author | nipkow |
Mon, 17 Mar 2003 18:38:50 +0100 | |
changeset 13867 | 1fdecd15437f |
parent 13726 | 9550a6f4ed4a |
child 14208 | 144f45277d5a |
permissions | -rw-r--r-- |
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(* Title: HOL/Transitive_Closure.thy |
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ID: $Id$ |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Copyright 1992 University of Cambridge |
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*) |
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header {* Reflexive and Transitive closure of a relation *} |
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theory Transitive_Closure = Inductive: |
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text {* |
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@{text rtrancl} is reflexive/transitive closure, |
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@{text trancl} is transitive closure, |
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@{text reflcl} is reflexive closure. |
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These postfix operators have \emph{maximum priority}, forcing their |
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operands to be atomic. |
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*} |
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consts |
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rtrancl :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_^*)" [1000] 999) |
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inductive "r^*" |
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intros |
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rtrancl_refl [intro!, CPure.intro!, simp]: "(a, a) : r^*" |
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rtrancl_into_rtrancl [CPure.intro]: "(a, b) : r^* ==> (b, c) : r ==> (a, c) : r^*" |
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consts |
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trancl :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_^+)" [1000] 999) |
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inductive "r^+" |
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intros |
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r_into_trancl [intro, CPure.intro]: "(a, b) : r ==> (a, b) : r^+" |
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trancl_into_trancl [CPure.intro]: "(a, b) : r^+ ==> (b, c) : r ==> (a,c) : r^+" |
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syntax |
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"_reflcl" :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_^=)" [1000] 999) |
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translations |
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"r^=" == "r \<union> Id" |
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syntax (xsymbols) |
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rtrancl :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_\\<^sup>*)" [1000] 999) |
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trancl :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_\\<^sup>+)" [1000] 999) |
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"_reflcl" :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_\\<^sup>=)" [1000] 999) |
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subsection {* Reflexive-transitive closure *} |
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lemma r_into_rtrancl [intro]: "!!p. p \<in> r ==> p \<in> r^*" |
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-- {* @{text rtrancl} of @{text r} contains @{text r} *} |
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apply (simp only: split_tupled_all) |
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apply (erule rtrancl_refl [THEN rtrancl_into_rtrancl]) |
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done |
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lemma rtrancl_mono: "r \<subseteq> s ==> r^* \<subseteq> s^*" |
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-- {* monotonicity of @{text rtrancl} *} |
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apply (rule subsetI) |
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apply (simp only: split_tupled_all) |
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apply (erule rtrancl.induct) |
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apply (rule_tac [2] rtrancl_into_rtrancl) |
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apply blast+ |
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done |
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theorem rtrancl_induct [consumes 1, induct set: rtrancl]: |
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assumes a: "(a, b) : r^*" |
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and cases: "P a" "!!y z. [| (a, y) : r^*; (y, z) : r; P y |] ==> P z" |
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shows "P b" |
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proof - |
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from a have "a = a --> P b" |
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by (induct "%x y. x = a --> P y" a b) (rules intro: cases)+ |
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thus ?thesis by rules |
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qed |
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ML_setup {* |
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bind_thm ("rtrancl_induct2", split_rule |
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(read_instantiate [("a","(ax,ay)"), ("b","(bx,by)")] (thm "rtrancl_induct"))); |
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*} |
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lemma trans_rtrancl: "trans(r^*)" |
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-- {* transitivity of transitive closure!! -- by induction *} |
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proof (rule transI) |
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fix x y z |
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assume "(x, y) \<in> r\<^sup>*" |
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assume "(y, z) \<in> r\<^sup>*" |
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thus "(x, z) \<in> r\<^sup>*" by induct (rules!)+ |
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qed |
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lemmas rtrancl_trans = trans_rtrancl [THEN transD, standard] |
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lemma rtranclE: |
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"[| (a::'a,b) : r^*; (a = b) ==> P; |
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!!y.[| (a,y) : r^*; (y,b) : r |] ==> P |
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|] ==> P" |
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-- {* elimination of @{text rtrancl} -- by induction on a special formula *} |
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proof - |
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assume major: "(a::'a,b) : r^*" |
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case rule_context |
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show ?thesis |
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apply (subgoal_tac "(a::'a) = b | (EX y. (a,y) : r^* & (y,b) : r)") |
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apply (rule_tac [2] major [THEN rtrancl_induct]) |
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prefer 2 apply (blast!) |
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prefer 2 apply (blast!) |
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apply (erule asm_rl exE disjE conjE prems)+ |
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done |
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qed |
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lemma converse_rtrancl_into_rtrancl: |
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"(a, b) \<in> r \<Longrightarrow> (b, c) \<in> r\<^sup>* \<Longrightarrow> (a, c) \<in> r\<^sup>*" |
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by (rule rtrancl_trans) rules+ |
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text {* |
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\medskip More @{term "r^*"} equations and inclusions. |
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*} |
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lemma rtrancl_idemp [simp]: "(r^*)^* = r^*" |
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apply auto |
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apply (erule rtrancl_induct) |
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apply (rule rtrancl_refl) |
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apply (blast intro: rtrancl_trans) |
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done |
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lemma rtrancl_idemp_self_comp [simp]: "R^* O R^* = R^*" |
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apply (rule set_ext) |
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apply (simp only: split_tupled_all) |
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apply (blast intro: rtrancl_trans) |
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done |
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lemma rtrancl_subset_rtrancl: "r \<subseteq> s^* ==> r^* \<subseteq> s^*" |
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apply (drule rtrancl_mono) |
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apply simp |
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done |
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lemma rtrancl_subset: "R \<subseteq> S ==> S \<subseteq> R^* ==> S^* = R^*" |
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apply (drule rtrancl_mono) |
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apply (drule rtrancl_mono) |
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apply simp |
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apply blast |
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done |
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lemma rtrancl_Un_rtrancl: "(R^* \<union> S^*)^* = (R \<union> S)^*" |
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by (blast intro!: rtrancl_subset intro: r_into_rtrancl rtrancl_mono [THEN subsetD]) |
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lemma rtrancl_reflcl [simp]: "(R^=)^* = R^*" |
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by (blast intro!: rtrancl_subset intro: r_into_rtrancl) |
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lemma rtrancl_r_diff_Id: "(r - Id)^* = r^*" |
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apply (rule sym) |
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apply (rule rtrancl_subset) |
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apply blast |
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apply clarify |
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apply (rename_tac a b) |
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apply (case_tac "a = b") |
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apply blast |
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apply (blast intro!: r_into_rtrancl) |
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done |
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theorem rtrancl_converseD: |
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assumes r: "(x, y) \<in> (r^-1)^*" |
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shows "(y, x) \<in> r^*" |
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proof - |
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from r show ?thesis |
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by induct (rules intro: rtrancl_trans dest!: converseD)+ |
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qed |
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theorem rtrancl_converseI: |
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assumes r: "(y, x) \<in> r^*" |
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shows "(x, y) \<in> (r^-1)^*" |
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proof - |
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from r show ?thesis |
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by induct (rules intro: rtrancl_trans converseI)+ |
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qed |
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lemma rtrancl_converse: "(r^-1)^* = (r^*)^-1" |
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by (fast dest!: rtrancl_converseD intro!: rtrancl_converseI) |
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theorem converse_rtrancl_induct: |
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assumes major: "(a, b) : r^*" |
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and cases: "P b" "!!y z. [| (y, z) : r; (z, b) : r^*; P z |] ==> P y" |
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shows "P a" |
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proof - |
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from rtrancl_converseI [OF major] |
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show ?thesis |
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by induct (rules intro: cases dest!: converseD rtrancl_converseD)+ |
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qed |
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ML_setup {* |
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bind_thm ("converse_rtrancl_induct2", split_rule |
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(read_instantiate [("a","(ax,ay)"),("b","(bx,by)")] (thm "converse_rtrancl_induct"))); |
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*} |
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lemma converse_rtranclE: |
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"[| (x,z):r^*; |
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x=z ==> P; |
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!!y. [| (x,y):r; (y,z):r^* |] ==> P |
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|] ==> P" |
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proof - |
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assume major: "(x,z):r^*" |
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case rule_context |
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show ?thesis |
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apply (subgoal_tac "x = z | (EX y. (x,y) : r & (y,z) : r^*)") |
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apply (rule_tac [2] major [THEN converse_rtrancl_induct]) |
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prefer 2 apply rules |
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prefer 2 apply rules |
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apply (erule asm_rl exE disjE conjE prems)+ |
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done |
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qed |
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ML_setup {* |
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bind_thm ("converse_rtranclE2", split_rule |
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(read_instantiate [("x","(xa,xb)"), ("z","(za,zb)")] (thm "converse_rtranclE"))); |
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*} |
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lemma r_comp_rtrancl_eq: "r O r^* = r^* O r" |
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by (blast elim: rtranclE converse_rtranclE |
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intro: rtrancl_into_rtrancl converse_rtrancl_into_rtrancl) |
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subsection {* Transitive closure *} |
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lemma trancl_mono: "!!p. p \<in> r^+ ==> r \<subseteq> s ==> p \<in> s^+" |
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apply (simp only: split_tupled_all) |
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apply (erule trancl.induct) |
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apply (rules dest: subsetD)+ |
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done |
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lemma r_into_trancl': "!!p. p : r ==> p : r^+" |
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by (simp only: split_tupled_all) (erule r_into_trancl) |
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text {* |
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\medskip Conversions between @{text trancl} and @{text rtrancl}. |
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*} |
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lemma trancl_into_rtrancl: "(a, b) \<in> r^+ ==> (a, b) \<in> r^*" |
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by (erule trancl.induct) rules+ |
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lemma rtrancl_into_trancl1: assumes r: "(a, b) \<in> r^*" |
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shows "!!c. (b, c) \<in> r ==> (a, c) \<in> r^+" using r |
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by induct rules+ |
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lemma rtrancl_into_trancl2: "[| (a,b) : r; (b,c) : r^* |] ==> (a,c) : r^+" |
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-- {* intro rule from @{text r} and @{text rtrancl} *} |
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apply (erule rtranclE) |
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apply rules |
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apply (rule rtrancl_trans [THEN rtrancl_into_trancl1]) |
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apply (assumption | rule r_into_rtrancl)+ |
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done |
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lemma trancl_induct [consumes 1, induct set: trancl]: |
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assumes a: "(a,b) : r^+" |
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and cases: "!!y. (a, y) : r ==> P y" |
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"!!y z. (a,y) : r^+ ==> (y, z) : r ==> P y ==> P z" |
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shows "P b" |
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-- {* Nice induction rule for @{text trancl} *} |
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proof - |
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from a have "a = a --> P b" |
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by (induct "%x y. x = a --> P y" a b) (rules intro: cases)+ |
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thus ?thesis by rules |
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qed |
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lemma trancl_trans_induct: |
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"[| (x,y) : r^+; |
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!!x y. (x,y) : r ==> P x y; |
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!!x y z. [| (x,y) : r^+; P x y; (y,z) : r^+; P y z |] ==> P x z |
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|] ==> P x y" |
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-- {* Another induction rule for trancl, incorporating transitivity *} |
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proof - |
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assume major: "(x,y) : r^+" |
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case rule_context |
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show ?thesis |
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by (rules intro: r_into_trancl major [THEN trancl_induct] prems) |
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qed |
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inductive_cases tranclE: "(a, b) : r^+" |
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lemma trans_trancl: "trans(r^+)" |
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-- {* Transitivity of @{term "r^+"} *} |
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proof (rule transI) |
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fix x y z |
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assume "(x, y) \<in> r^+" |
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assume "(y, z) \<in> r^+" |
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thus "(x, z) \<in> r^+" by induct (rules!)+ |
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qed |
12691 | 283 |
|
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lemmas trancl_trans = trans_trancl [THEN transD, standard] |
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||
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lemma rtrancl_trancl_trancl: assumes r: "(x, y) \<in> r^*" |
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shows "!!z. (y, z) \<in> r^+ ==> (x, z) \<in> r^+" using r |
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by induct (rules intro: trancl_trans)+ |
12691 | 289 |
|
290 |
lemma trancl_into_trancl2: "(a, b) \<in> r ==> (b, c) \<in> r^+ ==> (a, c) \<in> r^+" |
|
291 |
by (erule transD [OF trans_trancl r_into_trancl]) |
|
292 |
||
293 |
lemma trancl_insert: |
|
294 |
"(insert (y, x) r)^+ = r^+ \<union> {(a, b). (a, y) \<in> r^* \<and> (x, b) \<in> r^*}" |
|
295 |
-- {* primitive recursion for @{text trancl} over finite relations *} |
|
296 |
apply (rule equalityI) |
|
297 |
apply (rule subsetI) |
|
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apply (simp only: split_tupled_all) |
|
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apply (erule trancl_induct) |
|
300 |
apply blast |
|
301 |
apply (blast intro: rtrancl_into_trancl1 trancl_into_rtrancl r_into_trancl trancl_trans) |
|
302 |
apply (rule subsetI) |
|
303 |
apply (blast intro: trancl_mono rtrancl_mono |
|
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[THEN [2] rev_subsetD] rtrancl_trancl_trancl rtrancl_into_trancl2) |
|
305 |
done |
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306 |
||
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lemma trancl_converseI: "(x, y) \<in> (r^+)^-1 ==> (x, y) \<in> (r^-1)^+" |
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apply (drule converseD) |
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apply (erule trancl.induct) |
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apply (rules intro: converseI trancl_trans)+ |
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done |
312 |
||
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lemma trancl_converseD: "(x, y) \<in> (r^-1)^+ ==> (x, y) \<in> (r^+)^-1" |
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apply (rule converseI) |
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apply (erule trancl.induct) |
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apply (rules dest: converseD intro: trancl_trans)+ |
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done |
12691 | 318 |
|
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lemma trancl_converse: "(r^-1)^+ = (r^+)^-1" |
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by (fastsimp simp add: split_tupled_all |
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321 |
intro!: trancl_converseI trancl_converseD) |
12691 | 322 |
|
323 |
lemma converse_trancl_induct: |
|
324 |
"[| (a,b) : r^+; !!y. (y,b) : r ==> P(y); |
|
325 |
!!y z.[| (y,z) : r; (z,b) : r^+; P(z) |] ==> P(y) |] |
|
326 |
==> P(a)" |
|
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proof - |
|
328 |
assume major: "(a,b) : r^+" |
|
329 |
case rule_context |
|
330 |
show ?thesis |
|
331 |
apply (rule major [THEN converseI, THEN trancl_converseI [THEN trancl_induct]]) |
|
332 |
apply (rule prems) |
|
333 |
apply (erule converseD) |
|
334 |
apply (blast intro: prems dest!: trancl_converseD) |
|
335 |
done |
|
336 |
qed |
|
337 |
||
338 |
lemma tranclD: "(x, y) \<in> R^+ ==> EX z. (x, z) \<in> R \<and> (z, y) \<in> R^*" |
|
339 |
apply (erule converse_trancl_induct) |
|
340 |
apply auto |
|
341 |
apply (blast intro: rtrancl_trans) |
|
342 |
done |
|
343 |
||
13867 | 344 |
lemma irrefl_tranclI: "r^-1 \<inter> r^* = {} ==> (x, x) \<notin> r^+" |
345 |
by(blast elim: tranclE dest: trancl_into_rtrancl) |
|
12691 | 346 |
|
347 |
lemma irrefl_trancl_rD: "!!X. ALL x. (x, x) \<notin> r^+ ==> (x, y) \<in> r ==> x \<noteq> y" |
|
348 |
by (blast dest: r_into_trancl) |
|
349 |
||
350 |
lemma trancl_subset_Sigma_aux: |
|
351 |
"(a, b) \<in> r^* ==> r \<subseteq> A \<times> A ==> a = b \<or> a \<in> A" |
|
352 |
apply (erule rtrancl_induct) |
|
353 |
apply auto |
|
354 |
done |
|
355 |
||
356 |
lemma trancl_subset_Sigma: "r \<subseteq> A \<times> A ==> r^+ \<subseteq> A \<times> A" |
|
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apply (rule subsetI) |
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apply (simp only: split_tupled_all) |
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apply (erule tranclE) |
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360 |
apply (blast dest!: trancl_into_rtrancl trancl_subset_Sigma_aux)+ |
12691 | 361 |
done |
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362 |
|
11090 | 363 |
lemma reflcl_trancl [simp]: "(r^+)^= = r^*" |
11084 | 364 |
apply safe |
12691 | 365 |
apply (erule trancl_into_rtrancl) |
11084 | 366 |
apply (blast elim: rtranclE dest: rtrancl_into_trancl1) |
367 |
done |
|
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|
368 |
|
11090 | 369 |
lemma trancl_reflcl [simp]: "(r^=)^+ = r^*" |
11084 | 370 |
apply safe |
371 |
apply (drule trancl_into_rtrancl) |
|
372 |
apply simp |
|
373 |
apply (erule rtranclE) |
|
374 |
apply safe |
|
375 |
apply (rule r_into_trancl) |
|
376 |
apply simp |
|
377 |
apply (rule rtrancl_into_trancl1) |
|
378 |
apply (erule rtrancl_reflcl [THEN equalityD2, THEN subsetD]) |
|
379 |
apply fast |
|
380 |
done |
|
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|
381 |
|
11090 | 382 |
lemma trancl_empty [simp]: "{}^+ = {}" |
11084 | 383 |
by (auto elim: trancl_induct) |
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|
384 |
|
11090 | 385 |
lemma rtrancl_empty [simp]: "{}^* = Id" |
11084 | 386 |
by (rule subst [OF reflcl_trancl]) simp |
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|
387 |
|
11090 | 388 |
lemma rtranclD: "(a, b) \<in> R^* ==> a = b \<or> a \<noteq> b \<and> (a, b) \<in> R^+" |
11084 | 389 |
by (force simp add: reflcl_trancl [symmetric] simp del: reflcl_trancl) |
390 |
||
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|
391 |
|
12691 | 392 |
text {* @{text Domain} and @{text Range} *} |
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|
393 |
|
11090 | 394 |
lemma Domain_rtrancl [simp]: "Domain (R^*) = UNIV" |
11084 | 395 |
by blast |
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|
396 |
|
11090 | 397 |
lemma Range_rtrancl [simp]: "Range (R^*) = UNIV" |
11084 | 398 |
by blast |
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|
399 |
|
11090 | 400 |
lemma rtrancl_Un_subset: "(R^* \<union> S^*) \<subseteq> (R Un S)^*" |
11084 | 401 |
by (rule rtrancl_Un_rtrancl [THEN subst]) fast |
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|
402 |
|
11090 | 403 |
lemma in_rtrancl_UnI: "x \<in> R^* \<or> x \<in> S^* ==> x \<in> (R \<union> S)^*" |
11084 | 404 |
by (blast intro: subsetD [OF rtrancl_Un_subset]) |
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|
405 |
|
11090 | 406 |
lemma trancl_domain [simp]: "Domain (r^+) = Domain r" |
11084 | 407 |
by (unfold Domain_def) (blast dest: tranclD) |
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|
408 |
|
11090 | 409 |
lemma trancl_range [simp]: "Range (r^+) = Range r" |
11084 | 410 |
by (simp add: Range_def trancl_converse [symmetric]) |
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|
411 |
|
11115 | 412 |
lemma Not_Domain_rtrancl: |
12691 | 413 |
"x ~: Domain R ==> ((x, y) : R^*) = (x = y)" |
414 |
apply auto |
|
415 |
by (erule rev_mp, erule rtrancl_induct, auto) |
|
416 |
||
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|
417 |
|
12691 | 418 |
text {* More about converse @{text rtrancl} and @{text trancl}, should |
419 |
be merged with main body. *} |
|
12428
f3033eed309a
setup [trans] rules for calculational Isar reasoning
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diff
changeset
|
420 |
|
12691 | 421 |
lemma r_r_into_trancl: "(a, b) \<in> R ==> (b, c) \<in> R ==> (a, c) \<in> R^+" |
12428
f3033eed309a
setup [trans] rules for calculational Isar reasoning
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|
422 |
by (fast intro: trancl_trans) |
f3033eed309a
setup [trans] rules for calculational Isar reasoning
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changeset
|
423 |
|
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setup [trans] rules for calculational Isar reasoning
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|
424 |
lemma trancl_into_trancl [rule_format]: |
12691 | 425 |
"(a, b) \<in> r\<^sup>+ ==> (b, c) \<in> r --> (a,c) \<in> r\<^sup>+" |
426 |
apply (erule trancl_induct) |
|
12428
f3033eed309a
setup [trans] rules for calculational Isar reasoning
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changeset
|
427 |
apply (fast intro: r_r_into_trancl) |
f3033eed309a
setup [trans] rules for calculational Isar reasoning
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changeset
|
428 |
apply (fast intro: r_r_into_trancl trancl_trans) |
f3033eed309a
setup [trans] rules for calculational Isar reasoning
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changeset
|
429 |
done |
f3033eed309a
setup [trans] rules for calculational Isar reasoning
kleing
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11327
diff
changeset
|
430 |
|
f3033eed309a
setup [trans] rules for calculational Isar reasoning
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|
431 |
lemma trancl_rtrancl_trancl: |
12691 | 432 |
"(a, b) \<in> r\<^sup>+ ==> (b, c) \<in> r\<^sup>* ==> (a, c) \<in> r\<^sup>+" |
12428
f3033eed309a
setup [trans] rules for calculational Isar reasoning
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changeset
|
433 |
apply (drule tranclD) |
f3033eed309a
setup [trans] rules for calculational Isar reasoning
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11327
diff
changeset
|
434 |
apply (erule exE, erule conjE) |
f3033eed309a
setup [trans] rules for calculational Isar reasoning
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11327
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changeset
|
435 |
apply (drule rtrancl_trans, assumption) |
f3033eed309a
setup [trans] rules for calculational Isar reasoning
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changeset
|
436 |
apply (drule rtrancl_into_trancl2, assumption) |
f3033eed309a
setup [trans] rules for calculational Isar reasoning
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changeset
|
437 |
apply assumption |
f3033eed309a
setup [trans] rules for calculational Isar reasoning
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11327
diff
changeset
|
438 |
done |
f3033eed309a
setup [trans] rules for calculational Isar reasoning
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diff
changeset
|
439 |
|
12691 | 440 |
lemmas transitive_closure_trans [trans] = |
441 |
r_r_into_trancl trancl_trans rtrancl_trans |
|
442 |
trancl_into_trancl trancl_into_trancl2 |
|
443 |
rtrancl_into_rtrancl converse_rtrancl_into_rtrancl |
|
444 |
rtrancl_trancl_trancl trancl_rtrancl_trancl |
|
12428
f3033eed309a
setup [trans] rules for calculational Isar reasoning
kleing
parents:
11327
diff
changeset
|
445 |
|
f3033eed309a
setup [trans] rules for calculational Isar reasoning
kleing
parents:
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diff
changeset
|
446 |
declare trancl_into_rtrancl [elim] |
11327
cd2c27a23df1
Transitive closure is now defined via "inductive".
berghofe
parents:
11115
diff
changeset
|
447 |
|
cd2c27a23df1
Transitive closure is now defined via "inductive".
berghofe
parents:
11115
diff
changeset
|
448 |
declare rtranclE [cases set: rtrancl] |
cd2c27a23df1
Transitive closure is now defined via "inductive".
berghofe
parents:
11115
diff
changeset
|
449 |
declare tranclE [cases set: trancl] |
cd2c27a23df1
Transitive closure is now defined via "inductive".
berghofe
parents:
11115
diff
changeset
|
450 |
|
10213 | 451 |
end |