author | wenzelm |
Fri, 03 Aug 2007 16:28:22 +0200 | |
changeset 24143 | 90a9a6fe0d01 |
parent 23743 | 52fbc991039f |
child 25295 | 12985023be5e |
permissions | -rw-r--r-- |
(* Title: HOL/Transitive_Closure.thy ID: $Id$ 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 Predicate uses "~~/src/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. *} inductive_set rtrancl :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" ("(_^*)" [1000] 999) for r :: "('a \<times> '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 \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" ("(_^+)" [1000] 999) for r :: "('a \<times> '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) abbreviation reflclp :: "('a => 'a => bool) => 'a => 'a => bool" ("(_^==)" [1000] 1000) where "r^== == sup r op =" abbreviation reflcl :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_^=)" [1000] 999) where "r^= == r \<union> Id" notation (xsymbols) rtranclp ("(_\<^sup>*\<^sup>*)" [1000] 1000) and tranclp ("(_\<^sup>+\<^sup>+)" [1000] 1000) and reflclp ("(_\<^sup>=\<^sup>=)" [1000] 1000) and rtrancl ("(_\<^sup>*)" [1000] 999) and trancl ("(_\<^sup>+)" [1000] 999) and reflcl ("(_\<^sup>=)" [1000] 999) notation (HTML output) rtranclp ("(_\<^sup>*\<^sup>*)" [1000] 1000) and tranclp ("(_\<^sup>+\<^sup>+)" [1000] 1000) and reflclp ("(_\<^sup>=\<^sup>=)" [1000] 1000) and rtrancl ("(_\<^sup>*)" [1000] 999) and trancl ("(_\<^sup>+)" [1000] 999) and reflcl ("(_\<^sup>=)" [1000] 999) subsection {* Reflexive-transitive closure *} lemma reflcl_set_eq [pred_set_conv]: "(sup (\<lambda>x y. (x, y) \<in> r) op =) = (\<lambda>x y. (x, y) \<in> r Un Id)" by (simp add: expand_fun_eq) lemma r_into_rtrancl [intro]: "!!p. p \<in> r ==> p \<in> 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 r_into_rtranclp [intro]: "r x y ==> r^** x y" -- {* @{text rtrancl} of @{text r} contains @{text r} *} by (erule rtranclp.rtrancl_refl [THEN rtranclp.rtrancl_into_rtrancl]) lemma rtranclp_mono: "r \<le> s ==> r^** \<le> s^**" -- {* monotonicity of @{text rtrancl} *} apply (rule predicate2I) apply (erule rtranclp.induct) apply (rule_tac [2] rtranclp.rtrancl_into_rtrancl, blast+) done lemmas rtrancl_mono = rtranclp_mono [to_set] theorem rtranclp_induct [consumes 1, 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" 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_induct [consumes 1, 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 reflexive_rtrancl: "reflexive (r^*)" by (unfold refl_def) fast lemma trans_rtrancl: "trans(r^*)" -- {* transitivity of transitive closure!! -- by induction *} proof (rule transI) fix x y z assume "(x, y) \<in> r\<^sup>*" assume "(y, z) \<in> r\<^sup>*" thus "(x, z) \<in> r\<^sup>*" by induct (iprover!)+ qed lemmas rtrancl_trans = trans_rtrancl [THEN transD, standard] lemma rtranclp_trans: assumes xy: "r^** x y" and yz: "r^** y z" shows "r^** x z" using yz xy by induct iprover+ lemma rtranclE: assumes major: "(a::'a,b) : r^*" and cases: "(a = b) ==> P" "!!y. [| (a,y) : r^*; (y,b) : r |] ==> P" shows P -- {* elimination of @{text rtrancl} -- by induction on a special formula *} 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 cases)+ done lemma rtrancl_Int_subset: "[| Id \<subseteq> s; r O (r^* \<inter> s) \<subseteq> s|] ==> r^* \<subseteq> s" apply (rule subsetI) apply (rule_tac p="x" in PairE, clarify) apply (erule rtrancl_induct, auto) done lemma converse_rtranclp_into_rtranclp: "r a b \<Longrightarrow> r\<^sup>*\<^sup>* b c \<Longrightarrow> r\<^sup>*\<^sup>* a c" by (rule rtranclp_trans) iprover+ lemmas converse_rtrancl_into_rtrancl = converse_rtranclp_into_rtranclp [to_set] text {* \medskip 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_ext) apply (simp only: split_tupled_all) apply (blast intro: rtrancl_trans) done lemma rtrancl_subset_rtrancl: "r \<subseteq> s^* ==> r^* \<subseteq> s^*" by (drule rtrancl_mono, simp) lemma rtranclp_subset: "R \<le> S ==> S \<le> R^** ==> S^** = R^**" apply (drule rtranclp_mono) apply (drule rtranclp_mono, 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_reflcl [simp]: "(R^==)^** = R^**" by (blast intro!: rtranclp_subset) lemmas rtrancl_reflcl [simp] = rtranclp_reflcl [to_set] 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 lemma rtranclp_r_diff_Id: "(inf r op ~=)^** = r^**" apply (rule sym) apply (rule rtranclp_subset) apply blast+ done theorem rtranclp_converseD: assumes r: "(r^--1)^** x y" shows "r^** y x" proof - from r show ?thesis by induct (iprover intro: rtranclp_trans dest!: conversepD)+ qed lemmas rtrancl_converseD = rtranclp_converseD [to_set] theorem rtranclp_converseI: assumes r: "r^** y x" shows "(r^--1)^** x y" proof - from r show ?thesis by induct (iprover intro: rtranclp_trans conversepI)+ qed 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]: assumes major: "r^** a b" and cases: "P b" "!!y z. [| r y z; r^** z b; P z |] ==> P y" shows "P a" proof - from rtranclp_converseI [OF major] show ?thesis by induct (iprover intro: cases dest!: conversepD rtranclp_converseD)+ qed lemmas converse_rtrancl_induct[consumes 1] = 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: 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 | (EX 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 Un r O r^*" by (auto intro: rtrancl_into_rtrancl elim: rtranclE) subsection {* Transitive closure *} lemma trancl_mono: "!!p. p \<in> r^+ ==> r \<subseteq> s ==> p \<in> 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 {* \medskip Conversions between @{text trancl} and @{text 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: "r^** a b" shows "!!c. r b c ==> r^++ a c" using r by induct 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 @{text r} and @{text rtrancl} *} apply (erule rtranclp.cases, 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] lemma tranclp_induct [consumes 1, induct set: 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" -- {* 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 lemmas trancl_induct [consumes 1, 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] inductive_cases tranclE: "(a, b) : r^+" lemma trancl_Int_subset: "[| r \<subseteq> s; r O (r^+ \<inter> s) \<subseteq> s|] ==> r^+ \<subseteq> s" apply (rule subsetI) apply (rule_tac p="x" in PairE, clarify) apply (erule trancl_induct, auto) done lemma trancl_unfold: "r^+ = r Un r O r^+" by (auto intro: trancl_into_trancl elim: tranclE) lemma trans_trancl[simp]: "trans(r^+)" -- {* Transitivity of @{term "r^+"} *} proof (rule transI) fix x y z assume xy: "(x, y) \<in> r^+" assume "(y, z) \<in> r^+" thus "(x, z) \<in> r^+" by induct (insert xy, iprover)+ qed lemmas trancl_trans = trans_trancl [THEN transD, standard] lemma tranclp_trans: assumes xy: "r^++ x y" and yz: "r^++ y z" shows "r^++ x z" using yz xy by induct iprover+ lemma trancl_id[simp]: "trans r \<Longrightarrow> r^+ = r" apply(auto) apply(erule trancl_induct) apply assumption apply(unfold trans_def) apply(blast) done lemma rtranclp_tranclp_tranclp: assumes r: "r^** x y" shows "!!z. r^++ y z ==> r^++ x z" using r 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 trancl_insert: "(insert (y, x) r)^+ = r^+ \<union> {(a, b). (a, y) \<in> r^* \<and> (x, b) \<in> 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 tranclp_converseI: "(r^++)^--1 x y ==> (r^--1)^++ 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^--1)^++ x y ==> (r^++)^--1 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^--1)^++ = (r^++)^--1" by (fastsimp simp add: expand_fun_eq 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: 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: prems dest!: tranclp_converseD conversepD) done lemmas converse_trancl_induct = converse_tranclp_induct [to_set] lemma tranclpD: "R^++ x y ==> EX z. R x z \<and> R^** z y" apply (erule converse_tranclp_induct, auto) apply (blast intro: rtranclp_trans) done lemmas tranclD = tranclpD [to_set] lemma irrefl_tranclI: "r^-1 \<inter> r^* = {} ==> (x, x) \<notin> r^+" by (blast elim: tranclE dest: trancl_into_rtrancl) lemma irrefl_trancl_rD: "!!X. ALL x. (x, x) \<notin> r^+ ==> (x, y) \<in> r ==> x \<noteq> y" by (blast dest: r_into_trancl) lemma trancl_subset_Sigma_aux: "(a, b) \<in> r^* ==> r \<subseteq> A \<times> A ==> a = b \<or> a \<in> A" by (induct rule: rtrancl_induct) auto lemma trancl_subset_Sigma: "r \<subseteq> A \<times> A ==> r^+ \<subseteq> A \<times> 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_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] = reflcl_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 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 \<or> a \<noteq> b \<and> R^++ a b" by (force simp add: reflcl_tranclp [symmetric] simp del: reflcl_tranclp) lemmas rtranclD = rtranclpD [to_set] lemma rtrancl_eq_or_trancl: "(x,y) \<in> R\<^sup>* = (x=y \<or> x\<noteq>y \<and> (x,y) \<in> R\<^sup>+)" 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^* \<union> S^*) \<subseteq> (R Un S)^*" by (rule rtrancl_Un_rtrancl [THEN subst]) fast lemma in_rtrancl_UnI: "x \<in> R^* \<or> x \<in> S^* ==> x \<in> (R \<union> 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: "\<lbrakk> single_valued r; (x,y) \<in> r^*; (x,z) \<in> r^* \<rbrakk> \<Longrightarrow> (y,z) \<in> r^* \<or> (z,y) \<in> 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) \<in> R ==> (b, c) \<in> R ==> (a, c) \<in> R^+" by (fast intro: trancl_trans) lemma trancl_into_trancl [rule_format]: "(a, b) \<in> r\<^sup>+ ==> (b, c) \<in> r --> (a,c) \<in> r\<^sup>+" apply (erule trancl_induct) apply (fast intro: r_r_into_trancl) apply (fast intro: r_r_into_trancl trancl_trans) done lemma tranclp_rtranclp_tranclp: "r\<^sup>+\<^sup>+ a b ==> r\<^sup>*\<^sup>* b c ==> r\<^sup>+\<^sup>+ a c" apply (drule tranclpD) apply (erule exE, erule conjE) apply (drule rtranclp_trans, assumption) apply (drule rtranclp_into_tranclp2, assumption, assumption) 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] declare rtranclE [cases set: rtrancl] declare tranclE [cases set: trancl] subsection {* Setup of transitivity reasoner *} ML_setup {* structure Trancl_Tac = Trancl_Tac_Fun ( struct 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 (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); structure Tranclp_Tac = Trancl_Tac_Fun ( struct 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 (Trueprop $ t) = let fun dec (rel $ a $ b) = let fun decr (Const ("Transitive_Closure.rtranclp", _ ) $ r) = (r,"r*") | decr (Const ("Transitive_Closure.tranclp", _ ) $ r) = (r,"r+") | decr r = (r,"r"); val (rel,r) = decr rel; in SOME (a, b, Envir.beta_eta_contract rel, r) end | dec _ = NONE in dec t end; end); change_simpset (fn ss => ss addSolver (mk_solver "Trancl" (fn _ => Trancl_Tac.trancl_tac)) addSolver (mk_solver "Rtrancl" (fn _ => Trancl_Tac.rtrancl_tac)) addSolver (mk_solver "Tranclp" (fn _ => Tranclp_Tac.trancl_tac)) addSolver (mk_solver "Rtranclp" (fn _ => Tranclp_Tac.rtrancl_tac))); *} (* Optional methods *) method_setup trancl = {* Method.no_args (Method.SIMPLE_METHOD' Trancl_Tac.trancl_tac) *} {* simple transitivity reasoner *} method_setup rtrancl = {* Method.no_args (Method.SIMPLE_METHOD' Trancl_Tac.rtrancl_tac) *} {* simple transitivity reasoner *} method_setup tranclp = {* Method.no_args (Method.SIMPLE_METHOD' Tranclp_Tac.trancl_tac) *} {* simple transitivity reasoner (predicate version) *} method_setup rtranclp = {* Method.no_args (Method.SIMPLE_METHOD' Tranclp_Tac.rtrancl_tac) *} {* simple transitivity reasoner (predicate version) *} end