--- a/src/HOL/SizeChange/Graphs.thy Wed Jan 02 15:14:15 2008 +0100
+++ b/src/HOL/SizeChange/Graphs.thy Wed Jan 02 15:14:17 2008 +0100
@@ -14,7 +14,7 @@
datatype ('a, 'b) graph =
Graph "('a \<times> 'b \<times> 'a) set"
-fun dest_graph :: "('a, 'b) graph \<Rightarrow> ('a \<times> 'b \<times> 'a) set"
+primrec dest_graph :: "('a, 'b) graph \<Rightarrow> ('a \<times> 'b \<times> 'a) set"
where "dest_graph (Graph G) = G"
lemma graph_dest_graph[simp]:
@@ -60,29 +60,47 @@
by (cases G, cases H) (auto simp:split_paired_all has_edge_def)
-instance graph :: (type, type) "{comm_monoid_add}"
- graph_zero_def: "0 == Graph {}"
- graph_plus_def: "G + H == Graph (dest_graph G \<union> dest_graph H)"
-proof
+instantiation graph :: (type, type) comm_monoid_add
+begin
+
+definition
+ graph_zero_def: "0 = Graph {}"
+
+definition
+ graph_plus_def [code func del]: "G + H = Graph (dest_graph G \<union> dest_graph H)"
+
+instance proof
fix x y z :: "('a,'b) graph"
-
show "x + y + z = x + (y + z)"
and "x + y = y + x"
and "0 + x = x"
- unfolding graph_plus_def graph_zero_def
- by auto
+ unfolding graph_plus_def graph_zero_def by auto
qed
-lemmas [code func del] = graph_plus_def
+end
+
+instantiation graph :: (type, type) "{distrib_lattice, complete_lattice}"
+begin
+
+definition
+ graph_leq_def [code func del]: "G \<le> H \<longleftrightarrow> dest_graph G \<subseteq> dest_graph H"
+
+definition
+ graph_less_def [code func del]: "G < H \<longleftrightarrow> dest_graph G \<subset> dest_graph H"
-instance graph :: (type, type) "{distrib_lattice, complete_lattice}"
- graph_leq_def: "G \<le> H \<equiv> dest_graph G \<subseteq> dest_graph H"
- graph_less_def: "G < H \<equiv> dest_graph G \<subset> dest_graph H"
- "inf G H \<equiv> Graph (dest_graph G \<inter> dest_graph H)"
- "sup (G \<Colon> ('a, 'b) graph) H \<equiv> G + H"
- Inf_graph_def: "Inf \<equiv> \<lambda>Gs. Graph (\<Inter>(dest_graph ` Gs))"
- Sup_graph_def: "Sup \<equiv> \<lambda>Gs. Graph (\<Union>(dest_graph ` Gs))"
-proof
+definition
+ [code func del]: "inf G H = Graph (dest_graph G \<inter> dest_graph H)"
+
+definition
+ [code func del]: "sup (G \<Colon> ('a, 'b) graph) H = G + H"
+
+definition
+ Inf_graph_def [code func del]: "Inf = (\<lambda>Gs. Graph (\<Inter>(dest_graph ` Gs)))"
+
+definition
+ Sup_graph_def [code func del]: "Sup = (\<lambda>Gs. Graph (\<Union>(dest_graph ` Gs)))"
+
+instance proof
fix x y z :: "('a,'b) graph"
fix A :: "('a, 'b) graph set"
@@ -130,8 +148,7 @@
unfolding Sup_graph_def graph_leq_def by auto }
qed
-lemmas [code func del] = graph_leq_def graph_less_def
- inf_graph_def sup_graph_def Inf_graph_def Sup_graph_def
+end
lemma in_grplus:
"has_edge (G + H) p b q = (has_edge G p b q \<or> has_edge H p b q)"
@@ -144,9 +161,13 @@
subsubsection {* Multiplicative Structure *}
-instance graph :: (type, times) mult_zero
- graph_mult_def: "G * H == grcomp G H"
-proof
+instantiation graph :: (type, times) mult_zero
+begin
+
+definition
+ graph_mult_def [code func del]: "G * H = grcomp G H"
+
+instance proof
fix a :: "('a, 'b) graph"
show "0 * a = 0"
@@ -157,10 +178,17 @@
by (cases a) (simp add:grcomp.simps)
qed
-lemmas [code func del] = graph_mult_def
+end
+
+instantiation graph :: (type, one) one
+begin
-instance graph :: (type, one) one
- graph_one_def: "1 == Graph { (x, 1, x) |x. True}" ..
+definition
+ graph_one_def: "1 = Graph { (x, 1, x) |x. True}"
+
+instance ..
+
+end
lemma in_grcomp:
"has_edge (G * H) p b q
@@ -190,16 +218,18 @@
qed
qed
-fun grpow :: "nat \<Rightarrow> ('a::type, 'b::monoid_mult) graph \<Rightarrow> ('a, 'b) graph"
-where
- "grpow 0 A = 1"
-| "grpow (Suc n) A = A * (grpow n A)"
+instantiation graph :: (type, monoid_mult) "{semiring_1, idem_add, recpower, star}"
+begin
-instance graph :: (type, monoid_mult)
- "{semiring_1,idem_add,recpower,star}"
- graph_pow_def: "A ^ n == grpow n A"
- graph_star_def: "star (G \<Colon> ('a, 'b) graph) == (SUP n. G ^ n)"
-proof
+primrec power_graph :: "('a\<Colon>type, 'b\<Colon>monoid_mult) graph \<Rightarrow> nat => ('a, 'b) graph"
+where
+ "(A \<Colon> ('a, 'b) graph) ^ 0 = 1"
+| "(A \<Colon> ('a, 'b) graph) ^ Suc n = A * (A ^ n)"
+
+definition
+ graph_star_def: "star (G \<Colon> ('a, 'b) graph) = (SUP n. G ^ n)"
+
+instance proof
fix a b c :: "('a, 'b) graph"
show "1 * a = a"
@@ -219,9 +249,11 @@
show "a + a = a" unfolding graph_plus_def by simp
show "a ^ 0 = 1" "\<And>n. a ^ (Suc n) = a * a ^ n"
- unfolding graph_pow_def by simp_all
+ by simp_all
qed
+end
+
lemma graph_leqI:
assumes "\<And>n e n'. has_edge G n e n' \<Longrightarrow> has_edge H n e n'"
shows "G \<le> H"
@@ -309,7 +341,7 @@
lemma in_tcl:
"has_edge (tcl G) a x b = (\<exists>n>0. has_edge (G ^ n) a x b)"
- apply (auto simp: tcl_is_SUP in_SUP)
+ apply (auto simp: tcl_is_SUP in_SUP simp del: power_graph.simps)
apply (rule_tac x = "n - 1" in exI, auto)
done