Theory Trancl

(*  Title:      ZF/Trancl.thy
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1992  University of Cambridge
*)

sectionRelations: Their General Properties and Transitive Closure

theory Trancl imports Fixedpt Perm begin

definition
  refl     :: "[i,i]o"  where
    "refl(A,r)  (xA. x,x  r)"

definition
  irrefl   :: "[i,i]o"  where
    "irrefl(A,r)  xA. x,x  r"

definition
  sym      :: "io"  where
    "sym(r)  x y. x,y: r  y,x: r"

definition
  asym     :: "io"  where
    "asym(r)  x y. x,y:r  ¬ y,x:r"

definition
  antisym  :: "io"  where
    "antisym(r)  x y.x,y:r  y,x:r  x=y"

definition
  trans    :: "io"  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)  xA. yA. zA.
                          x,y: r  y,z: r  x,z: r"

definition
  rtrancl :: "ii"  ((_^*) [100] 100)  (*refl/transitive closure*)  where
    "r^*  lfp(field(r)*field(r), λs. id(field(r))  (r O s))"

definition
  trancl  :: "ii"  ((_^+) [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)"


subsectionGeneral properties of relations

subsubsectionirreflexivity

lemma irreflI:
    "x. x  A  x,x  r  irrefl(A,r)"
by (simp add: irrefl_def)

lemma irreflE: "irrefl(A,r);  x  A   x,x  r"
by (simp add: irrefl_def)

subsubsectionsymmetry

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)

subsubsectionantisymmetry

lemma antisymI:
     "x y.x,y: r;  y,x: r  x=y  antisym(r)"
by (simp add: antisym_def, blast)

lemma antisymE: "antisym(r); x,y: r;  y,x: r    x=y"
by (simp add: antisym_def, blast)

subsubsectiontransitivity

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)


subsectionTransitive 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 (simp add: relation_def)
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 (simp add: relation_def)
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 )
txtconverse direction
apply (frule rtrancl_type [THEN subsetD], clarify)
apply (erule rtrancl_induct)
apply (simp add: rtrancl_refl rtrancl_field)
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)
apply (auto simp add: trancl_converse)
done

end