(* Title: HOL/Library/SCT_Implementation.thy
ID: $Id$
Author: Alexander Krauss, TU Muenchen
*)
header ""
theory SCT_Implementation
imports ExecutableSet SCT_Definition
begin
fun edges_match :: "('n \<times> 'e \<times> 'n) \<times> ('n \<times> 'e \<times> 'n) \<Rightarrow> bool"
where
"edges_match ((n, e, m), (n',e',m')) = (m = n')"
fun connect_edges ::
"('n \<times> ('e::times) \<times> 'n) \<times> ('n \<times> 'e \<times> 'n)
\<Rightarrow> ('n \<times> 'e \<times> 'n)"
where
"connect_edges ((n,e,m), (n', e', m')) = (n, e * e', m')"
lemma grcomp_code [code]:
"grcomp (Graph G) (Graph H) = Graph (connect_edges ` { x \<in> G\<times>H. edges_match x })"
by (rule graph_ext) (auto simp:graph_mult_def has_edge_def image_def)
definition test_SCT :: "acg \<Rightarrow> bool"
where
"test_SCT \<A> =
(let \<T> = mk_tcl \<A> \<A>
in (\<T> \<noteq> 0 \<and>
(\<forall>(n,G,m)\<in>dest_graph \<T>.
n \<noteq> m \<or> G * G \<noteq> G \<or>
(\<exists>(p::nat,e,q)\<in>dest_graph G. p = q \<and> e = LESS))))"
lemma SCT'_exec:
assumes a: "test_SCT \<A>"
shows "SCT' \<A>"
proof -
from mk_tcl_correctness2 a
have "mk_tcl \<A> \<A> = tcl \<A>"
unfolding test_SCT_def Let_def by auto
with a
show ?thesis
unfolding SCT'_def no_bad_graphs_def test_SCT_def Let_def has_edge_def
by auto
qed
code_modulename SML
Implementation Graphs
lemma [code func]:
"(G\<Colon>('a\<Colon>eq, 'b\<Colon>eq) graph) \<le> H \<longleftrightarrow> dest_graph G \<subseteq> dest_graph H"
"(G\<Colon>('a\<Colon>eq, 'b\<Colon>eq) graph) < H \<longleftrightarrow> dest_graph G \<subset> dest_graph H"
unfolding graph_leq_def graph_less_def by rule+
lemma [code func]:
"(G\<Colon>('a\<Colon>eq, 'b\<Colon>eq) graph) + H = Graph (dest_graph G \<union> dest_graph H)"
unfolding graph_plus_def ..
lemma [code func]:
"(G\<Colon>('a\<Colon>eq, 'b\<Colon>{eq, times}) graph) * H = grcomp G H"
unfolding graph_mult_def ..
lemma SCT'_empty: "SCT' (Graph {})"
unfolding SCT'_def no_bad_graphs_def graph_zero_def[symmetric]
tcl_zero
by (simp add:in_grzero)
subsection {* Witness checking *}
definition test_SCT_witness :: "acg \<Rightarrow> acg \<Rightarrow> bool"
where
"test_SCT_witness A T =
(A \<le> T \<and> A * T \<le> T \<and>
(\<forall>(n,G,m)\<in>dest_graph T.
n \<noteq> m \<or> G * G \<noteq> G \<or>
(\<exists>(p::nat,e,q)\<in>dest_graph G. p = q \<and> e = LESS)))"
lemma no_bad_graphs_ucl:
assumes "A \<le> B"
assumes "no_bad_graphs B"
shows "no_bad_graphs A"
using prems
unfolding no_bad_graphs_def has_edge_def graph_leq_def
by blast
lemma SCT'_witness:
assumes a: "test_SCT_witness A T"
shows "SCT' A"
proof -
from a have "A \<le> T" "A * T \<le> T" by (auto simp:test_SCT_witness_def)
hence "A + A * T \<le> T"
by (subst add_idem[of T, symmetric], rule add_mono)
with star3' have "tcl A \<le> T" unfolding tcl_def .
moreover
from a have "no_bad_graphs T"
unfolding no_bad_graphs_def test_SCT_witness_def has_edge_def
by auto
ultimately
show ?thesis
unfolding SCT'_def
by (rule no_bad_graphs_ucl)
qed
code_modulename SML
Graphs SCT
Kleene_Algebras SCT
SCT_Implementation SCT
code_gen test_SCT (SML #)
end