theory Inductive_Examples

imports Simple_Inductive_Package

begin

section {* Transitive closure *}

simple_inductive

  trcl for r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"

where

  base: "trcl r x x"

| step: "trcl r x y \<Longrightarrow> r y z \<Longrightarrow> trcl r x z"

thm trcl_def

thm trcl.induct

thm base

thm step

thm trcl.intros

lemma trcl_strong_induct:

  assumes trcl: "trcl r x y"

  and I1: "\<And>x. P x x"

  and I2: "\<And>x y z. P x y \<Longrightarrow> trcl r x y \<Longrightarrow> r y z \<Longrightarrow> P x z"

  shows "P x y" 

proof -

  from trcl

  have "P x y \<and> trcl r x y"

  proof induct

    case (base x)

    from I1 and trcl.base show ?case ..

  next

    case (step x y z)

    then have "P x y" and "trcl r x y" by simp_all

    from `P x y` `trcl r x y` `r y z` have "P x z"

      by (rule I2)

    moreover from `trcl r x y` `r y z` have "trcl r x z"

      by (rule trcl.step)

    ultimately show ?case ..

  qed

  then show ?thesis ..

qed

section {* Even and odd numbers *}

simple_inductive

  even and odd

where

  even0: "even 0"

| evenS: "odd n \<Longrightarrow> even (Suc n)"

| oddS: "even n \<Longrightarrow> odd (Suc n)"

thm even_def odd_def

thm even.induct odd.induct

thm even0

thm evenS

thm oddS

thm even_odd.intros

section {* Accessible part *}

simple_inductive

  accpart for r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"

where

  accpartI: "(\<And>y. r y x \<Longrightarrow> accpart r y) \<Longrightarrow> accpart r x"

thm accpart_def

thm accpart.induct

thm accpartI

section {* Accessible part in locale *}

locale rel =

  fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"

simple_inductive (in rel) accpart'

where

  accpartI': "\<And>x. (\<And>y. r y x \<Longrightarrow> accpart' y) \<Longrightarrow> accpart' x"

context rel

begin

thm accpartI'

thm accpart'.induct

end

thm rel.accpartI'

thm rel.accpart'.induct

end