
```   186       \<and> J = (G * H) * I)"
```
```   187       by (simp only:in_grcomp) blast
```
```   188     show "\<dots> = has_edge (G1 * (G2 * G3)) p J q"
```
```   189       by (simp only:in_grcomp mult_assoc) blast
```
```   190   qed
```
```   191 qed
```
```   192
```
```   193 fun grpow :: "nat \<Rightarrow> ('a::type, 'b::monoid_mult) graph \<Rightarrow> ('a, 'b) graph"
```
```   194 where
```
```   195   "grpow 0 A = 1"
```
```   196 | "grpow (Suc n) A = A * (grpow n A)"
```
```   197
```
```   198 instance graph :: (type, monoid_mult)
```
```   199   "{semiring_1,idem_add,recpower,star}"
```
```   200   graph_pow_def: "A ^ n == grpow n A"
```
```   201   graph_star_def: "star G == (SUP n. G ^ n)"
```
```   202 proof
```
```   203   fix a b c :: "('a, 'b) graph"
```
```   204
```
```   205   show "1 * a = a"
```
```   206     by (rule graph_ext) (auto simp:in_grcomp in_grunit)
```
```   207   show "a * 1 = a"
```
```   208     by (rule graph_ext) (auto simp:in_grcomp in_grunit)
```
```   209
```
```   210   show "(a + b) * c = a * c + b * c"
```
```   211     by (rule graph_ext, simp add:in_grcomp in_grplus) blast
```
```   212
```
```   213   show "a * (b + c) = a * b + a * c"
```
```   214     by (rule graph_ext, simp add:in_grcomp in_grplus) blast
```
```   215
```
```   216   show "(0::('a,'b) graph) \<noteq> 1" unfolding graph_zero_def graph_one_def
```
```   217     by simp
```
```   218
```
```   219   show "a + a = a" unfolding graph_plus_def by simp
```
```   220
```
```   221   show "a ^ 0 = 1" "\<And>n. a ^ (Suc n) = a * a ^ n"
```
```   222     unfolding graph_pow_def by simp_all
```
```   223 qed
```
```   224
```
```   225 lemma graph_leqI:
```
```   226   assumes "\<And>n e n'. has_edge G n e n' \<Longrightarrow> has_edge H n e n'"
```
```   227   shows "G \<le> H"
```
```   228   using assms
```
```   229   unfolding graph_leq_def has_edge_def
```
```   230   by auto
```
```   231
```
```   232 lemma in_graph_plusE:
```
```   233   assumes "has_edge (G + H) n e n'"
```
```   234   assumes "has_edge G n e n' \<Longrightarrow> P"
```
```   235   assumes "has_edge H n e n' \<Longrightarrow> P"
```
```   236   shows P
```
```   237   using assms
```
```   238   by (auto simp: in_grplus)
```
```   239
```
```   240 lemma in_graph_compE:
```
```   241   assumes GH: "has_edge (G * H) n e n'"
```
```   242   obtains e1 k e2
```
```   243   where "has_edge G n e1 k" "has_edge H k e2 n'" "e = e1 * e2"
```
```   244   using GH
```
```   245   by (auto simp: in_grcomp)
```
```   246
```
```   247 lemma
```
```   248   assumes "x \<in> S k"
```
```   249   shows "x \<in> (\<Union>k. S k)"
```
```   250   using assms by blast
```
```   251
```
```   252 lemma graph_union_least:
```
```   253   assumes "\<And>n. Graph (G n) \<le> C"
```
```   254   shows "Graph (\<Union>n. G n) \<le> C"
```
```   255   using assms unfolding graph_leq_def
```
```   256   by auto
```
```   257
```
```   258 lemma Sup_graph_eq:
```
```   259   "(SUP n. Graph (G n)) = Graph (\<Union>n. G n)"
```
```   260 proof (rule order_antisym)
```
```   261   show "(SUP n. Graph (G n)) \<le> Graph (\<Union>n. G n)"
```
```   262     by  (rule SUP_leI) (auto simp add: graph_leq_def)
```
```   263
```
```   264   show "Graph (\<Union>n. G n) \<le> (SUP n. Graph (G n))"
```
```   265   by (rule graph_union_least, rule le_SUPI', rule)
```
```   266 qed
```
```   267
```
```   268 lemma has_edge_leq: "has_edge G p b q = (Graph {(p,b,q)} \<le> G)"
```
```   269   unfolding has_edge_def graph_leq_def
```
```   270   by (cases G) simp
```
```   271
```
```   272
```
```   273 lemma Sup_graph_eq2:
```
```   274   "(SUP n. G n) = Graph (\<Union>n. dest_graph (G n))"
```
```   275   using Sup_graph_eq[of "\<lambda>n. dest_graph (G n)", simplified]
```
```   276   by simp
```
```   277
```
```   278 lemma in_SUP:
```
```   279   "has_edge (SUP x. Gs x) p b q = (\<exists>x. has_edge (Gs x) p b q)"
```
```   280   unfolding Sup_graph_eq2 has_edge_leq graph_leq_def
```
```   281   by simp
```
```   282
```
```   283 instance graph :: (type, monoid_mult) kleene_by_complete_lattice
```
```   284 proof
```
```   285   fix a b c :: "('a, 'b) graph"
```
```   286
```
```   287   show "a \<le> b \<longleftrightarrow> a + b = b" unfolding graph_leq_def graph_plus_def
```
```   288     by (cases a, cases b) auto
```
```   289
```
```   290   from order_less_le show "a < b \<longleftrightarrow> a \<le> b \<and> a \<noteq> b" .
```
```   291
```
```   292   show "a * star b * c = (SUP n. a * b ^ n * c)"
```
```   293     unfolding graph_star_def
```
```   294     by (rule graph_ext) (force simp:in_SUP in_grcomp)
```
```   295 qed
```
```   296
```
```   297
```
```   298 lemma in_star:
```
```   299   "has_edge (star G) a x b = (\<exists>n. has_edge (G ^ n) a x b)"
```
```   300   by (auto simp:graph_star_def in_SUP)
```
```   301
```
```   302 lemma tcl_is_SUP:
```
```   303   "tcl (G::('a::type, 'b::monoid_mult) graph) =
```
```   304   (SUP n. G ^ (Suc n))"
```
```   305   unfolding tcl_def
```
```   306   using star_cont[of 1 G G]
```
```   307   by (simp add:power_Suc power_commutes)
```
```   308
```
```   309
```
```   310 lemma in_tcl:
```
```   311   "has_edge (tcl G) a x b = (\<exists>n>0. has_edge (G ^ n) a x b)"
```
```   312   apply (auto simp: tcl_is_SUP in_SUP)
```
```   313   apply (rule_tac x = "n - 1" in exI, auto)
```
```   314   done
```
```   315
```
```   316
```
```   317 subsection {* Infinite Paths *}
```
```   318
```
```   319 types ('n, 'e) ipath = "('n \<times> 'e) sequence"
```
```   320
```
```   321 definition has_ipath :: "('n, 'e) graph \<Rightarrow> ('n, 'e) ipath \<Rightarrow> bool"
```
```   322 where
```
```   323   "has_ipath G p =
```
```   324   (\<forall>i. has_edge G (fst (p i)) (snd (p i)) (fst (p (Suc i))))"
```
```   325
```
```   326
```
```   327 subsection {* Finite Paths *}
```
```   328
```
```   329 types ('n, 'e) fpath = "('n \<times> ('e \<times> 'n) list)"
```
```   330
```
```   331 inductive  has_fpath :: "('n, 'e) graph \<Rightarrow> ('n, 'e) fpath \<Rightarrow> bool"
```
```   332   for G :: "('n, 'e) graph"
```
```   333 where
```
```   334   has_fpath_empty: "has_fpath G (n, [])"
```
```   335 | 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)"
```
```   336
```
```   337 definition
```
```   338   "end_node p =
```
```   339   (if snd p = [] then fst p else snd (snd p ! (length (snd p) - 1)))"
```
```   340
```
```   341 definition path_nth :: "('n, 'e) fpath \<Rightarrow> nat \<Rightarrow> ('n \<times> 'e \<times> 'n)"
```
```   342 where
```
```   343   "path_nth p k = (if k = 0 then fst p else snd (snd p ! (k - 1)), snd p ! k)"
```
```   344
```
```   345 lemma endnode_nth:
```
```   346   assumes "length (snd p) = Suc k"
```
```   347   shows "end_node p = snd (snd (path_nth p k))"
```
```   348   using assms unfolding end_node_def path_nth_def
```
```   349   by auto
```
```   350
```
```   351 lemma path_nth_graph:
```
```   352   assumes "k < length (snd p)"
```
```   353   assumes "has_fpath G p"
```
```   354   shows "(\<lambda>(n,e,n'). has_edge G n e n') (path_nth p k)"
```
```   355 using assms
```
```   356 proof (induct k arbitrary: p)
```
```   357   case 0 thus ?case
```
```   358     unfolding path_nth_def by (auto elim:has_fpath.cases)
```
```   359 next
```
```   360   case (Suc k p)
```
```   361
```
```   362   from `has_fpath G p` show ?case
```
```   363   proof (rule has_fpath.cases)
```
```   364     case goal1 with Suc show ?case by simp
```
```   365   next
```
```   366     fix n e n' es
```
```   367     assume st: "p = (n, (e, n') # es)"
```
```   368        "G \<turnstile> n \<leadsto>\<^bsup>e\<^esup> n'"
```
```   369        "has_fpath G (n', es)"
```
```   370     with Suc
```
```   371     have "(\<lambda>(n, b, a). G \<turnstile> n \<leadsto>\<^bsup>b\<^esup> a) (path_nth (n', es) k)" by simp
```
```   372     with st show ?thesis by (cases k, auto simp:path_nth_def)
```
```   373   qed
```
```   374 qed
```
```   375
```
```   376 lemma path_nth_connected:
```
```   377   assumes "Suc k < length (snd p)"
```
```   378   shows "fst (path_nth p (Suc k)) = snd (snd (path_nth p k))"
```
```   379   using assms
```
```   380   unfolding path_nth_def
```
```   381   by auto
```
```   382
```
```   383 definition path_loop :: "('n, 'e) fpath \<Rightarrow> ('n, 'e) ipath" ("omega")
```
```   384 where
```
```   385   "omega p \<equiv> (\<lambda>i. (\<lambda>(n,e,n'). (n,e)) (path_nth p (i mod (length (snd p)))))"
```
```   386
```
```   387 lemma fst_p0: "fst (path_nth p 0) = fst p"
```
```   388   unfolding path_nth_def by simp
```
```   389
```
```   390 lemma path_loop_connect:
```
```   391   assumes "fst p = end_node p"
```
```   392   and "0 < length (snd p)" (is "0 < ?l")
```
```   393   shows "fst (path_nth p (Suc i mod (length (snd p))))
```
```   394   = snd (snd (path_nth p (i mod length (snd p))))"
```
```   395   (is "\<dots> = snd (snd (path_nth p ?k))")
```
```   396 proof -
```
```   397   from `0 < ?l` have "i mod ?l < ?l" (is "?k < ?l")
```
```   398     by simp
```
```   399
```
```   400   show ?thesis
```
```   401   proof (cases "Suc ?k < ?l")
```
```   402     case True
```
```   403     hence "Suc ?k \<noteq> ?l" by simp
```
```   404     with path_nth_connected[OF True]
```
```   405     show ?thesis
```
```   406       by (simp add:mod_Suc)
```
```   407   next
```
```   408     case False
```
```   409     with `?k < ?l` have wrap: "Suc ?k = ?l" by simp
```
```   410
```
```   411     hence "fst (path_nth p (Suc i mod ?l)) = fst (path_nth p 0)"
```
```   412       by (simp add: mod_Suc)
```
```   413     also from fst_p0 have "\<dots> = fst p" .
```
```   414     also have "\<dots> = end_node p" by fact
```
```   415     also have "\<dots> = snd (snd (path_nth p ?k))"
```
```   416       by (auto simp: endnode_nth wrap)
```
```   417     finally show ?thesis .
```
```   418   qed
```
```   419 qed
```
```   420
```
```   421 lemma path_loop_graph:
```
```   422   assumes "has_fpath G p"
```
```   423   and loop: "fst p = end_node p"
```
```   424   and nonempty: "0 < length (snd p)" (is "0 < ?l")
```
```   425   shows "has_ipath G (omega p)"
```
```   426 proof -
```
```   427   {
```
```   428     fix i
```
```   429     from `0 < ?l` have "i mod ?l < ?l" (is "?k < ?l")
```
```   430       by simp
```
```   431     from this and `has_fpath G p`
```
```   432     have pk_G: "(\<lambda>(n,e,n'). has_edge G n e n') (path_nth p ?k)"
```
```   433       by (rule path_nth_graph)
```
```   434
```
```   435     from path_loop_connect[OF loop nonempty] pk_G
```
```   436     have "has_edge G (fst (omega p i)) (snd (omega p i)) (fst (omega p (Suc i)))"
```
```   437       unfolding path_loop_def has_edge_def split_def
```
```   438       by simp
```
```   439   }
```
```   440   then show ?thesis by (auto simp:has_ipath_def)
```
```   441 qed
```
```   442
```
```   443 definition prod :: "('n, 'e::monoid_mult) fpath \<Rightarrow> 'e"
```
```   444 where
```
```   445   "prod p = foldr (op *) (map fst (snd p)) 1"
```
```   446
```
```   447 lemma prod_simps[simp]:
```
```   448   "prod (n, []) = 1"
```
```   449   "prod (n, (e,n')#es) = e * (prod (n',es))"
```
```   450 unfolding prod_def
```
```   451 by simp_all
```
```   452
```
```   453 lemma power_induces_path:
```
```   454   assumes a: "has_edge (A ^ k) n G m"
```
```   455   obtains p
```
```   456     where "has_fpath A p"
```
```   457       and "n = fst p" "m = end_node p"
```
```   458       and "G = prod p"
```
```   459       and "k = length (snd p)"
```
```   460   using a
```
```   461 proof (induct k arbitrary:m n G thesis)
```
```   462   case (0 m n G)
```
```   463   let ?p = "(n, [])"
```
```   464   from 0 have "has_fpath A ?p" "m = end_node ?p" "G = prod ?p"
```
```   465     by (auto simp:in_grunit end_node_def intro:has_fpath.intros)
```
```   466   thus ?case using 0 by (auto simp:end_node_def)
```
```   467 next
```
```   468   case (Suc k m n G)
```
```   469   hence "has_edge (A * A ^ k) n G m"
```
```   470     by (simp add:power_Suc power_commutes)
```
```   471   then obtain G' H j where
```
```   472     a_A: "has_edge A n G' j"
```
```   473     and H_pow: "has_edge (A ^ k) j H m"
```
```   474     and [simp]: "G = G' * H"
```
```   475     by (auto simp:in_grcomp)
```
```   476
```
```   477   from H_pow and Suc
```
```   478   obtain p
```
```   479     where p_path: "has_fpath A p"
```
```   480     and [simp]: "j = fst p" "m = end_node p" "H = prod p"
```
```   481     "k = length (snd p)"
```
```   482     by blast
```
```   483
```
```   484   let ?p' = "(n, (G', j)#snd p)"
```
```   485   from a_A and p_path
```
```   486   have "has_fpath A ?p'" "m = end_node ?p'" "G = prod ?p'"
```
```   487     by (auto simp:end_node_def nth.simps intro:has_fpath.intros split:nat.split)
```
```   488   thus ?case using Suc by auto
```
```   489 qed
```
```   490
```
```   491
```
```   492 subsection {* Sub-Paths *}
```
```   493
```
```   494 definition sub_path :: "('n, 'e) ipath \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> ('n, 'e) fpath"
```
```   495 ("(_\<langle>_,_\<rangle>)")
```
```   496 where
```
```   497   "p\<langle>i,j\<rangle> =
```
```   498   (fst (p i), map (\<lambda>k. (snd (p k), fst (p (Suc k)))) [i ..< j])"
```
```   499
```
```   500
```
```   501 lemma sub_path_is_path:
```
```   502   assumes ipath: "has_ipath G p"
```
```   503   assumes l: "i \<le> j"
```
```   504   shows "has_fpath G (p\<langle>i,j\<rangle>)"
```
```   505   using l
```
```   506 proof (induct i rule:inc_induct)
```
```   507   case base show ?case by (auto simp:sub_path_def intro:has_fpath.intros)
```
```   508 next
```
```   509   case (step i)
```
```   510   with ipath upt_rec[of i j]
```
```   511   show ?case
```
```   512     by (auto simp:sub_path_def has_ipath_def intro:has_fpath.intros)
```
```   513 qed
```
```   514
```
```   515
```
```   516 lemma sub_path_start[simp]:
```
```   517   "fst (p\<langle>i,j\<rangle>) = fst (p i)"
```
```   518   by (simp add:sub_path_def)
```
```   519
```
```   520 lemma nth_upto[simp]: "k < j - i \<Longrightarrow> [i ..< j] ! k = i + k"
```
```   521   by (induct k) auto
```
```   522
```
```   523 lemma sub_path_end[simp]:
```
```   524   "i < j \<Longrightarrow> end_node (p\<langle>i,j\<rangle>) = fst (p j)"
```
```   525   by (auto simp:sub_path_def end_node_def)
```
```   526
```
```   527 lemma foldr_map: "foldr f (map g xs) = foldr (f o g) xs"
```
```   528   by (induct xs) auto
```
```   529
```
```   530 lemma upto_append[simp]:
```
```   531   assumes "i \<le> j" "j \<le> k"
```
```   532   shows "[ i ..< j ] @ [j ..< k] = [i ..< k]"
```
```   533   using assms and upt_add_eq_append[of i j "k - j"]
```
```   534   by simp
```
```   535
```
```   536 lemma foldr_monoid: "foldr (op *) xs 1 * foldr (op *) ys 1
```
```   537   = foldr (op *) (xs @ ys) (1::'a::monoid_mult)"
```
```   538   by (induct xs) (auto simp:mult_assoc)
```
```   539
```
```   540 lemma sub_path_prod:
```
```   541   assumes "i < j"
```
```   542   assumes "j < k"
```
```   543   shows "prod (p\<langle>i,k\<rangle>) = prod (p\<langle>i,j\<rangle>) * prod (p\<langle>j,k\<rangle>)"
```
```   544   using assms
```
```   545   unfolding prod_def sub_path_def
```
```   546   by (simp add:map_compose[symmetric] comp_def)
```
```   547    (simp only:foldr_monoid map_append[symmetric] upto_append)
```
```   548
```
```   549
```
```   550 lemma path_acgpow_aux:
```
```   551   assumes "length es = l"
```
```   552   assumes "has_fpath G (n,es)"
```
```   553   shows "has_edge (G ^ l) n (prod (n,es)) (end_node (n,es))"
```
```   554 using assms
```
```   555 proof (induct l arbitrary:n es)
```
```   556   case 0 thus ?case
```
```   557     by (simp add:in_grunit end_node_def)
```
```   558 next
```
```   559   case (Suc l n es)
```
```   560   hence "es \<noteq> []" by auto
```
```   561   let ?n' = "snd (hd es)"
```
```   562   let ?es' = "tl es"
```
```   563   let ?e = "fst (hd es)"
```
```   564
```
```   565   from Suc have len: "length ?es' = l" by auto
```
```   566
```
```   567   from Suc
```
```   568   have [simp]: "end_node (n, es) = end_node (?n', ?es')"
```
```   569     by (cases es) (auto simp:end_node_def nth.simps split:nat.split)
```
```   570
```
```   571   from `has_fpath G (n,es)`
```
```   572   have "has_fpath G (?n', ?es')"
```
```   573     by (rule has_fpath.cases) (auto intro:has_fpath.intros)
```
```   574   with Suc len
```
```   575   have "has_edge (G ^ l) ?n' (prod (?n', ?es')) (end_node (?n', ?es'))"
```
```   576     by auto
```
```   577   moreover
```
```   578   from `es \<noteq> []`
```
```   579   have "prod (n, es) = ?e * (prod (?n', ?es'))"
```
```   580     by (cases es) auto
```
```   581   moreover
```
```   582   from `has_fpath G (n,es)` have c:"has_edge G n ?e ?n'"
```
```   583     by (rule has_fpath.cases) (insert `es \<noteq> []`, auto)
```
```   584
```
```   585   ultimately
```
```   586   show ?case
```
```   587      unfolding power_Suc
```
```   588      by (auto simp:in_grcomp)
```
```   589 qed
```
```   590
```
```   591
```
```   592 lemma path_acgpow:
```
```   593    "has_fpath G p
```
```   594   \<Longrightarrow> has_edge (G ^ length (snd p)) (fst p) (prod p) (end_node p)"
```
```   595 by (cases p)
```
```   596    (rule path_acgpow_aux[of "snd p" "length (snd p)" _ "fst p", simplified])
```
```   597
```
```   598
```
```   599 lemma star_paths:
```
```   600   "has_edge (star G) a x b =
```
```   601    (\<exists>p. has_fpath G p \<and> a = fst p \<and> b = end_node p \<and> x = prod p)"
```
```   602 proof
```
```   603   assume "has_edge (star G) a x b"
```
```   604   then obtain n where pow: "has_edge (G ^ n) a x b"
```
```   605     by (auto simp:in_star)
```
```   606
```
```   607   then obtain p where
```
```   608     "has_fpath G p" "a = fst p" "b = end_node p" "x = prod p"
```
```   609     by (rule power_induces_path)
```
```   610
```
```   611   thus "\<exists>p. has_fpath G p \<and> a = fst p \<and> b = end_node p \<and> x = prod p"
```
```   612     by blast
```
```   613 next
```
```   614   assume "\<exists>p. has_fpath G p \<and> a = fst p \<and> b = end_node p \<and> x = prod p"
```
```   615   then obtain p where
```
```   616     "has_fpath G p" "a = fst p" "b = end_node p" "x = prod p"
```
```   617     by blast
```
```   618
```
```   619   hence "has_edge (G ^ length (snd p)) a x b"
```
```   620     by (auto intro:path_acgpow)
```
```   621
```
```   622   thus "has_edge (star G) a x b"
```
```   623     by (auto simp:in_star)
```
```   624 qed
```
```   625
```
```   626
```
```   627 lemma plus_paths:
```
```   628   "has_edge (tcl G) a x b =
```
```   629    (\<exists>p. has_fpath G p \<and> a = fst p \<and> b = end_node p \<and> x = prod p \<and> 0 < length (snd p))"
```
```   630 proof
```
```   631   assume "has_edge (tcl G) a x b"
```
```   632
```
```   633   then obtain n where pow: "has_edge (G ^ n) a x b" and "0 < n"
```
```   634     by (auto simp:in_tcl)
```
```   635
```
```   636   from pow obtain p where
```
```   637     "has_fpath G p" "a = fst p" "b = end_node p" "x = prod p"
```
```   638     "n = length (snd p)"
```
```   639     by (rule power_induces_path)
```
```   640
```
```   641   with `0 < n`
```
```   642   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) "
```
```   643     by blast
```
```   644 next
```
```   645   assume "\<exists>p. has_fpath G p \<and> a = fst p \<and> b = end_node p \<and> x = prod p
```
```   646     \<and> 0 < length (snd p)"
```
```   647   then obtain p where
```
```   648     "has_fpath G p" "a = fst p" "b = end_node p" "x = prod p"
```
```   649     "0 < length (snd p)"
```
```   650     by blast
```
```   651
```
```   652   hence "has_edge (G ^ length (snd p)) a x b"
```
```   653     by (auto intro:path_acgpow)
```
```   654
```
```   655   with `0 < length (snd p)`
```
```   656   show "has_edge (tcl G) a x b"
```
```   657     by (auto simp:in_tcl)
```
```   658 qed
```
```   659
```
```   660
```
```   661 definition
```
```   662   "contract s p =
```
```   663   (\<lambda>i. (fst (p (s i)), prod (p\<langle>s i,s (Suc i)\<rangle>)))"
```
```   664
```
```   665 lemma ipath_contract:
```
```   666   assumes [simp]: "increasing s"
```
```   667   assumes ipath: "has_ipath G p"
```
```   668   shows "has_ipath (tcl G) (contract s p)"
```
```   669   unfolding has_ipath_def
```
```   670 proof
```
```   671   fix i
```
```   672   let ?p = "p\<langle>s i,s (Suc i)\<rangle>"
```
```   673
```
```   674   from increasing_strict
```
```   675 	have "fst (p (s (Suc i))) = end_node ?p" by simp
```
```   676   moreover
```
```   677   from increasing_strict[of s i "Suc i"] have "snd ?p \<noteq> []"
```
```   678     by (simp add:sub_path_def)
```
```   679   moreover
```
```   680   from ipath increasing_weak[of s] have "has_fpath G ?p"
```
```   681     by (rule sub_path_is_path) auto
```
```   682   ultimately
```
```   683   show "has_edge (tcl G)
```
```   684     (fst (contract s p i)) (snd (contract s p i)) (fst (contract s p (Suc i)))"
```
```   685     unfolding contract_def plus_paths
```
```   686     by (intro exI) auto
```
```   687 qed
```
```   688
```
```   689 lemma prod_unfold:
```
```   690   "i < j \<Longrightarrow> prod (p\<langle>i,j\<rangle>)
```
```   691   = snd (p i) * prod (p\<langle>Suc i, j\<rangle>)"
```
```   692   unfolding prod_def
```
```   693   by (simp add:sub_path_def upt_rec[of "i" j])
```
```   694
```
```   695
```
```   696 lemma sub_path_loop:
```
```   697   assumes "0 < k"
```
```   698   assumes k: "k = length (snd loop)"
```
```   699   assumes loop: "fst loop = end_node loop"
```
```   700   shows "(omega loop)\<langle>k * i,k * Suc i\<rangle> = loop" (is "?\<omega> = loop")
```
```   701 proof (rule prod_eqI)
```
```   702   show "fst ?\<omega> = fst loop"
```
```   703     by (auto simp:path_loop_def path_nth_def split_def k)
```
```   704
```
```   705   show "snd ?\<omega> = snd loop"
```
```   706   proof (rule nth_equalityI[rule_format])
```
```   707     show leneq: "length (snd ?\<omega>) = length (snd loop)"
```
```   708       unfolding sub_path_def k by simp
```
```   709
```
```   710     fix j assume "j < length (snd (?\<omega>))"
```
```   711     with leneq and k have "j < k" by simp
```
```   712
```
```   713     have a: "\<And>i. fst (path_nth loop (Suc i mod k))
```
```   714       = snd (snd (path_nth loop (i mod k)))"
```
```   715       unfolding k
```
```   716       apply (rule path_loop_connect[OF loop])
```
```   717       using `0 < k` and k
```
```   718       apply auto
```
```   719       done
```
```   720
```
```   721     from `j < k`
```
```   722     show "snd ?\<omega> ! j = snd loop ! j"
```
```   723       unfolding sub_path_def
```
```   724       apply (simp add:path_loop_def split_def add_ac)
```
```   725       apply (simp add:a k[symmetric])
```
```   726       apply (simp add:path_nth_def)
```
```   727       done
```
```   728   qed
```
```   729 qed
```
```   730
```
```   731 end
```