# Theory Transitive_Closure

theory Transitive_Closure
imports Relation
```(*  Title:      HOL/Transitive_Closure.thy
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright   1992  University of Cambridge
*)

section ‹Reflexive and Transitive closure of a relation›

theory Transitive_Closure
imports Relation
begin

ML_file "~~/src/Provers/trancl.ML"

text ‹
‹rtrancl› is reflexive/transitive closure,
‹trancl› is transitive closure,
‹reflcl› is reflexive closure.

These postfix operators have ∗‹maximum priority›, forcing their
operands to be atomic.
›

context notes [[inductive_internals]]
begin

inductive_set rtrancl :: "('a × 'a) set ⇒ ('a × 'a) set"  ("(_⇧*)" [1000] 999)
for r :: "('a × 'a) set"
where
rtrancl_refl [intro!, Pure.intro!, simp]: "(a, a) ∈ r⇧*"
| rtrancl_into_rtrancl [Pure.intro]: "(a, b) ∈ r⇧* ⟹ (b, c) ∈ r ⟹ (a, c) ∈ r⇧*"

inductive_set trancl :: "('a × 'a) set ⇒ ('a × 'a) set"  ("(_⇧+)" [1000] 999)
for r :: "('a × 'a) set"
where
r_into_trancl [intro, Pure.intro]: "(a, b) ∈ r ⟹ (a, b) ∈ r⇧+"
| trancl_into_trancl [Pure.intro]: "(a, b) ∈ r⇧+ ⟹ (b, c) ∈ r ⟹ (a, c) ∈ r⇧+"

notation
rtranclp  ("(_⇧*⇧*)" [1000] 1000) and
tranclp  ("(_⇧+⇧+)" [1000] 1000)

declare
rtrancl_def [nitpick_unfold del]
rtranclp_def [nitpick_unfold del]
trancl_def [nitpick_unfold del]
tranclp_def [nitpick_unfold del]

end

abbreviation reflcl :: "('a × 'a) set ⇒ ('a × 'a) set"  ("(_⇧=)" [1000] 999)
where "r⇧= ≡ r ∪ Id"

abbreviation reflclp :: "('a ⇒ 'a ⇒ bool) ⇒ 'a ⇒ 'a ⇒ bool"  ("(_⇧=⇧=)" [1000] 1000)
where "r⇧=⇧= ≡ sup r op ="

notation (ASCII)
rtrancl  ("(_^*)" [1000] 999) and
trancl  ("(_^+)" [1000] 999) and
reflcl  ("(_^=)" [1000] 999) and
rtranclp  ("(_^**)" [1000] 1000) and
tranclp  ("(_^++)" [1000] 1000) and
reflclp  ("(_^==)" [1000] 1000)

subsection ‹Reflexive closure›

lemma refl_reflcl[simp]: "refl (r⇧=)"
by (simp add: refl_on_def)

lemma antisym_reflcl[simp]: "antisym (r⇧=) = antisym r"
by (simp add: antisym_def)

lemma trans_reflclI[simp]: "trans r ⟹ trans (r⇧=)"
unfolding trans_def by blast

lemma reflclp_idemp [simp]: "(P⇧=⇧=)⇧=⇧= = P⇧=⇧="
by blast

subsection ‹Reflexive-transitive closure›

lemma reflcl_set_eq [pred_set_conv]: "(sup (λx y. (x, y) ∈ r) op =) = (λx y. (x, y) ∈ r ∪ Id)"
by (auto simp add: fun_eq_iff)

lemma r_into_rtrancl [intro]: "⋀p. p ∈ r ⟹ p ∈ r⇧*"
― ‹‹rtrancl› of ‹r› contains ‹r››
apply (simp only: split_tupled_all)
apply (erule rtrancl_refl [THEN rtrancl_into_rtrancl])
done

lemma r_into_rtranclp [intro]: "r x y ⟹ r⇧*⇧* x y"
― ‹‹rtrancl› of ‹r› contains ‹r››
by (erule rtranclp.rtrancl_refl [THEN rtranclp.rtrancl_into_rtrancl])

lemma rtranclp_mono: "r ≤ s ⟹ r⇧*⇧* ≤ s⇧*⇧*"
― ‹monotonicity of ‹rtrancl››
apply (rule predicate2I)
apply (erule rtranclp.induct)
apply (rule_tac [2] rtranclp.rtrancl_into_rtrancl, blast+)
done

lemma mono_rtranclp[mono]: "(⋀a b. x a b ⟶ y a b) ⟹ x⇧*⇧* a b ⟶ y⇧*⇧* a b"
using rtranclp_mono[of x y] by auto

lemmas rtrancl_mono = rtranclp_mono [to_set]

theorem rtranclp_induct [consumes 1, case_names base step, induct set: rtranclp]:
assumes a: "r⇧*⇧* a b"
and cases: "P a" "⋀y z. r⇧*⇧* a y ⟹ r y z ⟹ P y ⟹ P z"
shows "P b"
using a by (induct x≡a b) (rule cases)+

lemmas rtrancl_induct [induct set: rtrancl] = rtranclp_induct [to_set]

lemmas rtranclp_induct2 =
rtranclp_induct[of _ "(ax,ay)" "(bx,by)", split_rule, consumes 1, case_names refl step]

lemmas rtrancl_induct2 =
rtrancl_induct[of "(ax,ay)" "(bx,by)", split_format (complete), consumes 1, case_names refl step]

lemma refl_rtrancl: "refl (r⇧*)"
unfolding refl_on_def by fast

text ‹Transitivity of transitive closure.›
lemma trans_rtrancl: "trans (r⇧*)"
proof (rule transI)
fix x y z
assume "(x, y) ∈ r⇧*"
assume "(y, z) ∈ r⇧*"
then show "(x, z) ∈ r⇧*"
proof induct
case base
show "(x, y) ∈ r⇧*" by fact
next
case (step u v)
from ‹(x, u) ∈ r⇧*› and ‹(u, v) ∈ r›
show "(x, v) ∈ r⇧*" ..
qed
qed

lemmas rtrancl_trans = trans_rtrancl [THEN transD]

lemma rtranclp_trans:
assumes "r⇧*⇧* x y"
and "r⇧*⇧* y z"
shows "r⇧*⇧* x z"
using assms(2,1) by induct iprover+

lemma rtranclE [cases set: rtrancl]:
fixes a b :: 'a
assumes major: "(a, b) ∈ r⇧*"
obtains
(base) "a = b"
| (step) y where "(a, y) ∈ r⇧*" and "(y, b) ∈ r"
― ‹elimination of ‹rtrancl› -- by induction on a special formula›
apply (subgoal_tac "a = b ∨ (∃y. (a, y) ∈ r⇧* ∧ (y, b) ∈ r)")
apply (rule_tac [2] major [THEN rtrancl_induct])
prefer 2 apply blast
prefer 2 apply blast
apply (erule asm_rl exE disjE conjE base step)+
done

lemma rtrancl_Int_subset: "Id ⊆ s ⟹ (r⇧* ∩ s) O r ⊆ s ⟹ r⇧* ⊆ s"
apply (rule subsetI)
apply auto
apply (erule rtrancl_induct)
apply auto
done

lemma converse_rtranclp_into_rtranclp: "r a b ⟹ r⇧*⇧* b c ⟹ r⇧*⇧* a c"
by (rule rtranclp_trans) iprover+

lemmas converse_rtrancl_into_rtrancl = converse_rtranclp_into_rtranclp [to_set]

text ‹┉ More @{term "r⇧*"} equations and inclusions.›

lemma rtranclp_idemp [simp]: "(r⇧*⇧*)⇧*⇧* = r⇧*⇧*"
apply (auto intro!: order_antisym)
apply (erule rtranclp_induct)
apply (rule rtranclp.rtrancl_refl)
apply (blast intro: rtranclp_trans)
done

lemmas rtrancl_idemp [simp] = rtranclp_idemp [to_set]

lemma rtrancl_idemp_self_comp [simp]: "R⇧* O R⇧* = R⇧*"
apply (rule set_eqI)
apply (simp only: split_tupled_all)
apply (blast intro: rtrancl_trans)
done

lemma rtrancl_subset_rtrancl: "r ⊆ s⇧* ⟹ r⇧* ⊆ s⇧*"
apply (drule rtrancl_mono)
apply simp
done

lemma rtranclp_subset: "R ≤ S ⟹ S ≤ R⇧*⇧* ⟹ S⇧*⇧* = R⇧*⇧*"
apply (drule rtranclp_mono)
apply (drule rtranclp_mono)
apply simp
done

lemmas rtrancl_subset = rtranclp_subset [to_set]

lemma rtranclp_sup_rtranclp: "(sup (R⇧*⇧*) (S⇧*⇧*))⇧*⇧* = (sup R S)⇧*⇧*"
by (blast intro!: rtranclp_subset intro: rtranclp_mono [THEN predicate2D])

lemmas rtrancl_Un_rtrancl = rtranclp_sup_rtranclp [to_set]

lemma rtranclp_reflclp [simp]: "(R⇧=⇧=)⇧*⇧* = R⇧*⇧*"
by (blast intro!: rtranclp_subset)

lemmas rtrancl_reflcl [simp] = rtranclp_reflclp [to_set]

lemma rtrancl_r_diff_Id: "(r - Id)⇧* = r⇧*"
apply (rule sym)
apply (rule rtrancl_subset)
apply blast
apply clarify
apply (rename_tac a b)
apply (case_tac "a = b")
apply blast
apply blast
done

lemma rtranclp_r_diff_Id: "(inf r op ≠)⇧*⇧* = r⇧*⇧*"
apply (rule sym)
apply (rule rtranclp_subset)
apply blast+
done

theorem rtranclp_converseD:
assumes "(r¯¯)⇧*⇧* x y"
shows "r⇧*⇧* y x"
using assms by induct (iprover intro: rtranclp_trans dest!: conversepD)+

lemmas rtrancl_converseD = rtranclp_converseD [to_set]

theorem rtranclp_converseI:
assumes "r⇧*⇧* y x"
shows "(r¯¯)⇧*⇧* x y"
using assms by induct (iprover intro: rtranclp_trans conversepI)+

lemmas rtrancl_converseI = rtranclp_converseI [to_set]

lemma rtrancl_converse: "(r^-1)⇧* = (r⇧*)^-1"
by (fast dest!: rtrancl_converseD intro!: rtrancl_converseI)

lemma sym_rtrancl: "sym r ⟹ sym (r⇧*)"
by (simp only: sym_conv_converse_eq rtrancl_converse [symmetric])

theorem converse_rtranclp_induct [consumes 1, case_names base step]:
assumes major: "r⇧*⇧* a b"
and cases: "P b" "⋀y z. r y z ⟹ r⇧*⇧* z b ⟹ P z ⟹ P y"
shows "P a"
using rtranclp_converseI [OF major]
by induct (iprover intro: cases dest!: conversepD rtranclp_converseD)+

lemmas converse_rtrancl_induct = converse_rtranclp_induct [to_set]

lemmas converse_rtranclp_induct2 =
converse_rtranclp_induct [of _ "(ax, ay)" "(bx, by)", split_rule, consumes 1, case_names refl step]

lemmas converse_rtrancl_induct2 =
converse_rtrancl_induct [of "(ax, ay)" "(bx, by)", split_format (complete),
consumes 1, case_names refl step]

lemma converse_rtranclpE [consumes 1, case_names base step]:
assumes major: "r⇧*⇧* x z"
and cases: "x = z ⟹ P" "⋀y. r x y ⟹ r⇧*⇧* y z ⟹ P"
shows P
apply (subgoal_tac "x = z ∨ (∃y. r x y ∧ r⇧*⇧* y z)")
apply (rule_tac [2] major [THEN converse_rtranclp_induct])
prefer 2 apply iprover
prefer 2 apply iprover
apply (erule asm_rl exE disjE conjE cases)+
done

lemmas converse_rtranclE = converse_rtranclpE [to_set]

lemmas converse_rtranclpE2 = converse_rtranclpE [of _ "(xa,xb)" "(za,zb)", split_rule]

lemmas converse_rtranclE2 = converse_rtranclE [of "(xa,xb)" "(za,zb)", split_rule]

lemma r_comp_rtrancl_eq: "r O r⇧* = r⇧* O r"
by (blast elim: rtranclE converse_rtranclE
intro: rtrancl_into_rtrancl converse_rtrancl_into_rtrancl)

lemma rtrancl_unfold: "r⇧* = Id ∪ r⇧* O r"
by (auto intro: rtrancl_into_rtrancl elim: rtranclE)

lemma rtrancl_Un_separatorE:
"(a, b) ∈ (P ∪ Q)⇧* ⟹ ∀x y. (a, x) ∈ P⇧* ⟶ (x, y) ∈ Q ⟶ x = y ⟹ (a, b) ∈ P⇧*"
proof (induct rule: rtrancl.induct)
case rtrancl_refl
then show ?case by blast
next
case rtrancl_into_rtrancl
then show ?case by (blast intro: rtrancl_trans)
qed

lemma rtrancl_Un_separator_converseE:
"(a, b) ∈ (P ∪ Q)⇧* ⟹ ∀x y. (x, b) ∈ P⇧* ⟶ (y, x) ∈ Q ⟶ y = x ⟹ (a, b) ∈ P⇧*"
proof (induct rule: converse_rtrancl_induct)
case base
then show ?case by blast
next
case step
then show ?case by (blast intro: rtrancl_trans)
qed

lemma Image_closed_trancl:
assumes "r `` X ⊆ X"
shows "r⇧* `` X = X"
proof -
from assms have **: "{y. ∃x∈X. (x, y) ∈ r} ⊆ X"
by auto
have "x ∈ X" if 1: "(y, x) ∈ r⇧*" and 2: "y ∈ X" for x y
proof -
from 1 show "x ∈ X"
proof induct
case base
show ?case by (fact 2)
next
case step
with ** show ?case by auto
qed
qed
then show ?thesis by auto
qed

subsection ‹Transitive closure›

lemma trancl_mono: "⋀p. p ∈ r⇧+ ⟹ r ⊆ s ⟹ p ∈ s⇧+"
apply (simp add: split_tupled_all)
apply (erule trancl.induct)
apply (iprover dest: subsetD)+
done

lemma r_into_trancl': "⋀p. p ∈ r ⟹ p ∈ r⇧+"
by (simp only: split_tupled_all) (erule r_into_trancl)

text ‹┉ Conversions between ‹trancl› and ‹rtrancl›.›

lemma tranclp_into_rtranclp: "r⇧+⇧+ a b ⟹ r⇧*⇧* a b"
by (erule tranclp.induct) iprover+

lemmas trancl_into_rtrancl = tranclp_into_rtranclp [to_set]

lemma rtranclp_into_tranclp1:
assumes "r⇧*⇧* a b"
shows "r b c ⟹ r⇧+⇧+ a c"
using assms by (induct arbitrary: c) iprover+

lemmas rtrancl_into_trancl1 = rtranclp_into_tranclp1 [to_set]

lemma rtranclp_into_tranclp2: "r a b ⟹ r⇧*⇧* b c ⟹ r⇧+⇧+ a c"
― ‹intro rule from ‹r› and ‹rtrancl››
apply (erule rtranclp.cases)
apply iprover
apply (rule rtranclp_trans [THEN rtranclp_into_tranclp1])
apply (simp | rule r_into_rtranclp)+
done

lemmas rtrancl_into_trancl2 = rtranclp_into_tranclp2 [to_set]

text ‹Nice induction rule for ‹trancl››
lemma tranclp_induct [consumes 1, case_names base step, induct pred: tranclp]:
assumes a: "r⇧+⇧+ a b"
and cases: "⋀y. r a y ⟹ P y" "⋀y z. r⇧+⇧+ a y ⟹ r y z ⟹ P y ⟹ P z"
shows "P b"
using a by (induct x≡a b) (iprover intro: cases)+

lemmas trancl_induct [induct set: trancl] = tranclp_induct [to_set]

lemmas tranclp_induct2 =
tranclp_induct [of _ "(ax, ay)" "(bx, by)", split_rule, consumes 1, case_names base step]

lemmas trancl_induct2 =
trancl_induct [of "(ax, ay)" "(bx, by)", split_format (complete),
consumes 1, case_names base step]

lemma tranclp_trans_induct:
assumes major: "r⇧+⇧+ x y"
and cases: "⋀x y. r x y ⟹ P x y" "⋀x y z. r⇧+⇧+ x y ⟹ P x y ⟹ r⇧+⇧+ y z ⟹ P y z ⟹ P x z"
shows "P x y"
― ‹Another induction rule for trancl, incorporating transitivity›
by (iprover intro: major [THEN tranclp_induct] cases)

lemmas trancl_trans_induct = tranclp_trans_induct [to_set]

lemma tranclE [cases set: trancl]:
assumes "(a, b) ∈ r⇧+"
obtains
(base) "(a, b) ∈ r"
| (step) c where "(a, c) ∈ r⇧+" and "(c, b) ∈ r"
using assms by cases simp_all

lemma trancl_Int_subset: "r ⊆ s ⟹ (r⇧+ ∩ s) O r ⊆ s ⟹ r⇧+ ⊆ s"
apply (rule subsetI)
apply auto
apply (erule trancl_induct)
apply auto
done

lemma trancl_unfold: "r⇧+ = r ∪ r⇧+ O r"
by (auto intro: trancl_into_trancl elim: tranclE)

text ‹Transitivity of @{term "r⇧+"}›
lemma trans_trancl [simp]: "trans (r⇧+)"
proof (rule transI)
fix x y z
assume "(x, y) ∈ r⇧+"
assume "(y, z) ∈ r⇧+"
then show "(x, z) ∈ r⇧+"
proof induct
case (base u)
from ‹(x, y) ∈ r⇧+› and ‹(y, u) ∈ r›
show "(x, u) ∈ r⇧+" ..
next
case (step u v)
from ‹(x, u) ∈ r⇧+› and ‹(u, v) ∈ r›
show "(x, v) ∈ r⇧+" ..
qed
qed

lemmas trancl_trans = trans_trancl [THEN transD]

lemma tranclp_trans:
assumes "r⇧+⇧+ x y"
and "r⇧+⇧+ y z"
shows "r⇧+⇧+ x z"
using assms(2,1) by induct iprover+

lemma trancl_id [simp]: "trans r ⟹ r⇧+ = r"
apply auto
apply (erule trancl_induct)
apply assumption
apply (unfold trans_def)
apply blast
done

lemma rtranclp_tranclp_tranclp:
assumes "r⇧*⇧* x y"
shows "⋀z. r⇧+⇧+ y z ⟹ r⇧+⇧+ x z"
using assms by induct (iprover intro: tranclp_trans)+

lemmas rtrancl_trancl_trancl = rtranclp_tranclp_tranclp [to_set]

lemma tranclp_into_tranclp2: "r a b ⟹ r⇧+⇧+ b c ⟹ r⇧+⇧+ a c"
by (erule tranclp_trans [OF tranclp.r_into_trancl])

lemmas trancl_into_trancl2 = tranclp_into_tranclp2 [to_set]

lemma tranclp_converseI: "(r⇧+⇧+)¯¯ x y ⟹ (r¯¯)⇧+⇧+ x y"
apply (drule conversepD)
apply (erule tranclp_induct)
apply (iprover intro: conversepI tranclp_trans)+
done

lemmas trancl_converseI = tranclp_converseI [to_set]

lemma tranclp_converseD: "(r¯¯)⇧+⇧+ x y ⟹ (r⇧+⇧+)¯¯ x y"
apply (rule conversepI)
apply (erule tranclp_induct)
apply (iprover dest: conversepD intro: tranclp_trans)+
done

lemmas trancl_converseD = tranclp_converseD [to_set]

lemma tranclp_converse: "(r¯¯)⇧+⇧+ = (r⇧+⇧+)¯¯"
by (fastforce simp add: fun_eq_iff intro!: tranclp_converseI dest!: tranclp_converseD)

lemmas trancl_converse = tranclp_converse [to_set]

lemma sym_trancl: "sym r ⟹ sym (r⇧+)"
by (simp only: sym_conv_converse_eq trancl_converse [symmetric])

lemma converse_tranclp_induct [consumes 1, case_names base step]:
assumes major: "r⇧+⇧+ a b"
and cases: "⋀y. r y b ⟹ P y" "⋀y z. r y z ⟹ r⇧+⇧+ z b ⟹ P z ⟹ P y"
shows "P a"
apply (rule tranclp_induct [OF tranclp_converseI, OF conversepI, OF major])
apply (rule cases)
apply (erule conversepD)
apply (blast intro: assms dest!: tranclp_converseD)
done

lemmas converse_trancl_induct = converse_tranclp_induct [to_set]

lemma tranclpD: "R⇧+⇧+ x y ⟹ ∃z. R x z ∧ R⇧*⇧* z y"
apply (erule converse_tranclp_induct)
apply auto
apply (blast intro: rtranclp_trans)
done

lemmas tranclD = tranclpD [to_set]

lemma converse_tranclpE:
assumes major: "tranclp r x z"
and base: "r x z ⟹ P"
and step: "⋀y. r x y ⟹ tranclp r y z ⟹ P"
shows P
proof -
from tranclpD [OF major] obtain y where "r x y" and "rtranclp r y z"
by iprover
from this(2) show P
proof (cases rule: rtranclp.cases)
case rtrancl_refl
with ‹r x y› base show P
by iprover
next
case rtrancl_into_rtrancl
from this have "tranclp r y z"
by (iprover intro: rtranclp_into_tranclp1)
with ‹r x y› step show P
by iprover
qed
qed

lemmas converse_tranclE = converse_tranclpE [to_set]

lemma tranclD2: "(x, y) ∈ R⇧+ ⟹ ∃z. (x, z) ∈ R⇧* ∧ (z, y) ∈ R"
by (blast elim: tranclE intro: trancl_into_rtrancl)

lemma irrefl_tranclI: "r¯ ∩ r⇧* = {} ⟹ (x, x) ∉ r⇧+"
by (blast elim: tranclE dest: trancl_into_rtrancl)

lemma irrefl_trancl_rD: "∀x. (x, x) ∉ r⇧+ ⟹ (x, y) ∈ r ⟹ x ≠ y"
by (blast dest: r_into_trancl)

lemma trancl_subset_Sigma_aux: "(a, b) ∈ r⇧* ⟹ r ⊆ A × A ⟹ a = b ∨ a ∈ A"
by (induct rule: rtrancl_induct) auto

lemma trancl_subset_Sigma: "r ⊆ A × A ⟹ r⇧+ ⊆ A × A"
apply (rule subsetI)
apply (simp only: split_tupled_all)
apply (erule tranclE)
apply (blast dest!: trancl_into_rtrancl trancl_subset_Sigma_aux)+
done

lemma reflclp_tranclp [simp]: "(r⇧+⇧+)⇧=⇧= = r⇧*⇧*"
apply (safe intro!: order_antisym)
apply (erule tranclp_into_rtranclp)
apply (blast elim: rtranclp.cases dest: rtranclp_into_tranclp1)
done

lemmas reflcl_trancl [simp] = reflclp_tranclp [to_set]

lemma trancl_reflcl [simp]: "(r⇧=)⇧+ = r⇧*"
apply safe
apply (drule trancl_into_rtrancl, simp)
apply (erule rtranclE, safe)
apply (rule r_into_trancl, simp)
apply (rule rtrancl_into_trancl1)
apply (erule rtrancl_reflcl [THEN equalityD2, THEN subsetD], fast)
done

lemma rtrancl_trancl_reflcl [code]: "r⇧* = (r⇧+)⇧="
by simp

lemma trancl_empty [simp]: "{}⇧+ = {}"
by (auto elim: trancl_induct)

lemma rtrancl_empty [simp]: "{}⇧* = Id"
by (rule subst [OF reflcl_trancl]) simp

lemma rtranclpD: "R⇧*⇧* a b ⟹ a = b ∨ a ≠ b ∧ R⇧+⇧+ a b"
by (force simp add: reflclp_tranclp [symmetric] simp del: reflclp_tranclp)

lemmas rtranclD = rtranclpD [to_set]

lemma rtrancl_eq_or_trancl: "(x,y) ∈ R⇧* ⟷ x = y ∨ x ≠ y ∧ (x, y) ∈ R⇧+"
by (fast elim: trancl_into_rtrancl dest: rtranclD)

lemma trancl_unfold_right: "r⇧+ = r⇧* O r"
by (auto dest: tranclD2 intro: rtrancl_into_trancl1)

lemma trancl_unfold_left: "r⇧+ = r O r⇧*"
by (auto dest: tranclD intro: rtrancl_into_trancl2)

lemma trancl_insert: "(insert (y, x) r)⇧+ = r⇧+ ∪ {(a, b). (a, y) ∈ r⇧* ∧ (x, b) ∈ r⇧*}"
― ‹primitive recursion for ‹trancl› over finite relations›
apply (rule equalityI)
apply (rule subsetI)
apply (simp only: split_tupled_all)
apply (erule trancl_induct, blast)
apply (blast intro: rtrancl_into_trancl1 trancl_into_rtrancl trancl_trans)
apply (rule subsetI)
apply (blast intro: trancl_mono rtrancl_mono
[THEN [2] rev_subsetD] rtrancl_trancl_trancl rtrancl_into_trancl2)
done

lemma trancl_insert2:
"(insert (a, b) r)⇧+ = r⇧+ ∪ {(x, y). ((x, a) ∈ r⇧+ ∨ x = a) ∧ ((b, y) ∈ r⇧+ ∨ y = b)}"
by (auto simp add: trancl_insert rtrancl_eq_or_trancl)

lemma rtrancl_insert: "(insert (a,b) r)⇧* = r⇧* ∪ {(x, y). (x, a) ∈ r⇧* ∧ (b, y) ∈ r⇧*}"
using trancl_insert[of a b r]
by (simp add: rtrancl_trancl_reflcl del: reflcl_trancl) blast

text ‹Simplifying nested closures›

lemma rtrancl_trancl_absorb[simp]: "(R⇧*)⇧+ = R⇧*"
by (simp add: trans_rtrancl)

lemma trancl_rtrancl_absorb[simp]: "(R⇧+)⇧* = R⇧*"
by (subst reflcl_trancl[symmetric]) simp

lemma rtrancl_reflcl_absorb[simp]: "(R⇧*)⇧= = R⇧*"
by auto

text ‹‹Domain› and ‹Range››

lemma Domain_rtrancl [simp]: "Domain (R⇧*) = UNIV"
by blast

lemma Range_rtrancl [simp]: "Range (R⇧*) = UNIV"
by blast

lemma rtrancl_Un_subset: "(R⇧* ∪ S⇧*) ⊆ (R ∪ S)⇧*"
by (rule rtrancl_Un_rtrancl [THEN subst]) fast

lemma in_rtrancl_UnI: "x ∈ R⇧* ∨ x ∈ S⇧* ⟹ x ∈ (R ∪ S)⇧*"
by (blast intro: subsetD [OF rtrancl_Un_subset])

lemma trancl_domain [simp]: "Domain (r⇧+) = Domain r"
by (unfold Domain_unfold) (blast dest: tranclD)

lemma trancl_range [simp]: "Range (r⇧+) = Range r"
unfolding Domain_converse [symmetric] by (simp add: trancl_converse [symmetric])

lemma Not_Domain_rtrancl: "x ∉ Domain R ⟹ (x, y) ∈ R⇧* ⟷ x = y"
apply auto
apply (erule rev_mp)
apply (erule rtrancl_induct)
apply auto
done

lemma trancl_subset_Field2: "r⇧+ ⊆ Field r × Field r"
apply clarify
apply (erule trancl_induct)
apply (auto simp add: Field_def)
done

lemma finite_trancl[simp]: "finite (r⇧+) = finite r"
apply auto
prefer 2
apply (rule trancl_subset_Field2 [THEN finite_subset])
apply (rule finite_SigmaI)
prefer 3
apply (blast intro: r_into_trancl' finite_subset)
apply (auto simp add: finite_Field)
done

text ‹More about converse ‹rtrancl› and ‹trancl›, should
be merged with main body.›

lemma single_valued_confluent:
"single_valued r ⟹ (x, y) ∈ r⇧* ⟹ (x, z) ∈ r⇧* ⟹ (y, z) ∈ r⇧* ∨ (z, y) ∈ r⇧*"
apply (erule rtrancl_induct)
apply simp
apply (erule disjE)
apply (blast elim:converse_rtranclE dest:single_valuedD)
apply (blast intro:rtrancl_trans)
done

lemma r_r_into_trancl: "(a, b) ∈ R ⟹ (b, c) ∈ R ⟹ (a, c) ∈ R⇧+"
by (fast intro: trancl_trans)

lemma trancl_into_trancl: "(a, b) ∈ r⇧+ ⟹ (b, c) ∈ r ⟹ (a, c) ∈ r⇧+"
by (induct rule: trancl_induct) (fast intro: r_r_into_trancl trancl_trans)+

lemma tranclp_rtranclp_tranclp: "r⇧+⇧+ a b ⟹ r⇧*⇧* b c ⟹ r⇧+⇧+ a c"
apply (drule tranclpD)
apply (elim exE conjE)
apply (drule rtranclp_trans, assumption)
apply (drule (2) rtranclp_into_tranclp2)
done

lemmas trancl_rtrancl_trancl = tranclp_rtranclp_tranclp [to_set]

lemmas transitive_closure_trans [trans] =
r_r_into_trancl trancl_trans rtrancl_trans
trancl.trancl_into_trancl trancl_into_trancl2
rtrancl.rtrancl_into_rtrancl converse_rtrancl_into_rtrancl
rtrancl_trancl_trancl trancl_rtrancl_trancl

lemmas transitive_closurep_trans' [trans] =
tranclp_trans rtranclp_trans
tranclp.trancl_into_trancl tranclp_into_tranclp2
rtranclp.rtrancl_into_rtrancl converse_rtranclp_into_rtranclp
rtranclp_tranclp_tranclp tranclp_rtranclp_tranclp

declare trancl_into_rtrancl [elim]

subsection ‹The power operation on relations›

text ‹‹R ^^ n = R O … O R›, the n-fold composition of ‹R››

overloading
relpow ≡ "compow :: nat ⇒ ('a × 'a) set ⇒ ('a × 'a) set"
relpowp ≡ "compow :: nat ⇒ ('a ⇒ 'a ⇒ bool) ⇒ ('a ⇒ 'a ⇒ bool)"
begin

primrec relpow :: "nat ⇒ ('a × 'a) set ⇒ ('a × 'a) set"
where
"relpow 0 R = Id"
| "relpow (Suc n) R = (R ^^ n) O R"

primrec relpowp :: "nat ⇒ ('a ⇒ 'a ⇒ bool) ⇒ ('a ⇒ 'a ⇒ bool)"
where
"relpowp 0 R = HOL.eq"
| "relpowp (Suc n) R = (R ^^ n) OO R"

end

lemma relpowp_relpow_eq [pred_set_conv]:
"(λx y. (x, y) ∈ R) ^^ n = (λx y. (x, y) ∈ R ^^ n)" for R :: "'a rel"
by (induct n) (simp_all add: relcompp_relcomp_eq)

text ‹For code generation:›

definition relpow :: "nat ⇒ ('a × 'a) set ⇒ ('a × 'a) set"
where relpow_code_def [code_abbrev]: "relpow = compow"

definition relpowp :: "nat ⇒ ('a ⇒ 'a ⇒ bool) ⇒ ('a ⇒ 'a ⇒ bool)"
where relpowp_code_def [code_abbrev]: "relpowp = compow"

lemma [code]:
"relpow (Suc n) R = (relpow n R) O R"
"relpow 0 R = Id"
by (simp_all add: relpow_code_def)

lemma [code]:
"relpowp (Suc n) R = (R ^^ n) OO R"
"relpowp 0 R = HOL.eq"
by (simp_all add: relpowp_code_def)

hide_const (open) relpow
hide_const (open) relpowp

lemma relpow_1 [simp]: "R ^^ 1 = R"
for R :: "('a × 'a) set"
by simp

lemma relpowp_1 [simp]: "P ^^ 1 = P"
for P :: "'a ⇒ 'a ⇒ bool"
by (fact relpow_1 [to_pred])

lemma relpow_0_I: "(x, x) ∈ R ^^ 0"
by simp

lemma relpowp_0_I: "(P ^^ 0) x x"
by (fact relpow_0_I [to_pred])

lemma relpow_Suc_I: "(x, y) ∈  R ^^ n ⟹ (y, z) ∈ R ⟹ (x, z) ∈ R ^^ Suc n"
by auto

lemma relpowp_Suc_I: "(P ^^ n) x y ⟹ P y z ⟹ (P ^^ Suc n) x z"
by (fact relpow_Suc_I [to_pred])

lemma relpow_Suc_I2: "(x, y) ∈ R ⟹ (y, z) ∈ R ^^ n ⟹ (x, z) ∈ R ^^ Suc n"
by (induct n arbitrary: z) (simp, fastforce)

lemma relpowp_Suc_I2: "P x y ⟹ (P ^^ n) y z ⟹ (P ^^ Suc n) x z"
by (fact relpow_Suc_I2 [to_pred])

lemma relpow_0_E: "(x, y) ∈ R ^^ 0 ⟹ (x = y ⟹ P) ⟹ P"
by simp

lemma relpowp_0_E: "(P ^^ 0) x y ⟹ (x = y ⟹ Q) ⟹ Q"
by (fact relpow_0_E [to_pred])

lemma relpow_Suc_E: "(x, z) ∈ R ^^ Suc n ⟹ (⋀y. (x, y) ∈ R ^^ n ⟹ (y, z) ∈ R ⟹ P) ⟹ P"
by auto

lemma relpowp_Suc_E: "(P ^^ Suc n) x z ⟹ (⋀y. (P ^^ n) x y ⟹ P y z ⟹ Q) ⟹ Q"
by (fact relpow_Suc_E [to_pred])

lemma relpow_E:
"(x, z) ∈  R ^^ n ⟹
(n = 0 ⟹ x = z ⟹ P) ⟹
(⋀y m. n = Suc m ⟹ (x, y) ∈  R ^^ m ⟹ (y, z) ∈ R ⟹ P) ⟹ P"
by (cases n) auto

lemma relpowp_E:
"(P ^^ n) x z ⟹
(n = 0 ⟹ x = z ⟹ Q) ⟹
(⋀y m. n = Suc m ⟹ (P ^^ m) x y ⟹ P y z ⟹ Q) ⟹ Q"
by (fact relpow_E [to_pred])

lemma relpow_Suc_D2: "(x, z) ∈ R ^^ Suc n ⟹ (∃y. (x, y) ∈ R ∧ (y, z) ∈ R ^^ n)"
by (induct n arbitrary: x z)
(blast intro: relpow_0_I relpow_Suc_I elim: relpow_0_E relpow_Suc_E)+

lemma relpowp_Suc_D2: "(P ^^ Suc n) x z ⟹ ∃y. P x y ∧ (P ^^ n) y z"
by (fact relpow_Suc_D2 [to_pred])

lemma relpow_Suc_E2: "(x, z) ∈ R ^^ Suc n ⟹ (⋀y. (x, y) ∈ R ⟹ (y, z) ∈ R ^^ n ⟹ P) ⟹ P"
by (blast dest: relpow_Suc_D2)

lemma relpowp_Suc_E2: "(P ^^ Suc n) x z ⟹ (⋀y. P x y ⟹ (P ^^ n) y z ⟹ Q) ⟹ Q"
by (fact relpow_Suc_E2 [to_pred])

lemma relpow_Suc_D2': "∀x y z. (x, y) ∈ R ^^ n ∧ (y, z) ∈ R ⟶ (∃w. (x, w) ∈ R ∧ (w, z) ∈ R ^^ n)"
by (induct n) (simp_all, blast)

lemma relpowp_Suc_D2': "∀x y z. (P ^^ n) x y ∧ P y z ⟶ (∃w. P x w ∧ (P ^^ n) w z)"
by (fact relpow_Suc_D2' [to_pred])

lemma relpow_E2:
"(x, z) ∈ R ^^ n ⟹
(n = 0 ⟹ x = z ⟹ P) ⟹
(⋀y m. n = Suc m ⟹ (x, y) ∈ R ⟹ (y, z) ∈ R ^^ m ⟹ P) ⟹ P"
apply (cases n)
apply simp
apply (rename_tac nat)
apply (cut_tac n=nat and R=R in relpow_Suc_D2')
apply simp
apply blast
done

lemma relpowp_E2:
"(P ^^ n) x z ⟹
(n = 0 ⟹ x = z ⟹ Q) ⟹
(⋀y m. n = Suc m ⟹ P x y ⟹ (P ^^ m) y z ⟹ Q) ⟹ Q"
by (fact relpow_E2 [to_pred])

lemma relpow_add: "R ^^ (m + n) = R^^m O R^^n"
by (induct n) auto

lemma relpowp_add: "P ^^ (m + n) = P ^^ m OO P ^^ n"
by (fact relpow_add [to_pred])

lemma relpow_commute: "R O R ^^ n = R ^^ n O R"
by (induct n) (simp_all add: O_assoc [symmetric])

lemma relpowp_commute: "P OO P ^^ n = P ^^ n OO P"
by (fact relpow_commute [to_pred])

lemma relpow_empty: "0 < n ⟹ ({} :: ('a × 'a) set) ^^ n = {}"
by (cases n) auto

lemma relpowp_bot: "0 < n ⟹ (⊥ :: 'a ⇒ 'a ⇒ bool) ^^ n = ⊥"
by (fact relpow_empty [to_pred])

lemma rtrancl_imp_UN_relpow:
assumes "p ∈ R⇧*"
shows "p ∈ (⋃n. R ^^ n)"
proof (cases p)
case (Pair x y)
with assms have "(x, y) ∈ R⇧*" by simp
then have "(x, y) ∈ (⋃n. R ^^ n)"
proof induct
case base
show ?case by (blast intro: relpow_0_I)
next
case step
then show ?case by (blast intro: relpow_Suc_I)
qed
with Pair show ?thesis by simp
qed

lemma rtranclp_imp_Sup_relpowp:
assumes "(P⇧*⇧*) x y"
shows "(⨆n. P ^^ n) x y"
using assms and rtrancl_imp_UN_relpow [of "(x, y)", to_pred] by simp

lemma relpow_imp_rtrancl:
assumes "p ∈ R ^^ n"
shows "p ∈ R⇧*"
proof (cases p)
case (Pair x y)
with assms have "(x, y) ∈ R ^^ n" by simp
then have "(x, y) ∈ R⇧*"
proof (induct n arbitrary: x y)
case 0
then show ?case by simp
next
case Suc
then show ?case
by (blast elim: relpow_Suc_E intro: rtrancl_into_rtrancl)
qed
with Pair show ?thesis by simp
qed

lemma relpowp_imp_rtranclp: "(P ^^ n) x y ⟹ (P⇧*⇧*) x y"
using relpow_imp_rtrancl [of "(x, y)", to_pred] by simp

lemma rtrancl_is_UN_relpow: "R⇧* = (⋃n. R ^^ n)"
by (blast intro: rtrancl_imp_UN_relpow relpow_imp_rtrancl)

lemma rtranclp_is_Sup_relpowp: "P⇧*⇧* = (⨆n. P ^^ n)"
using rtrancl_is_UN_relpow [to_pred, of P] by auto

lemma rtrancl_power: "p ∈ R⇧* ⟷ (∃n. p ∈ R ^^ n)"
by (simp add: rtrancl_is_UN_relpow)

lemma rtranclp_power: "(P⇧*⇧*) x y ⟷ (∃n. (P ^^ n) x y)"
by (simp add: rtranclp_is_Sup_relpowp)

lemma trancl_power: "p ∈ R⇧+ ⟷ (∃n > 0. p ∈ R ^^ n)"
apply (cases p)
apply simp
apply (rule iffI)
apply (drule tranclD2)
apply (clarsimp simp: rtrancl_is_UN_relpow)
apply (rule_tac x="Suc x" in exI)
apply (clarsimp simp: relcomp_unfold)
apply fastforce
apply clarsimp
apply (case_tac n)
apply simp
apply clarsimp
apply (drule relpow_imp_rtrancl)
apply (drule rtrancl_into_trancl1)
apply auto
done

lemma tranclp_power: "(P⇧+⇧+) x y ⟷ (∃n > 0. (P ^^ n) x y)"
using trancl_power [to_pred, of P "(x, y)"] by simp

lemma rtrancl_imp_relpow: "p ∈ R⇧* ⟹ ∃n. p ∈ R ^^ n"
by (auto dest: rtrancl_imp_UN_relpow)

lemma rtranclp_imp_relpowp: "(P⇧*⇧*) x y ⟹ ∃n. (P ^^ n) x y"
by (auto dest: rtranclp_imp_Sup_relpowp)

text ‹By Sternagel/Thiemann:›
lemma relpow_fun_conv: "(a, b) ∈ R ^^ n ⟷ (∃f. f 0 = a ∧ f n = b ∧ (∀i<n. (f i, f (Suc i)) ∈ R))"
proof (induct n arbitrary: b)
case 0
show ?case by auto
next
case (Suc n)
show ?case
proof (simp add: relcomp_unfold Suc)
show "(∃y. (∃f. f 0 = a ∧ f n = y ∧ (∀i<n. (f i,f(Suc i)) ∈ R)) ∧ (y,b) ∈ R) ⟷
(∃f. f 0 = a ∧ f(Suc n) = b ∧ (∀i<Suc n. (f i, f (Suc i)) ∈ R))"
(is "?l = ?r")
proof
assume ?l
then obtain c f
where 1: "f 0 = a"  "f n = c"  "⋀i. i < n ⟹ (f i, f (Suc i)) ∈ R"  "(c,b) ∈ R"
by auto
let ?g = "λ m. if m = Suc n then b else f m"
show ?r by (rule exI[of _ ?g]) (simp add: 1)
next
assume ?r
then obtain f where 1: "f 0 = a"  "b = f (Suc n)"  "⋀i. i < Suc n ⟹ (f i, f (Suc i)) ∈ R"
by auto
show ?l by (rule exI[of _ "f n"], rule conjI, rule exI[of _ f], insert 1, auto)
qed
qed
qed

lemma relpowp_fun_conv: "(P ^^ n) x y ⟷ (∃f. f 0 = x ∧ f n = y ∧ (∀i<n. P (f i) (f (Suc i))))"
by (fact relpow_fun_conv [to_pred])

lemma relpow_finite_bounded1:
fixes R :: "('a × 'a) set"
assumes "finite R" and "k > 0"
shows "R^^k ⊆ (⋃n∈{n. 0 < n ∧ n ≤ card R}. R^^n)"
(is "_ ⊆ ?r")
proof -
have "(a, b) ∈ R^^(Suc k) ⟹ ∃n. 0 < n ∧ n ≤ card R ∧ (a, b) ∈ R^^n" for a b k
proof (induct k arbitrary: b)
case 0
then have "R ≠ {}" by auto
with card_0_eq[OF ‹finite R›] have "card R ≥ Suc 0" by auto
then show ?case using 0 by force
next
case (Suc k)
then obtain a' where "(a, a') ∈ R^^(Suc k)" and "(a', b) ∈ R"
by auto
from Suc(1)[OF ‹(a, a') ∈ R^^(Suc k)›] obtain n where "n ≤ card R" and "(a, a') ∈ R ^^ n"
by auto
have "(a, b) ∈ R^^(Suc n)"
using ‹(a, a') ∈ R^^n› and ‹(a', b)∈ R› by auto
from ‹n ≤ card R› consider "n < card R" | "n = card R" by force
then show ?case
proof cases
case 1
then show ?thesis
using ‹(a, b) ∈ R^^(Suc n)› Suc_leI[OF ‹n < card R›] by blast
next
case 2
from ‹(a, b) ∈ R ^^ (Suc n)› [unfolded relpow_fun_conv]
obtain f where "f 0 = a" and "f (Suc n) = b"
and steps: "⋀i. i ≤ n ⟹ (f i, f (Suc i)) ∈ R" by auto
let ?p = "λi. (f i, f(Suc i))"
let ?N = "{i. i ≤ n}"
have "?p ` ?N ⊆ R"
using steps by auto
from card_mono[OF assms(1) this] have "card (?p ` ?N) ≤ card R" .
also have "… < card ?N"
using ‹n = card R› by simp
finally have "¬ inj_on ?p ?N"
by (rule pigeonhole)
then obtain i j where i: "i ≤ n" and j: "j ≤ n" and ij: "i ≠ j" and pij: "?p i = ?p j"
by (auto simp: inj_on_def)
let ?i = "min i j"
let ?j = "max i j"
have i: "?i ≤ n" and j: "?j ≤ n" and pij: "?p ?i = ?p ?j" and ij: "?i < ?j"
using i j ij pij unfolding min_def max_def by auto
from i j pij ij obtain i j where i: "i ≤ n" and j: "j ≤ n" and ij: "i < j"
and pij: "?p i = ?p j"
by blast
let ?g = "λl. if l ≤ i then f l else f (l + (j - i))"
let ?n = "Suc (n - (j - i))"
have abl: "(a, b) ∈ R ^^ ?n"
unfolding relpow_fun_conv
proof (rule exI[of _ ?g], intro conjI impI allI)
show "?g ?n = b"
using ‹f(Suc n) = b› j ij by auto
next
fix k
assume "k < ?n"
show "(?g k, ?g (Suc k)) ∈ R"
proof (cases "k < i")
case True
with i have "k ≤ n"
by auto
from steps[OF this] show ?thesis
using True by simp
next
case False
then have "i ≤ k" by auto
show ?thesis
proof (cases "k = i")
case True
then show ?thesis
using ij pij steps[OF i] by simp
next
case False
with ‹i ≤ k› have "i < k" by auto
then have small: "k + (j - i) ≤ n"
using ‹k<?n› by arith
show ?thesis
using steps[OF small] ‹i<k› by auto
qed
qed
qed (simp add: ‹f 0 = a›)
moreover have "?n ≤ n"
using i j ij by arith
ultimately show ?thesis
using ‹n = card R› by blast
qed
qed
then show ?thesis
using gr0_implies_Suc[OF ‹k > 0›] by auto
qed

lemma relpow_finite_bounded:
fixes R :: "('a × 'a) set"
assumes "finite R"
shows "R^^k ⊆ (UN n:{n. n ≤ card R}. R^^n)"
apply (cases k)
apply force
apply (use relpow_finite_bounded1[OF assms, of k] in auto)
done

lemma rtrancl_finite_eq_relpow: "finite R ⟹ R⇧* = (⋃n∈{n. n ≤ card R}. R^^n)"
by (fastforce simp: rtrancl_power dest: relpow_finite_bounded)

lemma trancl_finite_eq_relpow: "finite R ⟹ R⇧+ = (⋃n∈{n. 0 < n ∧ n ≤ card R}. R^^n)"
apply (auto simp: trancl_power)
apply (auto dest: relpow_finite_bounded1)
done

lemma finite_relcomp[simp,intro]:
assumes "finite R" and "finite S"
shows "finite (R O S)"
proof-
have "R O S = (⋃(x, y)∈R. ⋃(u, v)∈S. if u = y then {(x, v)} else {})"
by (force simp add: split_def image_constant_conv split: if_splits)
then show ?thesis
using assms by clarsimp
qed

lemma finite_relpow [simp, intro]:
fixes R :: "('a × 'a) set"
assumes "finite R"
shows "n > 0 ⟹ finite (R^^n)"
proof (induct n)
case 0
then show ?case by simp
next
case (Suc n)
then show ?case by (cases n) (use assms in simp_all)
qed

lemma single_valued_relpow:
fixes R :: "('a × 'a) set"
shows "single_valued R ⟹ single_valued (R ^^ n)"
proof (induct n arbitrary: R)
case 0
then show ?case by simp
next
case (Suc n)
show ?case
by (rule single_valuedI)
(use Suc in ‹fast dest: single_valuedD elim: relpow_Suc_E›)
qed

subsection ‹Bounded transitive closure›

definition ntrancl :: "nat ⇒ ('a × 'a) set ⇒ ('a × 'a) set"
where "ntrancl n R = (⋃i∈{i. 0 < i ∧ i ≤ Suc n}. R ^^ i)"

lemma ntrancl_Zero [simp, code]: "ntrancl 0 R = R"
proof
show "R ⊆ ntrancl 0 R"
unfolding ntrancl_def by fastforce
have "0 < i ∧ i ≤ Suc 0 ⟷ i = 1" for i
by auto
then show "ntrancl 0 R ≤ R"
unfolding ntrancl_def by auto
qed

lemma ntrancl_Suc [simp]: "ntrancl (Suc n) R = ntrancl n R O (Id ∪ R)"
proof
have "(a, b) ∈ ntrancl n R O (Id ∪ R)" if "(a, b) ∈ ntrancl (Suc n) R" for a b
proof -
from that obtain i where "0 < i" "i ≤ Suc (Suc n)" "(a, b) ∈ R ^^ i"
unfolding ntrancl_def by auto
show ?thesis
proof (cases "i = 1")
case True
from this ‹(a, b) ∈ R ^^ i› show ?thesis
by (auto simp: ntrancl_def)
next
case False
with ‹0 < i› obtain j where j: "i = Suc j" "0 < j"
by (cases i) auto
with ‹(a, b) ∈ R ^^ i› obtain c where c1: "(a, c) ∈ R ^^ j" and c2: "(c, b) ∈ R"
by auto
from c1 j ‹i ≤ Suc (Suc n)› have "(a, c) ∈ ntrancl n R"
by (fastforce simp: ntrancl_def)
with c2 show ?thesis by fastforce
qed
qed
then show "ntrancl (Suc n) R ⊆ ntrancl n R O (Id ∪ R)"
by auto
show "ntrancl n R O (Id ∪ R) ⊆ ntrancl (Suc n) R"
by (fastforce simp: ntrancl_def)
qed

lemma [code]: "ntrancl (Suc n) r = (let r' = ntrancl n r in r' ∪ r' O r)"
by (auto simp: Let_def)

lemma finite_trancl_ntranl: "finite R ⟹ trancl R = ntrancl (card R - 1) R"
by (cases "card R") (auto simp add: trancl_finite_eq_relpow relpow_empty ntrancl_def)

subsection ‹Acyclic relations›

definition acyclic :: "('a × 'a) set ⇒ bool"
where "acyclic r ⟷ (∀x. (x,x) ∉ r⇧+)"

abbreviation acyclicP :: "('a ⇒ 'a ⇒ bool) ⇒ bool"
where "acyclicP r ≡ acyclic {(x, y). r x y}"

lemma acyclic_irrefl [code]: "acyclic r ⟷ irrefl (r⇧+)"
by (simp add: acyclic_def irrefl_def)

lemma acyclicI: "∀x. (x, x) ∉ r⇧+ ⟹ acyclic r"
by (simp add: acyclic_def)

lemma (in order) acyclicI_order:
assumes *: "⋀a b. (a, b) ∈ r ⟹ f b < f a"
shows "acyclic r"
proof -
have "f b < f a" if "(a, b) ∈ r⇧+" for a b
using that by induct (auto intro: * less_trans)
then show ?thesis
by (auto intro!: acyclicI)
qed

lemma acyclic_insert [iff]: "acyclic (insert (y, x) r) ⟷ acyclic r ∧ (x, y) ∉ r⇧*"
by (simp add: acyclic_def trancl_insert) (blast intro: rtrancl_trans)

lemma acyclic_converse [iff]: "acyclic (r¯) ⟷ acyclic r"
by (simp add: acyclic_def trancl_converse)

lemmas acyclicP_converse [iff] = acyclic_converse [to_pred]

lemma acyclic_impl_antisym_rtrancl: "acyclic r ⟹ antisym (r⇧*)"
by (simp add: acyclic_def antisym_def)
(blast elim: rtranclE intro: rtrancl_into_trancl1 rtrancl_trancl_trancl)

(* Other direction:
acyclic = no loops
antisym = only self loops
Goalw [acyclic_def,antisym_def] "antisym( r⇧* ) ⟹ acyclic(r - Id)
⟹ antisym( r⇧* ) = acyclic(r - Id)";
*)

lemma acyclic_subset: "acyclic s ⟹ r ⊆ s ⟹ acyclic r"
unfolding acyclic_def by (blast intro: trancl_mono)

subsection ‹Setup of transitivity reasoner›

ML ‹
structure Trancl_Tac = Trancl_Tac
(
val r_into_trancl = @{thm trancl.r_into_trancl};
val trancl_trans  = @{thm trancl_trans};
val rtrancl_refl = @{thm rtrancl.rtrancl_refl};
val r_into_rtrancl = @{thm r_into_rtrancl};
val trancl_into_rtrancl = @{thm trancl_into_rtrancl};
val rtrancl_trancl_trancl = @{thm rtrancl_trancl_trancl};
val trancl_rtrancl_trancl = @{thm trancl_rtrancl_trancl};
val rtrancl_trans = @{thm rtrancl_trans};

fun decomp (@{const Trueprop} \$ t) =
let
fun dec (Const (@{const_name Set.member}, _) \$ (Const (@{const_name Pair}, _) \$ a \$ b) \$ rel) =
let
fun decr (Const (@{const_name rtrancl}, _ ) \$ r) = (r,"r*")
| decr (Const (@{const_name trancl}, _ ) \$ r)  = (r,"r+")
| decr r = (r,"r");
val (rel,r) = decr (Envir.beta_eta_contract rel);
in SOME (a,b,rel,r) end
| dec _ =  NONE
in dec t end
| decomp _ = NONE;
);

structure Tranclp_Tac = Trancl_Tac
(
val r_into_trancl = @{thm tranclp.r_into_trancl};
val trancl_trans  = @{thm tranclp_trans};
val rtrancl_refl = @{thm rtranclp.rtrancl_refl};
val r_into_rtrancl = @{thm r_into_rtranclp};
val trancl_into_rtrancl = @{thm tranclp_into_rtranclp};
val rtrancl_trancl_trancl = @{thm rtranclp_tranclp_tranclp};
val trancl_rtrancl_trancl = @{thm tranclp_rtranclp_tranclp};
val rtrancl_trans = @{thm rtranclp_trans};

fun decomp (@{const Trueprop} \$ t) =
let
fun dec (rel \$ a \$ b) =
let
fun decr (Const (@{const_name rtranclp}, _ ) \$ r) = (r,"r*")
| decr (Const (@{const_name tranclp}, _ ) \$ r)  = (r,"r+")
| decr r = (r,"r");
val (rel,r) = decr rel;
in SOME (a, b, rel, r) end
| dec _ =  NONE
in dec t end
| decomp _ = NONE;
);
›

setup ‹
map_theory_simpset (fn ctxt => ctxt
addSolver (mk_solver "Trancl" Trancl_Tac.trancl_tac)
addSolver (mk_solver "Rtrancl" Trancl_Tac.rtrancl_tac)
addSolver (mk_solver "Tranclp" Tranclp_Tac.trancl_tac)
addSolver (mk_solver "Rtranclp" Tranclp_Tac.rtrancl_tac))
›

text ‹Optional methods.›

method_setup trancl =
‹Scan.succeed (SIMPLE_METHOD' o Trancl_Tac.trancl_tac)›
‹simple transitivity reasoner›
method_setup rtrancl =
‹Scan.succeed (SIMPLE_METHOD' o Trancl_Tac.rtrancl_tac)›
‹simple transitivity reasoner›
method_setup tranclp =
‹Scan.succeed (SIMPLE_METHOD' o Tranclp_Tac.trancl_tac)›
‹simple transitivity reasoner (predicate version)›
method_setup rtranclp =
‹Scan.succeed (SIMPLE_METHOD' o Tranclp_Tac.rtrancl_tac)›
‹simple transitivity reasoner (predicate version)›

end
```