# Theory Trancl

```(*  Title:      ZF/Trancl.thy
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
*)

section‹Relations: Their General Properties and Transitive Closure›

theory Trancl imports Fixedpt Perm begin

definition
refl     :: "[i,i]⇒o"  where
"refl(A,r) ≡ (∀x∈A. ⟨x,x⟩ ∈ r)"

definition
irrefl   :: "[i,i]⇒o"  where
"irrefl(A,r) ≡ ∀x∈A. ⟨x,x⟩ ∉ r"

definition
sym      :: "i⇒o"  where
"sym(r) ≡ ∀x y. ⟨x,y⟩: r ⟶ ⟨y,x⟩: r"

definition
asym     :: "i⇒o"  where
"asym(r) ≡ ∀x y. ⟨x,y⟩:r ⟶ ¬ ⟨y,x⟩:r"

definition
antisym  :: "i⇒o"  where
"antisym(r) ≡ ∀x y.⟨x,y⟩:r ⟶ ⟨y,x⟩:r ⟶ x=y"

definition
trans    :: "i⇒o"  where
"trans(r) ≡ ∀x y z. ⟨x,y⟩: r ⟶ ⟨y,z⟩: r ⟶ ⟨x,z⟩: r"

definition
trans_on :: "[i,i]⇒o"  (‹trans[_]'(_')›)  where
"trans[A](r) ≡ ∀x∈A. ∀y∈A. ∀z∈A.
⟨x,y⟩: r ⟶ ⟨y,z⟩: r ⟶ ⟨x,z⟩: r"

definition
rtrancl :: "i⇒i"  (‹(_^*)› [100] 100)  (*refl/transitive closure*)  where
"r^* ≡ lfp(field(r)*field(r), λs. id(field(r)) ∪ (r O s))"

definition
trancl  :: "i⇒i"  (‹(_^+)› [100] 100)  (*transitive closure*)  where
"r^+ ≡ r O r^*"

definition
equiv    :: "[i,i]⇒o"  where
"equiv(A,r) ≡ r ⊆ A*A ∧ refl(A,r) ∧ sym(r) ∧ trans(r)"

subsection‹General properties of relations›

subsubsection‹irreflexivity›

lemma irreflI:
"⟦⋀x. x ∈ A ⟹ ⟨x,x⟩ ∉ r⟧ ⟹ irrefl(A,r)"

lemma irreflE: "⟦irrefl(A,r);  x ∈ A⟧ ⟹  ⟨x,x⟩ ∉ r"

subsubsection‹symmetry›

lemma symI:
"⟦⋀x y.⟨x,y⟩: r ⟹ ⟨y,x⟩: r⟧ ⟹ sym(r)"
by (unfold sym_def, blast)

lemma symE: "⟦sym(r); ⟨x,y⟩: r⟧  ⟹  ⟨y,x⟩: r"
by (unfold sym_def, blast)

subsubsection‹antisymmetry›

lemma antisymI:
"⟦⋀x y.⟦⟨x,y⟩: r;  ⟨y,x⟩: r⟧ ⟹ x=y⟧ ⟹ antisym(r)"

lemma antisymE: "⟦antisym(r); ⟨x,y⟩: r;  ⟨y,x⟩: r⟧  ⟹  x=y"

subsubsection‹transitivity›

lemma transD: "⟦trans(r);  ⟨a,b⟩:r;  ⟨b,c⟩:r⟧ ⟹ ⟨a,c⟩:r"
by (unfold trans_def, blast)

lemma trans_onD:
"⟦trans[A](r);  ⟨a,b⟩:r;  ⟨b,c⟩:r;  a ∈ A;  b ∈ A;  c ∈ A⟧ ⟹ ⟨a,c⟩:r"
by (unfold trans_on_def, blast)

lemma trans_imp_trans_on: "trans(r) ⟹ trans[A](r)"
by (unfold trans_def trans_on_def, blast)

lemma trans_on_imp_trans: "⟦trans[A](r); r ⊆ A*A⟧ ⟹ trans(r)"
by (simp add: trans_on_def trans_def, blast)

subsection‹Transitive closure of a relation›

lemma rtrancl_bnd_mono:
"bnd_mono(field(r)*field(r), λs. id(field(r)) ∪ (r O s))"
by (rule bnd_monoI, blast+)

lemma rtrancl_mono: "r<=s ⟹ r^* ⊆ s^*"
unfolding rtrancl_def
apply (rule lfp_mono)
apply (rule rtrancl_bnd_mono)+
apply blast
done

(* @{term"r^* = id(field(r)) ∪ ( r O r^* )"}    *)
lemmas rtrancl_unfold =
rtrancl_bnd_mono [THEN rtrancl_def [THEN def_lfp_unfold]]

(** The relation rtrancl **)

(*  @{term"r^* ⊆ field(r) * field(r)"}  *)
lemmas rtrancl_type = rtrancl_def [THEN def_lfp_subset]

lemma relation_rtrancl: "relation(r^*)"
apply (blast dest: rtrancl_type [THEN subsetD])
done

(*Reflexivity of rtrancl*)
lemma rtrancl_refl: "⟦a ∈ field(r)⟧ ⟹ ⟨a,a⟩ ∈ r^*"
apply (rule rtrancl_unfold [THEN ssubst])
apply (erule idI [THEN UnI1])
done

(*Closure under composition with r  *)
lemma rtrancl_into_rtrancl: "⟦⟨a,b⟩ ∈ r^*;  ⟨b,c⟩ ∈ r⟧ ⟹ ⟨a,c⟩ ∈ r^*"
apply (rule rtrancl_unfold [THEN ssubst])
apply (rule compI [THEN UnI2], assumption, assumption)
done

(*rtrancl of r contains all pairs in r  *)
lemma r_into_rtrancl: "⟨a,b⟩ ∈ r ⟹ ⟨a,b⟩ ∈ r^*"
by (rule rtrancl_refl [THEN rtrancl_into_rtrancl], blast+)

(*The premise ensures that r consists entirely of pairs*)
lemma r_subset_rtrancl: "relation(r) ⟹ r ⊆ r^*"
by (simp add: relation_def, blast intro: r_into_rtrancl)

lemma rtrancl_field: "field(r^*) = field(r)"
by (blast intro: r_into_rtrancl dest!: rtrancl_type [THEN subsetD])

(** standard induction rule **)

lemma rtrancl_full_induct [case_names initial step, consumes 1]:
"⟦⟨a,b⟩ ∈ r^*;
⋀x. x ∈ field(r) ⟹ P(⟨x,x⟩);
⋀x y z.⟦P(⟨x,y⟩); ⟨x,y⟩: r^*; ⟨y,z⟩: r⟧  ⟹  P(⟨x,z⟩)⟧
⟹  P(⟨a,b⟩)"
by (erule def_induct [OF rtrancl_def rtrancl_bnd_mono], blast)

(*nice induction rule.
Tried adding the typing hypotheses y,z ∈ field(r), but these
caused expensive case splits!*)
lemma rtrancl_induct [case_names initial step, induct set: rtrancl]:
"⟦⟨a,b⟩ ∈ r^*;
P(a);
⋀y z.⟦⟨a,y⟩ ∈ r^*;  ⟨y,z⟩ ∈ r;  P(y)⟧ ⟹ P(z)
⟧ ⟹ P(b)"
(*by induction on this formula*)
apply (subgoal_tac "∀y. ⟨a,b⟩ = ⟨a,y⟩ ⟶ P (y) ")
(*now solve first subgoal: this formula is sufficient*)
apply (erule spec [THEN mp], rule refl)
(*now do the induction*)
apply (erule rtrancl_full_induct, blast+)
done

(*transitivity of transitive closure⋀-- by induction.*)
lemma trans_rtrancl: "trans(r^*)"
unfolding trans_def
apply (intro allI impI)
apply (erule_tac b = z in rtrancl_induct, assumption)
apply (blast intro: rtrancl_into_rtrancl)
done

lemmas rtrancl_trans = trans_rtrancl [THEN transD]

(*elimination of rtrancl -- by induction on a special formula*)
lemma rtranclE:
"⟦⟨a,b⟩ ∈ r^*;  (a=b) ⟹ P;
⋀y.⟦⟨a,y⟩ ∈ r^*;   ⟨y,b⟩ ∈ r⟧ ⟹ P⟧
⟹ P"
apply (subgoal_tac "a = b | (∃y. ⟨a,y⟩ ∈ r^* ∧ ⟨y,b⟩ ∈ r) ")
(*see HOL/trancl*)
apply blast
apply (erule rtrancl_induct, blast+)
done

(**** The relation trancl ****)

(*Transitivity of r^+ is proved by transitivity of r^*  *)
lemma trans_trancl: "trans(r^+)"
unfolding trans_def trancl_def
apply (blast intro: rtrancl_into_rtrancl
trans_rtrancl [THEN transD, THEN compI])
done

lemmas trans_on_trancl = trans_trancl [THEN trans_imp_trans_on]

lemmas trancl_trans = trans_trancl [THEN transD]

(** Conversions between trancl and rtrancl **)

lemma trancl_into_rtrancl: "⟨a,b⟩ ∈ r^+ ⟹ ⟨a,b⟩ ∈ r^*"
unfolding trancl_def
apply (blast intro: rtrancl_into_rtrancl)
done

(*r^+ contains all pairs in r  *)
lemma r_into_trancl: "⟨a,b⟩ ∈ r ⟹ ⟨a,b⟩ ∈ r^+"
unfolding trancl_def
apply (blast intro!: rtrancl_refl)
done

(*The premise ensures that r consists entirely of pairs*)
lemma r_subset_trancl: "relation(r) ⟹ r ⊆ r^+"
by (simp add: relation_def, blast intro: r_into_trancl)

(*intro rule by definition: from r^* and r  *)
lemma rtrancl_into_trancl1: "⟦⟨a,b⟩ ∈ r^*;  ⟨b,c⟩ ∈ r⟧   ⟹  ⟨a,c⟩ ∈ r^+"
by (unfold trancl_def, blast)

(*intro rule from r and r^*  *)
lemma rtrancl_into_trancl2:
"⟦⟨a,b⟩ ∈ r;  ⟨b,c⟩ ∈ r^*⟧   ⟹  ⟨a,c⟩ ∈ r^+"
apply (erule rtrancl_induct)
apply (erule r_into_trancl)
apply (blast intro: r_into_trancl trancl_trans)
done

(*Nice induction rule for trancl*)
lemma trancl_induct [case_names initial step, induct set: trancl]:
"⟦⟨a,b⟩ ∈ r^+;
⋀y.  ⟦⟨a,y⟩ ∈ r⟧ ⟹ P(y);
⋀y z.⟦⟨a,y⟩ ∈ r^+;  ⟨y,z⟩ ∈ r;  P(y)⟧ ⟹ P(z)
⟧ ⟹ P(b)"
apply (rule compEpair)
apply (unfold trancl_def, assumption)
(*by induction on this formula*)
apply (subgoal_tac "∀z. ⟨y,z⟩ ∈ r ⟶ P (z) ")
(*now solve first subgoal: this formula is sufficient*)
apply blast
apply (erule rtrancl_induct)
apply (blast intro: rtrancl_into_trancl1)+
done

(*elimination of r^+ -- NOT an induction rule*)
lemma tranclE:
"⟦⟨a,b⟩ ∈ r^+;
⟨a,b⟩ ∈ r ⟹ P;
⋀y.⟦⟨a,y⟩ ∈ r^+; ⟨y,b⟩ ∈ r⟧ ⟹ P
⟧ ⟹ P"
apply (subgoal_tac "⟨a,b⟩ ∈ r | (∃y. ⟨a,y⟩ ∈ r^+ ∧ ⟨y,b⟩ ∈ r) ")
apply blast
apply (rule compEpair)
apply (unfold trancl_def, assumption)
apply (erule rtranclE)
apply (blast intro: rtrancl_into_trancl1)+
done

lemma trancl_type: "r^+ ⊆ field(r)*field(r)"
unfolding trancl_def
apply (blast elim: rtrancl_type [THEN subsetD, THEN SigmaE2])
done

lemma relation_trancl: "relation(r^+)"
apply (blast dest: trancl_type [THEN subsetD])
done

lemma trancl_subset_times: "r ⊆ A * A ⟹ r^+ ⊆ A * A"
by (insert trancl_type [of r], blast)

lemma trancl_mono: "r<=s ⟹ r^+ ⊆ s^+"
by (unfold trancl_def, intro comp_mono rtrancl_mono)

lemma trancl_eq_r: "⟦relation(r); trans(r)⟧ ⟹ r^+ = r"
apply (rule equalityI)
prefer 2 apply (erule r_subset_trancl, clarify)
apply (frule trancl_type [THEN subsetD], clarify)
apply (erule trancl_induct, assumption)
apply (blast dest: transD)
done

(** Suggested by Sidi Ould Ehmety **)

lemma rtrancl_idemp [simp]: "(r^*)^* = r^*"
apply (rule equalityI, auto)
prefer 2
apply (frule rtrancl_type [THEN subsetD])
apply (blast intro: r_into_rtrancl )
txt‹converse direction›
apply (frule rtrancl_type [THEN subsetD], clarify)
apply (erule rtrancl_induct)
apply (blast intro: rtrancl_trans)
done

lemma rtrancl_subset: "⟦R ⊆ S; S ⊆ R^*⟧ ⟹ S^* = R^*"
apply (drule rtrancl_mono)
apply (drule rtrancl_mono, simp_all, blast)
done

lemma rtrancl_Un_rtrancl:
"⟦relation(r); relation(s)⟧ ⟹ (r^* ∪ s^*)^* = (r ∪ s)^*"
apply (rule rtrancl_subset)
apply (blast dest: r_subset_rtrancl)
apply (blast intro: rtrancl_mono [THEN subsetD])
done

(*** "converse" laws by Sidi Ould Ehmety ***)

(** rtrancl **)

lemma rtrancl_converseD: "⟨x,y⟩:converse(r)^* ⟹ ⟨x,y⟩:converse(r^*)"
apply (rule converseI)
apply (frule rtrancl_type [THEN subsetD])
apply (erule rtrancl_induct)
apply (blast intro: rtrancl_refl)
apply (blast intro: r_into_rtrancl rtrancl_trans)
done

lemma rtrancl_converseI: "⟨x,y⟩:converse(r^*) ⟹ ⟨x,y⟩:converse(r)^*"
apply (drule converseD)
apply (frule rtrancl_type [THEN subsetD])
apply (erule rtrancl_induct)
apply (blast intro: rtrancl_refl)
apply (blast intro: r_into_rtrancl rtrancl_trans)
done

lemma rtrancl_converse: "converse(r)^* = converse(r^*)"
apply (safe intro!: equalityI)
apply (frule rtrancl_type [THEN subsetD])
apply (safe dest!: rtrancl_converseD intro!: rtrancl_converseI)
done

(** trancl **)

lemma trancl_converseD: "⟨a, b⟩:converse(r)^+ ⟹ ⟨a, b⟩:converse(r^+)"
apply (erule trancl_induct)
apply (auto intro: r_into_trancl trancl_trans)
done

lemma trancl_converseI: "⟨x,y⟩:converse(r^+) ⟹ ⟨x,y⟩:converse(r)^+"
apply (drule converseD)
apply (erule trancl_induct)
apply (auto intro: r_into_trancl trancl_trans)
done

lemma trancl_converse: "converse(r)^+ = converse(r^+)"
apply (safe intro!: equalityI)
apply (frule trancl_type [THEN subsetD])
apply (safe dest!: trancl_converseD intro!: trancl_converseI)
done

lemma converse_trancl_induct [case_names initial step, consumes 1]:
"⟦⟨a, b⟩:r^+; ⋀y. ⟨y, b⟩ :r ⟹ P(y);
⋀y z. ⟦⟨y, z⟩ ∈ r; ⟨z, b⟩ ∈ r^+; P(z)⟧ ⟹ P(y)⟧
⟹ P(a)"
apply (drule converseI)
apply (simp (no_asm_use) add: trancl_converse [symmetric])
apply (erule trancl_induct)