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theory Transitive_Closure(* Title: HOL/Transitive_Closure.thy ID: $Id: Transitive_Closure.thy,v 1.34 2005/09/22 21:56:16 nipkow Exp $ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1992 University of Cambridge *) header {* Reflexive and Transitive closure of a relation *} theory Transitive_Closure imports Inductive uses ("../Provers/trancl.ML") begin text {* @{text rtrancl} is reflexive/transitive closure, @{text trancl} is transitive closure, @{text reflcl} is reflexive closure. These postfix operators have \emph{maximum priority}, forcing their operands to be atomic. *} consts rtrancl :: "('a × 'a) set => ('a × 'a) set" ("(_^*)" [1000] 999) inductive "r^*" intros rtrancl_refl [intro!, Pure.intro!, simp]: "(a, a) : r^*" rtrancl_into_rtrancl [Pure.intro]: "(a, b) : r^* ==> (b, c) : r ==> (a, c) : r^*" consts trancl :: "('a × 'a) set => ('a × 'a) set" ("(_^+)" [1000] 999) inductive "r^+" intros 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^+" syntax "_reflcl" :: "('a × 'a) set => ('a × 'a) set" ("(_^=)" [1000] 999) translations "r^=" == "r ∪ Id" syntax (xsymbols) rtrancl :: "('a × 'a) set => ('a × 'a) set" ("(_*)" [1000] 999) trancl :: "('a × 'a) set => ('a × 'a) set" ("(_+)" [1000] 999) "_reflcl" :: "('a × 'a) set => ('a × 'a) set" ("(_=)" [1000] 999) syntax (HTML output) rtrancl :: "('a × 'a) set => ('a × 'a) set" ("(_*)" [1000] 999) trancl :: "('a × 'a) set => ('a × 'a) set" ("(_+)" [1000] 999) "_reflcl" :: "('a × 'a) set => ('a × 'a) set" ("(_=)" [1000] 999) subsection {* Reflexive-transitive closure *} lemma r_into_rtrancl [intro]: "!!p. p ∈ r ==> p ∈ r^*" -- {* @{text rtrancl} of @{text r} contains @{text r} *} apply (simp only: split_tupled_all) apply (erule rtrancl_refl [THEN rtrancl_into_rtrancl]) done lemma rtrancl_mono: "r ⊆ s ==> r^* ⊆ s^*" -- {* monotonicity of @{text rtrancl} *} apply (rule subsetI) apply (simp only: split_tupled_all) apply (erule rtrancl.induct) apply (rule_tac [2] rtrancl_into_rtrancl, blast+) done theorem rtrancl_induct [consumes 1, induct set: rtrancl]: assumes a: "(a, b) : r^*" and cases: "P a" "!!y z. [| (a, y) : r^*; (y, z) : r; P y |] ==> P z" shows "P b" proof - from a have "a = a --> P b" by (induct "%x y. x = a --> P y" a b) (iprover intro: cases)+ thus ?thesis by iprover qed lemmas rtrancl_induct2 = rtrancl_induct[of "(ax,ay)" "(bx,by)", split_format (complete), consumes 1, case_names refl step] lemma trans_rtrancl: "trans(r^*)" -- {* transitivity of transitive closure!! -- by induction *} proof (rule transI) fix x y z assume "(x, y) ∈ r*" assume "(y, z) ∈ r*" thus "(x, z) ∈ r*" by induct (iprover!)+ qed lemmas rtrancl_trans = trans_rtrancl [THEN transD, standard] lemma rtranclE: "[| (a::'a,b) : r^*; (a = b) ==> P; !!y.[| (a,y) : r^*; (y,b) : r |] ==> P |] ==> P" -- {* elimination of @{text rtrancl} -- by induction on a special formula *} proof - assume major: "(a::'a,b) : r^*" case rule_context show ?thesis apply (subgoal_tac "(a::'a) = b | (EX 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 prems)+ done qed lemma converse_rtrancl_into_rtrancl: "(a, b) ∈ r ==> (b, c) ∈ r* ==> (a, c) ∈ r*" by (rule rtrancl_trans) iprover+ text {* \medskip More @{term "r^*"} equations and inclusions. *} lemma rtrancl_idemp [simp]: "(r^*)^* = r^*" apply auto apply (erule rtrancl_induct) apply (rule rtrancl_refl) apply (blast intro: rtrancl_trans) done lemma rtrancl_idemp_self_comp [simp]: "R^* O R^* = R^*" apply (rule set_ext) apply (simp only: split_tupled_all) apply (blast intro: rtrancl_trans) done lemma rtrancl_subset_rtrancl: "r ⊆ s^* ==> r^* ⊆ s^*" by (drule rtrancl_mono, simp) lemma rtrancl_subset: "R ⊆ S ==> S ⊆ R^* ==> S^* = R^*" apply (drule rtrancl_mono) apply (drule rtrancl_mono, simp) done lemma rtrancl_Un_rtrancl: "(R^* ∪ S^*)^* = (R ∪ S)^*" by (blast intro!: rtrancl_subset intro: r_into_rtrancl rtrancl_mono [THEN subsetD]) lemma rtrancl_reflcl [simp]: "(R^=)^* = R^*" by (blast intro!: rtrancl_subset intro: r_into_rtrancl) lemma rtrancl_r_diff_Id: "(r - Id)^* = r^*" apply (rule sym) apply (rule rtrancl_subset, blast, clarify) apply (rename_tac a b) apply (case_tac "a = b", blast) apply (blast intro!: r_into_rtrancl) done theorem rtrancl_converseD: assumes r: "(x, y) ∈ (r^-1)^*" shows "(y, x) ∈ r^*" proof - from r show ?thesis by induct (iprover intro: rtrancl_trans dest!: converseD)+ qed theorem rtrancl_converseI: assumes r: "(y, x) ∈ r^*" shows "(x, y) ∈ (r^-1)^*" proof - from r show ?thesis by induct (iprover intro: rtrancl_trans converseI)+ qed lemma rtrancl_converse: "(r^-1)^* = (r^*)^-1" by (fast dest!: rtrancl_converseD intro!: rtrancl_converseI) theorem converse_rtrancl_induct[consumes 1]: assumes major: "(a, b) : r^*" and cases: "P b" "!!y z. [| (y, z) : r; (z, b) : r^*; P z |] ==> P y" shows "P a" proof - from rtrancl_converseI [OF major] show ?thesis by induct (iprover intro: cases dest!: converseD rtrancl_converseD)+ qed lemmas converse_rtrancl_induct2 = converse_rtrancl_induct[of "(ax,ay)" "(bx,by)", split_format (complete), consumes 1, case_names refl step] lemma converse_rtranclE: "[| (x,z):r^*; x=z ==> P; !!y. [| (x,y):r; (y,z):r^* |] ==> P |] ==> P" proof - assume major: "(x,z):r^*" case rule_context show ?thesis apply (subgoal_tac "x = z | (EX y. (x,y) : r & (y,z) : r^*)") apply (rule_tac [2] major [THEN converse_rtrancl_induct]) prefer 2 apply iprover prefer 2 apply iprover apply (erule asm_rl exE disjE conjE prems)+ done qed ML_setup {* bind_thm ("converse_rtranclE2", split_rule (read_instantiate [("x","(xa,xb)"), ("z","(za,zb)")] (thm "converse_rtranclE"))); *} 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 Un (r O r^*)" by (auto intro: rtrancl_into_rtrancl elim: rtranclE) subsection {* Transitive closure *} lemma trancl_mono: "!!p. p ∈ r^+ ==> r ⊆ s ==> p ∈ s^+" apply (simp only: 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 {* \medskip Conversions between @{text trancl} and @{text rtrancl}. *} lemma trancl_into_rtrancl: "(a, b) ∈ r^+ ==> (a, b) ∈ r^*" by (erule trancl.induct) iprover+ lemma rtrancl_into_trancl1: assumes r: "(a, b) ∈ r^*" shows "!!c. (b, c) ∈ r ==> (a, c) ∈ r^+" using r by induct iprover+ lemma rtrancl_into_trancl2: "[| (a,b) : r; (b,c) : r^* |] ==> (a,c) : r^+" -- {* intro rule from @{text r} and @{text rtrancl} *} apply (erule rtranclE, iprover) apply (rule rtrancl_trans [THEN rtrancl_into_trancl1]) apply (assumption | rule r_into_rtrancl)+ done lemma trancl_induct [consumes 1, induct set: trancl]: assumes a: "(a,b) : r^+" and cases: "!!y. (a, y) : r ==> P y" "!!y z. (a,y) : r^+ ==> (y, z) : r ==> P y ==> P z" shows "P b" -- {* Nice induction rule for @{text trancl} *} proof - from a have "a = a --> P b" by (induct "%x y. x = a --> P y" a b) (iprover intro: cases)+ thus ?thesis by iprover qed lemma trancl_trans_induct: "[| (x,y) : r^+; !!x y. (x,y) : r ==> P x y; !!x y z. [| (x,y) : r^+; P x y; (y,z) : r^+; P y z |] ==> P x z |] ==> P x y" -- {* Another induction rule for trancl, incorporating transitivity *} proof - assume major: "(x,y) : r^+" case rule_context show ?thesis by (iprover intro: r_into_trancl major [THEN trancl_induct] prems) qed inductive_cases tranclE: "(a, b) : r^+" lemma trancl_unfold: "r^+ = r Un (r O r^+)" by (auto intro: trancl_into_trancl elim: tranclE) lemma trans_trancl: "trans(r^+)" -- {* Transitivity of @{term "r^+"} *} proof (rule transI) fix x y z assume "(x, y) ∈ r^+" assume "(y, z) ∈ r^+" thus "(x, z) ∈ r^+" by induct (iprover!)+ qed lemmas trancl_trans = trans_trancl [THEN transD, standard] lemma rtrancl_trancl_trancl: assumes r: "(x, y) ∈ r^*" shows "!!z. (y, z) ∈ r^+ ==> (x, z) ∈ r^+" using r by induct (iprover intro: trancl_trans)+ lemma trancl_into_trancl2: "(a, b) ∈ r ==> (b, c) ∈ r^+ ==> (a, c) ∈ r^+" by (erule transD [OF trans_trancl r_into_trancl]) lemma trancl_insert: "(insert (y, x) r)^+ = r^+ ∪ {(a, b). (a, y) ∈ r^* ∧ (x, b) ∈ r^*}" -- {* primitive recursion for @{text 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 r_into_trancl 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_converseI: "(x, y) ∈ (r^+)^-1 ==> (x, y) ∈ (r^-1)^+" apply (drule converseD) apply (erule trancl.induct) apply (iprover intro: converseI trancl_trans)+ done lemma trancl_converseD: "(x, y) ∈ (r^-1)^+ ==> (x, y) ∈ (r^+)^-1" apply (rule converseI) apply (erule trancl.induct) apply (iprover dest: converseD intro: trancl_trans)+ done lemma trancl_converse: "(r^-1)^+ = (r^+)^-1" by (fastsimp simp add: split_tupled_all intro!: trancl_converseI trancl_converseD) lemma converse_trancl_induct: "[| (a,b) : r^+; !!y. (y,b) : r ==> P(y); !!y z.[| (y,z) : r; (z,b) : r^+; P(z) |] ==> P(y) |] ==> P(a)" proof - assume major: "(a,b) : r^+" case rule_context show ?thesis apply (rule major [THEN converseI, THEN trancl_converseI [THEN trancl_induct]]) apply (rule prems) apply (erule converseD) apply (blast intro: prems dest!: trancl_converseD) done qed lemma tranclD: "(x, y) ∈ R^+ ==> EX z. (x, z) ∈ R ∧ (z, y) ∈ R^*" apply (erule converse_trancl_induct, auto) apply (blast intro: rtrancl_trans) done lemma irrefl_tranclI: "r^-1 ∩ r^* = {} ==> (x, x) ∉ r^+" by(blast elim: tranclE dest: trancl_into_rtrancl) lemma irrefl_trancl_rD: "!!X. ALL 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" apply (erule rtrancl_induct, auto) done 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 reflcl_trancl [simp]: "(r^+)^= = r^*" apply safe apply (erule trancl_into_rtrancl) apply (blast elim: rtranclE dest: rtrancl_into_trancl1) done 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 trancl_empty [simp]: "{}^+ = {}" by (auto elim: trancl_induct) lemma rtrancl_empty [simp]: "{}^* = Id" by (rule subst [OF reflcl_trancl]) simp lemma rtranclD: "(a, b) ∈ R^* ==> a = b ∨ a ≠ b ∧ (a, b) ∈ R^+" by (force simp add: reflcl_trancl [symmetric] simp del: reflcl_trancl) lemma rtrancl_eq_or_trancl: "(x,y) ∈ R* = (x=y ∨ x≠y ∧ (x,y) ∈ R+)" by (fast elim: trancl_into_rtrancl dest: rtranclD) text {* @{text Domain} and @{text 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 Un 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_def) (blast dest: tranclD) lemma trancl_range [simp]: "Range (r^+) = Range r" by (simp add: Range_def trancl_converse [symmetric]) lemma Not_Domain_rtrancl: "x ~: Domain R ==> ((x, y) : R^*) = (x = y)" apply auto by (erule rev_mp, erule rtrancl_induct, auto) text {* More about converse @{text rtrancl} and @{text 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 [rule_format]: "(a, b) ∈ r+ ==> (b, c) ∈ r --> (a,c) ∈ r+" apply (erule trancl_induct) apply (fast intro: r_r_into_trancl) apply (fast intro: r_r_into_trancl trancl_trans) done lemma trancl_rtrancl_trancl: "(a, b) ∈ r+ ==> (b, c) ∈ r* ==> (a, c) ∈ r+" apply (drule tranclD) apply (erule exE, erule conjE) apply (drule rtrancl_trans, assumption) apply (drule rtrancl_into_trancl2, assumption, assumption) done lemmas transitive_closure_trans [trans] = r_r_into_trancl trancl_trans rtrancl_trans trancl_into_trancl trancl_into_trancl2 rtrancl_into_rtrancl converse_rtrancl_into_rtrancl rtrancl_trancl_trancl trancl_rtrancl_trancl declare trancl_into_rtrancl [elim] declare rtranclE [cases set: rtrancl] declare tranclE [cases set: trancl] subsection {* Setup of transitivity reasoner *} use "../Provers/trancl.ML"; ML_setup {* structure Trancl_Tac = Trancl_Tac_Fun ( struct val r_into_trancl = thm "r_into_trancl"; val trancl_trans = thm "trancl_trans"; val rtrancl_refl = thm "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 (Trueprop $ t) = let fun dec (Const ("op :", _) $ (Const ("Pair", _) $ a $ b) $ rel ) = let fun decr (Const ("Transitive_Closure.rtrancl", _ ) $ r) = (r,"r*") | decr (Const ("Transitive_Closure.trancl", _ ) $ 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; end); (* struct *) simpset_ref() := simpset () addSolver (mk_solver "Trancl" (fn _ => Trancl_Tac.trancl_tac)) addSolver (mk_solver "Rtrancl" (fn _ => Trancl_Tac.rtrancl_tac)); *} (* Optional methods method_setup trancl = {* Method.no_args (Method.SIMPLE_METHOD' HEADGOAL (trancl_tac)) *} {* simple transitivity reasoner *} method_setup rtrancl = {* Method.no_args (Method.SIMPLE_METHOD' HEADGOAL (rtrancl_tac)) *} {* simple transitivity reasoner *} *) end
lemma r_into_rtrancl:
p ∈ r ==> p ∈ r*
lemma rtrancl_mono:
r ⊆ s ==> r* ⊆ s*
theorem rtrancl_induct:
[| (a, b) ∈ r*; P a; !!y z. [| (a, y) ∈ r*; (y, z) ∈ r; P y |] ==> P z |] ==> P b
lemmas rtrancl_induct2:
[| ((ax, ay), bx, by) ∈ r*; P ax ay; !!a b aa ba. [| ((ax, ay), a, b) ∈ r*; ((a, b), aa, ba) ∈ r; P a b |] ==> P aa ba |] ==> P bx by
lemmas rtrancl_induct2:
[| ((ax, ay), bx, by) ∈ r*; P ax ay; !!a b aa ba. [| ((ax, ay), a, b) ∈ r*; ((a, b), aa, ba) ∈ r; P a b |] ==> P aa ba |] ==> P bx by
lemma trans_rtrancl:
trans (r*)
lemmas rtrancl_trans:
[| (a, b) ∈ r*; (b, c) ∈ r* |] ==> (a, c) ∈ r*
lemmas rtrancl_trans:
[| (a, b) ∈ r*; (b, c) ∈ r* |] ==> (a, c) ∈ r*
lemma rtranclE:
[| (a, b) ∈ r*; a = b ==> P; !!y. [| (a, y) ∈ r*; (y, b) ∈ r |] ==> P |] ==> P
lemma converse_rtrancl_into_rtrancl:
[| (a, b) ∈ r; (b, c) ∈ r* |] ==> (a, c) ∈ r*
lemma rtrancl_idemp:
(r*)* = r*
lemma rtrancl_idemp_self_comp:
R* O R* = R*
lemma rtrancl_subset_rtrancl:
r ⊆ s* ==> r* ⊆ s*
lemma rtrancl_subset:
[| R ⊆ S; S ⊆ R* |] ==> S* = R*
lemma rtrancl_Un_rtrancl:
(R* ∪ S*)* = (R ∪ S)*
lemma rtrancl_reflcl:
(R=)* = R*
lemma rtrancl_r_diff_Id:
(r - Id)* = r*
theorem rtrancl_converseD:
(x, y) ∈ (r^-1)* ==> (y, x) ∈ r*
theorem rtrancl_converseI:
(y, x) ∈ r* ==> (x, y) ∈ (r^-1)*
lemma rtrancl_converse:
(r^-1)* = (r*)^-1
theorem converse_rtrancl_induct:
[| (a, b) ∈ r*; P b; !!y z. [| (y, z) ∈ r; (z, b) ∈ r*; P z |] ==> P y |] ==> P a
lemmas converse_rtrancl_induct2:
[| ((ax, ay), bx, by) ∈ r*; P bx by; !!a b aa ba. [| ((a, b), aa, ba) ∈ r; ((aa, ba), bx, by) ∈ r*; P aa ba |] ==> P a b |] ==> P ax ay
lemmas converse_rtrancl_induct2:
[| ((ax, ay), bx, by) ∈ r*; P bx by; !!a b aa ba. [| ((a, b), aa, ba) ∈ r; ((aa, ba), bx, by) ∈ r*; P aa ba |] ==> P a b |] ==> P ax ay
lemma converse_rtranclE:
[| (x, z) ∈ r*; x = z ==> P; !!y. [| (x, y) ∈ r; (y, z) ∈ r* |] ==> P |] ==> P
theorem converse_rtranclE2:
[| ((xa, xb), za, zb) ∈ r*; (xa, xb) = (za, zb) ==> P; !!a b. [| ((xa, xb), a, b) ∈ r; ((a, b), za, zb) ∈ r* |] ==> P |] ==> P
lemma r_comp_rtrancl_eq:
r O r* = r* O r
lemma rtrancl_unfold:
r* = Id ∪ (r O r*)
lemma trancl_mono:
[| p ∈ r+; r ⊆ s |] ==> p ∈ s+
lemma r_into_trancl':
p ∈ r ==> p ∈ r+
lemma trancl_into_rtrancl:
(a, b) ∈ r+ ==> (a, b) ∈ r*
lemma rtrancl_into_trancl1:
[| (a, b) ∈ r*; (b, c) ∈ r |] ==> (a, c) ∈ r+
lemma rtrancl_into_trancl2:
[| (a, b) ∈ r; (b, c) ∈ r* |] ==> (a, c) ∈ r+
lemma trancl_induct:
[| (a, b) ∈ r+; !!y. (a, y) ∈ r ==> P y; !!y z. [| (a, y) ∈ r+; (y, z) ∈ r; P y |] ==> P z |] ==> P b
lemma trancl_trans_induct:
[| (x, y) ∈ r+; !!x y. (x, y) ∈ r ==> P x y; !!x y z. [| (x, y) ∈ r+; P x y; (y, z) ∈ r+; P y z |] ==> P x z |] ==> P x y
lemmas tranclE:
[| (a, b) ∈ r+; (a, b) ∈ r ==> P; !!b. [| (a, b) ∈ r+; (b, b) ∈ r |] ==> P |] ==> P
lemma trancl_unfold:
r+ = r ∪ (r O r+)
lemma trans_trancl:
trans (r+)
lemmas trancl_trans:
[| (a, b) ∈ r+; (b, c) ∈ r+ |] ==> (a, c) ∈ r+
lemmas trancl_trans:
[| (a, b) ∈ r+; (b, c) ∈ r+ |] ==> (a, c) ∈ r+
lemma rtrancl_trancl_trancl:
[| (x, y) ∈ r*; (y, z) ∈ r+ |] ==> (x, z) ∈ r+
lemma trancl_into_trancl2:
[| (a, b) ∈ r; (b, c) ∈ r+ |] ==> (a, c) ∈ r+
lemma trancl_insert:
(insert (y, x) r)+ = r+ ∪ {(a, b). (a, y) ∈ r* ∧ (x, b) ∈ r*}
lemma trancl_converseI:
(x, y) ∈ (r+)^-1 ==> (x, y) ∈ (r^-1)+
lemma trancl_converseD:
(x, y) ∈ (r^-1)+ ==> (x, y) ∈ (r+)^-1
lemma trancl_converse:
(r^-1)+ = (r+)^-1
lemma converse_trancl_induct:
[| (a, b) ∈ r+; !!y. (y, b) ∈ r ==> P y; !!y z. [| (y, z) ∈ r; (z, b) ∈ r+; P z |] ==> P y |] ==> P a
lemma tranclD:
(x, y) ∈ R+ ==> ∃z. (x, z) ∈ R ∧ (z, y) ∈ R*
lemma irrefl_tranclI:
r^-1 ∩ r* = {} ==> (x, x) ∉ r+
lemma irrefl_trancl_rD:
[| ∀x. (x, x) ∉ r+; (x, y) ∈ r |] ==> x ≠ y
lemma trancl_subset_Sigma_aux:
[| (a, b) ∈ r*; r ⊆ A × A |] ==> a = b ∨ a ∈ A
lemma trancl_subset_Sigma:
r ⊆ A × A ==> r+ ⊆ A × A
lemma reflcl_trancl:
(r+)= = r*
lemma trancl_reflcl:
(r=)+ = r*
lemma trancl_empty:
{}+ = {}
lemma rtrancl_empty:
{}* = Id
lemma rtranclD:
(a, b) ∈ R* ==> a = b ∨ a ≠ b ∧ (a, b) ∈ R+
lemma rtrancl_eq_or_trancl:
((x, y) ∈ R*) = (x = y ∨ x ≠ y ∧ (x, y) ∈ R+)
lemma Domain_rtrancl:
Domain (R*) = UNIV
lemma Range_rtrancl:
Range (R*) = UNIV
lemma rtrancl_Un_subset:
R* ∪ S* ⊆ (R ∪ S)*
lemma in_rtrancl_UnI:
x ∈ R* ∨ x ∈ S* ==> x ∈ (R ∪ S)*
lemma trancl_domain:
Domain (r+) = Domain r
lemma trancl_range:
Range (r+) = Range r
lemma Not_Domain_rtrancl:
x ∉ Domain R ==> ((x, y) ∈ R*) = (x = y)
lemma single_valued_confluent:
[| single_valued r; (x, y) ∈ r*; (x, z) ∈ r* |] ==> (y, z) ∈ r* ∨ (z, y) ∈ r*
lemma r_r_into_trancl:
[| (a, b) ∈ R; (b, c) ∈ R |] ==> (a, c) ∈ R+
lemma trancl_into_trancl:
[| (a, b) ∈ r+; (b, c) ∈ r |] ==> (a, c) ∈ r+
lemma trancl_rtrancl_trancl:
[| (a, b) ∈ r+; (b, c) ∈ r* |] ==> (a, c) ∈ r+
lemmas transitive_closure_trans:
[| (a, b) ∈ R; (b, c) ∈ R |] ==> (a, c) ∈ R+
[| (a, b) ∈ r+; (b, c) ∈ r+ |] ==> (a, c) ∈ r+
[| (a, b) ∈ r*; (b, c) ∈ r* |] ==> (a, c) ∈ r*
[| (a, b) ∈ r+; (b, c) ∈ r |] ==> (a, c) ∈ r+
[| (a, b) ∈ r; (b, c) ∈ r+ |] ==> (a, c) ∈ r+
[| (a, b) ∈ r*; (b, c) ∈ r |] ==> (a, c) ∈ r*
[| (a, b) ∈ r; (b, c) ∈ r* |] ==> (a, c) ∈ r*
[| (x, y) ∈ r*; (y, z) ∈ r+ |] ==> (x, z) ∈ r+
[| (a, b) ∈ r+; (b, c) ∈ r* |] ==> (a, c) ∈ r+
lemmas transitive_closure_trans:
[| (a, b) ∈ R; (b, c) ∈ R |] ==> (a, c) ∈ R+
[| (a, b) ∈ r+; (b, c) ∈ r+ |] ==> (a, c) ∈ r+
[| (a, b) ∈ r*; (b, c) ∈ r* |] ==> (a, c) ∈ r*
[| (a, b) ∈ r+; (b, c) ∈ r |] ==> (a, c) ∈ r+
[| (a, b) ∈ r; (b, c) ∈ r+ |] ==> (a, c) ∈ r+
[| (a, b) ∈ r*; (b, c) ∈ r |] ==> (a, c) ∈ r*
[| (a, b) ∈ r; (b, c) ∈ r* |] ==> (a, c) ∈ r*
[| (x, y) ∈ r*; (y, z) ∈ r+ |] ==> (x, z) ∈ r+
[| (a, b) ∈ r+; (b, c) ∈ r* |] ==> (a, c) ∈ r+