# HG changeset patch # User berghofe # Date 1184145037 -7200 # Node ID 52fbc991039f985ac92cf09267596dacf6e542f5 # Parent d6349ac8b153a06c28b74415fd92cc98aa41abcb rtrancl and trancl are now defined using inductive_set. diff -r d6349ac8b153 -r 52fbc991039f src/HOL/Transitive_Closure.thy --- a/src/HOL/Transitive_Closure.thy Wed Jul 11 11:09:15 2007 +0200 +++ b/src/HOL/Transitive_Closure.thy Wed Jul 11 11:10:37 2007 +0200 @@ -20,83 +20,74 @@ operands to be atomic. *} -inductive2 - rtrancl :: "('a \ 'a \ bool) \ 'a \ 'a \ bool" ("(_^**)" [1000] 1000) - for r :: "'a \ 'a \ bool" +inductive_set + rtrancl :: "('a \ 'a) set \ ('a \ 'a) set" ("(_^*)" [1000] 999) + for r :: "('a \ 'a) set" where - rtrancl_refl [intro!, Pure.intro!, simp]: "r^** a a" - | rtrancl_into_rtrancl [Pure.intro]: "r^** a b ==> r b c ==> r^** a c" + rtrancl_refl [intro!, Pure.intro!, simp]: "(a, a) : r^*" + | rtrancl_into_rtrancl [Pure.intro]: "(a, b) : r^* ==> (b, c) : r ==> (a, c) : r^*" -inductive2 - trancl :: "('a \ 'a \ bool) \ 'a \ 'a \ bool" ("(_^++)" [1000] 1000) - for r :: "'a \ 'a \ bool" +inductive_set + trancl :: "('a \ 'a) set \ ('a \ 'a) set" ("(_^+)" [1000] 999) + for r :: "('a \ 'a) set" where - r_into_trancl [intro, Pure.intro]: "r a b ==> r^++ a b" - | trancl_into_trancl [Pure.intro]: "r^++ a b ==> r b c ==> r^++ a c" + 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^+" -constdefs - rtrancl_set :: "('a \ 'a) set => ('a \ 'a) set" ("(_^*)" [1000] 999) - "r^* == Collect2 (member2 r)^**" - - trancl_set :: "('a \ 'a) set => ('a \ 'a) set" ("(_^+)" [1000] 999) - "r^+ == Collect2 (member2 r)^++" +notation + rtranclp ("(_^**)" [1000] 1000) and + tranclp ("(_^++)" [1000] 1000) abbreviation - reflcl :: "('a => 'a => bool) => 'a => 'a => bool" ("(_^==)" [1000] 1000) where + reflclp :: "('a => 'a => bool) => 'a => 'a => bool" ("(_^==)" [1000] 1000) where "r^== == sup r op =" abbreviation - reflcl_set :: "('a \ 'a) set => ('a \ 'a) set" ("(_^=)" [1000] 999) where + reflcl :: "('a \ 'a) set => ('a \ 'a) set" ("(_^=)" [1000] 999) where "r^= == r \ Id" notation (xsymbols) - rtrancl ("(_\<^sup>*\<^sup>*)" [1000] 1000) and - trancl ("(_\<^sup>+\<^sup>+)" [1000] 1000) and - reflcl ("(_\<^sup>=\<^sup>=)" [1000] 1000) and - rtrancl_set ("(_\<^sup>*)" [1000] 999) and - trancl_set ("(_\<^sup>+)" [1000] 999) and - reflcl_set ("(_\<^sup>=)" [1000] 999) + 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) - rtrancl ("(_\<^sup>*\<^sup>*)" [1000] 1000) and - trancl ("(_\<^sup>+\<^sup>+)" [1000] 1000) and - reflcl ("(_\<^sup>=\<^sup>=)" [1000] 1000) and - rtrancl_set ("(_\<^sup>*)" [1000] 999) and - trancl_set ("(_\<^sup>+)" [1000] 999) and - reflcl_set ("(_\<^sup>=)" [1000] 999) + 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 rtrancl_set_eq [pred_set_conv]: "(member2 r)^** = member2 (r^*)" - by (simp add: rtrancl_set_def) - -lemma reflcl_set_eq [pred_set_conv]: "(sup (member2 r) op =) = member2 (r Un Id)" +lemma reflcl_set_eq [pred_set_conv]: "(sup (\x y. (x, y) \ r) op =) = (\x y. (x, y) \ r Un Id)" by (simp add: expand_fun_eq) -lemmas rtrancl_refl [intro!, Pure.intro!, simp] = rtrancl_refl [to_set] -lemmas rtrancl_into_rtrancl [Pure.intro] = rtrancl_into_rtrancl [to_set] - 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 r_into_rtrancl' [intro]: "r x y ==> r^** x y" +lemma r_into_rtranclp [intro]: "r x y ==> r^** x y" -- {* @{text rtrancl} of @{text r} contains @{text r} *} - by (erule rtrancl.rtrancl_refl [THEN rtrancl.rtrancl_into_rtrancl]) + by (erule rtranclp.rtrancl_refl [THEN rtranclp.rtrancl_into_rtrancl]) -lemma rtrancl_mono': "r \ s ==> r^** \ s^**" +lemma rtranclp_mono: "r \ s ==> r^** \ s^**" -- {* monotonicity of @{text rtrancl} *} apply (rule predicate2I) - apply (erule rtrancl.induct) - apply (rule_tac [2] rtrancl.rtrancl_into_rtrancl, blast+) + apply (erule rtranclp.induct) + apply (rule_tac [2] rtranclp.rtrancl_into_rtrancl, blast+) done -lemmas rtrancl_mono = rtrancl_mono' [to_set] +lemmas rtrancl_mono = rtranclp_mono [to_set] -theorem rtrancl_induct' [consumes 1, induct set: rtrancl]: +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" @@ -106,10 +97,10 @@ thus ?thesis by iprover qed -lemmas rtrancl_induct [consumes 1, induct set: rtrancl_set] = rtrancl_induct' [to_set] +lemmas rtrancl_induct [consumes 1, induct set: rtrancl] = rtranclp_induct [to_set] -lemmas rtrancl_induct2' = - rtrancl_induct'[of _ "(ax,ay)" "(bx,by)", split_rule, +lemmas rtranclp_induct2 = + rtranclp_induct[of _ "(ax,ay)" "(bx,by)", split_rule, consumes 1, case_names refl step] lemmas rtrancl_induct2 = @@ -130,7 +121,7 @@ lemmas rtrancl_trans = trans_rtrancl [THEN transD, standard] -lemma rtrancl_trans': +lemma rtranclp_trans: assumes xy: "r^** x y" and yz: "r^** y z" shows "r^** x z" using yz xy @@ -155,24 +146,24 @@ apply (erule rtrancl_induct, auto) done -lemma converse_rtrancl_into_rtrancl': +lemma converse_rtranclp_into_rtranclp: "r a b \ r\<^sup>*\<^sup>* b c \ r\<^sup>*\<^sup>* a c" - by (rule rtrancl_trans') iprover+ + by (rule rtranclp_trans) iprover+ -lemmas converse_rtrancl_into_rtrancl = converse_rtrancl_into_rtrancl' [to_set] +lemmas converse_rtrancl_into_rtrancl = converse_rtranclp_into_rtranclp [to_set] text {* \medskip More @{term "r^*"} equations and inclusions. *} -lemma rtrancl_idemp' [simp]: "(r^**)^** = r^**" +lemma rtranclp_idemp [simp]: "(r^**)^** = r^**" apply (auto intro!: order_antisym) - apply (erule rtrancl_induct') - apply (rule rtrancl.rtrancl_refl) - apply (blast intro: rtrancl_trans') + apply (erule rtranclp_induct) + apply (rule rtranclp.rtrancl_refl) + apply (blast intro: rtranclp_trans) done -lemmas rtrancl_idemp [simp] = rtrancl_idemp' [to_set] +lemmas rtrancl_idemp [simp] = rtranclp_idemp [to_set] lemma rtrancl_idemp_self_comp [simp]: "R^* O R^* = R^*" apply (rule set_ext) @@ -183,22 +174,22 @@ 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) +lemma rtranclp_subset: "R \ S ==> S \ R^** ==> S^** = R^**" + apply (drule rtranclp_mono) + apply (drule rtranclp_mono, simp) done -lemmas rtrancl_subset = rtrancl_subset' [to_set] +lemmas rtrancl_subset = rtranclp_subset [to_set] -lemma rtrancl_Un_rtrancl': "(sup (R^**) (S^**))^** = (sup R S)^**" - by (blast intro!: rtrancl_subset' intro: rtrancl_mono' [THEN predicate2D]) +lemma rtranclp_sup_rtranclp: "(sup (R^**) (S^**))^** = (sup R S)^**" + by (blast intro!: rtranclp_subset intro: rtranclp_mono [THEN predicate2D]) -lemmas rtrancl_Un_rtrancl = rtrancl_Un_rtrancl' [to_set] +lemmas rtrancl_Un_rtrancl = rtranclp_sup_rtranclp [to_set] -lemma rtrancl_reflcl' [simp]: "(R^==)^** = R^**" - by (blast intro!: rtrancl_subset') +lemma rtranclp_reflcl [simp]: "(R^==)^** = R^**" + by (blast intro!: rtranclp_subset) -lemmas rtrancl_reflcl [simp] = rtrancl_reflcl' [to_set] +lemmas rtrancl_reflcl [simp] = rtranclp_reflcl [to_set] lemma rtrancl_r_diff_Id: "(r - Id)^* = r^*" apply (rule sym) @@ -208,31 +199,31 @@ apply (blast intro!: r_into_rtrancl) done -lemma rtrancl_r_diff_Id': "(inf r op ~=)^** = r^**" +lemma rtranclp_r_diff_Id: "(inf r op ~=)^** = r^**" apply (rule sym) - apply (rule rtrancl_subset') + apply (rule rtranclp_subset) apply blast+ done -theorem rtrancl_converseD': +theorem rtranclp_converseD: assumes r: "(r^--1)^** x y" shows "r^** y x" proof - from r show ?thesis - by induct (iprover intro: rtrancl_trans' dest!: conversepD)+ + by induct (iprover intro: rtranclp_trans dest!: conversepD)+ qed -lemmas rtrancl_converseD = rtrancl_converseD' [to_set] +lemmas rtrancl_converseD = rtranclp_converseD [to_set] -theorem rtrancl_converseI': +theorem rtranclp_converseI: assumes r: "r^** y x" shows "(r^--1)^** x y" proof - from r show ?thesis - by induct (iprover intro: rtrancl_trans' conversepI)+ + by induct (iprover intro: rtranclp_trans conversepI)+ qed -lemmas rtrancl_converseI = rtrancl_converseI' [to_set] +lemmas rtrancl_converseI = rtranclp_converseI [to_set] lemma rtrancl_converse: "(r^-1)^* = (r^*)^-1" by (fast dest!: rtrancl_converseD intro!: rtrancl_converseI) @@ -240,41 +231,41 @@ lemma sym_rtrancl: "sym r ==> sym (r^*)" by (simp only: sym_conv_converse_eq rtrancl_converse [symmetric]) -theorem converse_rtrancl_induct'[consumes 1]: +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 rtrancl_converseI' [OF major] + from rtranclp_converseI [OF major] show ?thesis - by induct (iprover intro: cases dest!: conversepD rtrancl_converseD')+ + by induct (iprover intro: cases dest!: conversepD rtranclp_converseD)+ qed -lemmas converse_rtrancl_induct[consumes 1] = converse_rtrancl_induct' [to_set] +lemmas converse_rtrancl_induct[consumes 1] = converse_rtranclp_induct [to_set] -lemmas converse_rtrancl_induct2' = - converse_rtrancl_induct'[of _ "(ax,ay)" "(bx,by)", split_rule, +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_rtranclE': +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_rtrancl_induct']) + 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_rtranclE' [to_set] +lemmas converse_rtranclE = converse_rtranclpE [to_set] -lemmas converse_rtranclE2' = converse_rtranclE' [of _ "(xa,xb)" "(za,zb)", split_rule] +lemmas converse_rtranclpE2 = converse_rtranclpE [of _ "(xa,xb)" "(za,zb)", split_rule] lemmas converse_rtranclE2 = converse_rtranclE [of "(xa,xb)" "(za,zb)", split_rule] @@ -288,14 +279,8 @@ subsection {* Transitive closure *} -lemma trancl_set_eq [pred_set_conv]: "(member2 r)^++ = member2 (r^+)" - by (simp add: trancl_set_def) - -lemmas r_into_trancl [intro, Pure.intro] = r_into_trancl [to_set] -lemmas trancl_into_trancl [Pure.intro] = trancl_into_trancl [to_set] - lemma trancl_mono: "!!p. p \ r^+ ==> r \ s ==> p \ s^+" - apply (simp add: split_tupled_all trancl_set_def) + apply (simp add: split_tupled_all) apply (erule trancl.induct) apply (iprover dest: subsetD)+ done @@ -307,27 +292,27 @@ \medskip Conversions between @{text trancl} and @{text rtrancl}. *} -lemma trancl_into_rtrancl': "r^++ a b ==> r^** a b" - by (erule trancl.induct) iprover+ +lemma tranclp_into_rtranclp: "r^++ a b ==> r^** a b" + by (erule tranclp.induct) iprover+ -lemmas trancl_into_rtrancl = trancl_into_rtrancl' [to_set] +lemmas trancl_into_rtrancl = tranclp_into_rtranclp [to_set] -lemma rtrancl_into_trancl1': assumes r: "r^** a b" +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 = rtrancl_into_trancl1' [to_set] +lemmas rtrancl_into_trancl1 = rtranclp_into_tranclp1 [to_set] -lemma rtrancl_into_trancl2': "[| r a b; r^** b c |] ==> r^++ a c" +lemma rtranclp_into_tranclp2: "[| r a b; r^** b c |] ==> r^++ a c" -- {* intro rule from @{text r} and @{text rtrancl} *} - apply (erule rtrancl.cases, iprover) - apply (rule rtrancl_trans' [THEN rtrancl_into_trancl1']) - apply (simp | rule r_into_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 = rtrancl_into_trancl2' [to_set] +lemmas rtrancl_into_trancl2 = rtranclp_into_tranclp2 [to_set] -lemma trancl_induct' [consumes 1, induct set: trancl]: +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" @@ -339,32 +324,27 @@ thus ?thesis by iprover qed -lemmas trancl_induct [consumes 1, induct set: trancl_set] = trancl_induct' [to_set] +lemmas trancl_induct [consumes 1, induct set: trancl] = tranclp_induct [to_set] -lemmas trancl_induct2' = - trancl_induct'[of _ "(ax,ay)" "(bx,by)", split_rule, +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 trancl_trans_induct': +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 trancl_induct'] cases) - -lemmas trancl_trans_induct = trancl_trans_induct' [to_set] + by (iprover intro: major [THEN tranclp_induct] cases) -lemma tranclE: - assumes H: "(a, b) : r^+" - and cases: "(a, b) : r ==> P" "\c. (a, c) : r^+ ==> (c, b) : r ==> P" - shows P - using H [simplified trancl_set_def, simplified] - by cases (auto intro: cases [simplified trancl_set_def, simplified]) +lemmas trancl_trans_induct = tranclp_trans_induct [to_set] + +inductive_cases tranclE: "(a, b) : r^+" lemma trancl_Int_subset: "[| r \ s; r O (r^+ \ s) \ s|] ==> r^+ \ s" apply (rule subsetI) @@ -386,7 +366,7 @@ lemmas trancl_trans = trans_trancl [THEN transD, standard] -lemma trancl_trans': +lemma tranclp_trans: assumes xy: "r^++ x y" and yz: "r^++ y z" shows "r^++ x z" using yz xy @@ -400,16 +380,16 @@ apply(blast) done -lemma rtrancl_trancl_trancl': assumes r: "r^** x y" +lemma rtranclp_tranclp_tranclp: assumes r: "r^** x y" shows "!!z. r^++ y z ==> r^++ x z" using r - by induct (iprover intro: trancl_trans')+ + by induct (iprover intro: tranclp_trans)+ -lemmas rtrancl_trancl_trancl = rtrancl_trancl_trancl' [to_set] +lemmas rtrancl_trancl_trancl = rtranclp_tranclp_tranclp [to_set] -lemma trancl_into_trancl2': "r a b ==> r^++ b c ==> r^++ a c" - by (erule trancl_trans' [OF trancl.r_into_trancl]) +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 = trancl_into_trancl2' [to_set] +lemmas trancl_into_trancl2 = tranclp_into_tranclp2 [to_set] lemma trancl_insert: "(insert (y, x) r)^+ = r^+ \ {(a, b). (a, y) \ r^* \ (x, b) \ r^*}" @@ -424,50 +404,50 @@ [THEN [2] rev_subsetD] rtrancl_trancl_trancl rtrancl_into_trancl2) done -lemma trancl_converseI': "(r^++)^--1 x y ==> (r^--1)^++ x y" +lemma tranclp_converseI: "(r^++)^--1 x y ==> (r^--1)^++ x y" apply (drule conversepD) - apply (erule trancl_induct') - apply (iprover intro: conversepI trancl_trans')+ + apply (erule tranclp_induct) + apply (iprover intro: conversepI tranclp_trans)+ done -lemmas trancl_converseI = trancl_converseI' [to_set] +lemmas trancl_converseI = tranclp_converseI [to_set] -lemma trancl_converseD': "(r^--1)^++ x y ==> (r^++)^--1 x y" +lemma tranclp_converseD: "(r^--1)^++ x y ==> (r^++)^--1 x y" apply (rule conversepI) - apply (erule trancl_induct') - apply (iprover dest: conversepD intro: trancl_trans')+ + apply (erule tranclp_induct) + apply (iprover dest: conversepD intro: tranclp_trans)+ done -lemmas trancl_converseD = trancl_converseD' [to_set] +lemmas trancl_converseD = tranclp_converseD [to_set] -lemma trancl_converse': "(r^--1)^++ = (r^++)^--1" +lemma tranclp_converse: "(r^--1)^++ = (r^++)^--1" by (fastsimp simp add: expand_fun_eq - intro!: trancl_converseI' dest!: trancl_converseD') + intro!: tranclp_converseI dest!: tranclp_converseD) -lemmas trancl_converse = trancl_converse' [to_set] +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_trancl_induct': +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 trancl_induct' [OF trancl_converseI', OF conversepI, OF major]) + apply (rule tranclp_induct [OF tranclp_converseI, OF conversepI, OF major]) apply (rule cases) apply (erule conversepD) - apply (blast intro: prems dest!: trancl_converseD' conversepD) + apply (blast intro: prems dest!: tranclp_converseD conversepD) done -lemmas converse_trancl_induct = converse_trancl_induct' [to_set] +lemmas converse_trancl_induct = converse_tranclp_induct [to_set] -lemma tranclD': "R^++ x y ==> EX z. R x z \ R^** z y" - apply (erule converse_trancl_induct', auto) - apply (blast intro: rtrancl_trans') +lemma tranclpD: "R^++ x y ==> EX z. R x z \ R^** z y" + apply (erule converse_tranclp_induct, auto) + apply (blast intro: rtranclp_trans) done -lemmas tranclD = tranclD' [to_set] +lemmas tranclD = tranclpD [to_set] lemma irrefl_tranclI: "r^-1 \ r^* = {} ==> (x, x) \ r^+" by (blast elim: tranclE dest: trancl_into_rtrancl) @@ -486,13 +466,13 @@ apply (blast dest!: trancl_into_rtrancl trancl_subset_Sigma_aux)+ done -lemma reflcl_trancl' [simp]: "(r^++)^== = r^**" +lemma reflcl_tranclp [simp]: "(r^++)^== = r^**" apply (safe intro!: order_antisym) - apply (erule trancl_into_rtrancl') - apply (blast elim: rtrancl.cases dest: rtrancl_into_trancl1') + apply (erule tranclp_into_rtranclp) + apply (blast elim: rtranclp.cases dest: rtranclp_into_tranclp1) done -lemmas reflcl_trancl [simp] = reflcl_trancl' [to_set] +lemmas reflcl_trancl [simp] = reflcl_tranclp [to_set] lemma trancl_reflcl [simp]: "(r^=)^+ = r^*" apply safe @@ -509,10 +489,10 @@ lemma rtrancl_empty [simp]: "{}^* = Id" by (rule subst [OF reflcl_trancl]) simp -lemma rtranclD': "R^** a b ==> a = b \ a \ b \ R^++ a b" - by (force simp add: reflcl_trancl' [symmetric] simp del: reflcl_trancl') +lemma rtranclpD: "R^** a b ==> a = b \ a \ b \ R^++ a b" + by (force simp add: reflcl_tranclp [symmetric] simp del: reflcl_tranclp) -lemmas rtranclD = rtranclD' [to_set] +lemmas rtranclD = rtranclpD [to_set] lemma rtrancl_eq_or_trancl: "(x,y) \ R\<^sup>* = (x=y \ x\y \ (x,y) \ R\<^sup>+)" @@ -567,32 +547,32 @@ apply (fast intro: r_r_into_trancl trancl_trans) done -lemma trancl_rtrancl_trancl': +lemma tranclp_rtranclp_tranclp: "r\<^sup>+\<^sup>+ a b ==> r\<^sup>*\<^sup>* b c ==> r\<^sup>+\<^sup>+ a c" - apply (drule tranclD') + apply (drule tranclpD) apply (erule exE, erule conjE) - apply (drule rtrancl_trans', assumption) - apply (drule rtrancl_into_trancl2', assumption, assumption) + apply (drule rtranclp_trans, assumption) + apply (drule rtranclp_into_tranclp2, assumption, assumption) done -lemmas trancl_rtrancl_trancl = trancl_rtrancl_trancl' [to_set] +lemmas trancl_rtrancl_trancl = tranclp_rtranclp_tranclp [to_set] 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 + trancl.trancl_into_trancl trancl_into_trancl2 + rtrancl.rtrancl_into_rtrancl converse_rtrancl_into_rtrancl rtrancl_trancl_trancl trancl_rtrancl_trancl -lemmas transitive_closure_trans' [trans] = - 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_set] -declare tranclE [cases set: trancl_set] +declare rtranclE [cases set: rtrancl] +declare tranclE [cases set: trancl] @@ -604,9 +584,9 @@ structure Trancl_Tac = Trancl_Tac_Fun ( struct - val r_into_trancl = thm "r_into_trancl"; + val r_into_trancl = thm "trancl.r_into_trancl"; val trancl_trans = thm "trancl_trans"; - val rtrancl_refl = thm "rtrancl_refl"; + 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"; @@ -615,8 +595,8 @@ fun decomp (Trueprop $ t) = let fun dec (Const ("op :", _) $ (Const ("Pair", _) $ a $ b) $ rel ) = - let fun decr (Const ("Transitive_Closure.rtrancl_set", _ ) $ r) = (r,"r*") - | decr (Const ("Transitive_Closure.trancl_set", _ ) $ r) = (r,"r+") + 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 @@ -627,19 +607,19 @@ structure Tranclp_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'"; + 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.rtrancl", _ ) $ r) = (r,"r*") - | decr (Const ("Transitive_Closure.trancl", _ ) $ r) = (r,"r+") + 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