(* Title: HOL/Metis_Examples/Trans_Closure.thy
Author: Lawrence C. Paulson, Cambridge University Computer Laboratory
Author: Jasmin Blanchette, TU Muenchen
Metis example featuring the transitive closure.
*)
section \<open>Metis Example Featuring the Transitive Closure\<close>
theory Trans_Closure
imports Main
begin
declare [[metis_new_skolem]]
type_synonym addr = nat
datatype val
= Unit \<comment> \<open>dummy result value of void expressions\<close>
| Null \<comment> \<open>null reference\<close>
| Bool bool \<comment> \<open>Boolean value\<close>
| Intg int \<comment> \<open>integer value\<close>
| Addr addr \<comment> \<open>addresses of objects in the heap\<close>
consts R :: "(addr \<times> addr) set"
consts f :: "addr \<Rightarrow> val"
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>
\<Longrightarrow> \<exists>c. (b, c) \<in> R \<and> (a, c) \<in> R\<^sup>*"
(* sledgehammer *)
proof -
assume A1: "f c = Intg x"
assume A2: "\<forall>y. f b = Intg y \<longrightarrow> y \<noteq> x"
assume A3: "(a, b) \<in> R\<^sup>*"
assume A4: "(b, c) \<in> R\<^sup>*"
have F1: "f c \<noteq> f b" using A2 A1 by metis
have F2: "\<forall>u. (b, u) \<in> R \<longrightarrow> (a, u) \<in> R\<^sup>*" using A3 by (metis transitive_closure_trans(6))
have F3: "\<exists>x. (b, x b c R) \<in> R \<or> c = b" using A4 by (metis converse_rtranclE)
have "c \<noteq> b" using F1 by metis
hence "\<exists>u. (b, u) \<in> R" using F3 by metis
thus "\<exists>c. (b, c) \<in> R \<and> (a, c) \<in> R\<^sup>*" using F2 by metis
qed
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>
\<Longrightarrow> \<exists>c. (b, c) \<in> R \<and> (a, c) \<in> R\<^sup>*"
(* sledgehammer [isar_proofs, compress = 2] *)
proof -
assume A1: "f c = Intg x"
assume A2: "\<forall>y. f b = Intg y \<longrightarrow> y \<noteq> x"
assume A3: "(a, b) \<in> R\<^sup>*"
assume A4: "(b, c) \<in> R\<^sup>*"
have "b \<noteq> c" using A1 A2 by metis
hence "\<exists>x\<^sub>1. (b, x\<^sub>1) \<in> R" using A4 by (metis converse_rtranclE)
thus "\<exists>c. (b, c) \<in> R \<and> (a, c) \<in> R\<^sup>*" using A3 by (metis transitive_closure_trans(6))
qed
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>
\<Longrightarrow> \<exists>c. (b, c) \<in> R \<and> (a, c) \<in> R\<^sup>*"
apply (erule_tac x = b in converse_rtranclE)
apply metis
by (metis transitive_closure_trans(6))
end