Homework 4.2

Definitions File

theory Defs
imports Main
begin

inductive is_path :: "('v \<Rightarrow> 'v \<Rightarrow> bool) \<Rightarrow> 'v \<Rightarrow> 'v list \<Rightarrow> 'v \<Rightarrow> bool"
for E where
  NilI: "is_path E u [] u"
| ConsI: "\<lbrakk> E u v; is_path E v l w \<rbrakk> \<Longrightarrow> is_path E u (u#l) w"

fun path :: "('v \<Rightarrow> 'v \<Rightarrow> bool) \<Rightarrow> 'v \<Rightarrow> 'v list \<Rightarrow> 'v \<Rightarrow> bool" where
  "path E u [] v \<longleftrightarrow> v = u" |
  "path E u [v] w \<longleftrightarrow> u = v \<and> E v w" |
  "path E u (x # y # xs) v \<longleftrightarrow> (u = x \<and> E x y \<and> path E y (y # xs) v)"

lemma path_Nil: "is_path E u [] v \<longleftrightarrow> u=v"
  by (auto intro: is_path.intros elim: is_path.cases)

lemma path_Cons: "is_path E u (v#p) w \<longleftrightarrow> 
  (\<exists>vh. u=v \<and> E v vh \<and> is_path E vh p w)"
  by (auto intro: is_path.intros elim: is_path.cases)

end

Template File

theory Submission
  imports Defs
begin

theorem path_distinct:
  assumes "is_path E u p v"
  shows "\<exists>p'. distinct p' \<and> is_path E u p' v"
  using assms
proof (induction p rule: length_induct)
  case step: (1 p)
  note IH = step.IH
  note prems = step.prems
  show ?case proof (cases "distinct p")
    case True
    then show ?thesis sorry
  next
    case False
    then show ?thesis sorry
  qed
qed

end

Check File

theory Check
  imports Submission
begin

theorem path_distinct:
  assumes "is_path E u p v"
  shows "\<exists>p'. distinct p' \<and> is_path E u p' v"
  using assms by (rule Submission.path_distinct)

end