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Commit e4bd8fd9 by haftmann

### avoid name clash

parent 1a42ddd5dbbc
 ... ... @@ -121,7 +121,7 @@ proof unfold_locales also have "... \ top * --x * top" by (simp add: minarc_below mult_isotone) finally show "top * x * top = top" using 1 antisym by simp using 1 order.antisym by simp qed lemma minarc_bot: ... ...
 ... ... @@ -387,7 +387,7 @@ proof also have "... = (if i = ?h \ j = i then h (i,j) + (\\<^sub>k \\<^sub>l f (k,l)) else h (i,j) + zero)" by simp also have "... \ (if i = ?h \ j = i then h (i,j) + (\\<^sub>k \\<^sub>l g (k,l)) else h (i,j) + zero)" using 4 inf.eq_iff by auto using 4 order.eq_iff by auto also have "... = h (i,j) + (if i = ?h \ j = i then \\<^sub>k \\<^sub>l g (k,l) else zero)" by simp finally show "h (i,j) + (sum\<^sub>M f) (i,j) \ h (i,j) + (sum\<^sub>M g) (i,j)" ... ... @@ -715,7 +715,7 @@ lemma regular_bot_top_2: sublocale heyting: heyting_stone_algebra where implies = "\x y . if x \ y then top else y" apply unfold_locales apply (simp add: antisym) apply (simp add: order.antisym) by auto end ... ...
 ... ... @@ -72,7 +72,7 @@ proof (intro_classes, unfold_locales; unfold compare_complex_def) assume "?c (Re x, Im x) (Re y, Im y) = Lt" "?c (Re y, Im y) (Re z, Im z) = Lt" thus "?c (Re x, Im x) (Re z, Im z) = Lt" by (rule trans) by (rule comp_trans) } { assume "?c (Re x, Im x) (Re y, Im y) = Eq" ... ...
 ... ... @@ -115,7 +115,7 @@ proof - qed lemma fdia_arden_eq: "\x = 0 \ d p = d q + |x\ p \ d p = |x\<^sup>\\ q" by (simp add: fdia_arden dka.fdia_star_induct_eq eq_iff) by (simp add: fdia_arden dka.fdia_star_induct_eq order.eq_iff) lemma fdia_arden_iff: "\x = 0 \ (d p = d q + |x\ p \ d p = |x\<^sup>\\ q)" by (metis fdia_arden_eq dka.fdia_d_simp dka.fdia_star_unfold_var) ... ... @@ -137,7 +137,7 @@ proof - qed lemma fbox_arden_eq: "\x = 0 \ d q \ |x] p = d p \ |x\<^sup>\] q = d p" by (simp add: fbox_arden antisym fbox_star_induct_eq) by (simp add: fbox_arden order.antisym fbox_star_induct_eq) lemma fbox_arden_iff: "\x = 0 \ (d p = d q \ |x] p \ d p = |x\<^sup>\] q)" by (metis fbox_arden_eq fbox_simp fbox_star_unfold_var) ... ...
 ... ... @@ -112,7 +112,7 @@ lemma a_d_iff: "a x = 1 \ -(x \ \)" by (clarsimp simp: a_def dblo.d_def inf_sup_distrib1) lemma topr: "-(x \ \) \ \ = -(x \ \)" proof (rule antisym) proof (rule order.antisym) show "-(x \ \) \ -(x \ \) \ \" by (metis mult_isol_var mult_oner order_refl top_greatest) have "-(x \ \) \ (x \ \) = \" ... ... @@ -180,7 +180,7 @@ lemma ar_r_iff: "ar x = 1 \ -(\ \ x)" by (simp add: ar_def inf_sup_distrib1 r_def) lemma topl: "\\(-(\ \ x)) = -(\ \ x)" proof (rule antisym) proof (rule order.antisym) show "\ \ - (\ \ x) \ - (\ \ x)" by (metis bot_annir' compl_inf_bot inf_bot_iff_le ldv') show "- (\ \ x) \ \ \ - (\ \ x)" ... ... @@ -254,7 +254,7 @@ sublocale dioid_one_zero "(\)" "(\)" "1" "\" "(\)" "( by (standard, simp add: le_iff_sup, auto) lemma Sup_sup_pred: "x \ Sup{y. P y} = Sup{y. y = x \ P y}" apply (rule antisym) apply (rule order.antisym) apply (simp add: Collect_mono Sup_subset_mono Sup_upper) using Sup_least Sup_upper le_supI2 by fastforce ... ...
 ... ... @@ -22,7 +22,7 @@ next from this have "mset_set (set_mset (M + {#x#})) = mset_set (set_mset M)" by (simp add: insert_absorb) from this add.hyps show ?thesis using subset_mset.order.trans by fastforce using subset_mset.trans by fastforce next assume "\ x \# M" from this add.hyps have "{#x#} + mset_set (set_mset M) \# M + {#x#}" ... ...
 ... ... @@ -109,7 +109,7 @@ proof (rule ccontr) with \chain C\ \C \ {}\ have "chain (C - A)" "C - A \ {}" by (auto intro: chain_Diff) moreover have "Sup C = Sup (C - A)" proof (safe intro!: antisym ccpo_Sup_least \chain C\ chain_Diff) proof (safe intro!: order.antisym ccpo_Sup_least \chain C\ chain_Diff) fix c assume "c \ C" with * obtain c' where "c' \ C" "c \ c'" "c' \ A" by auto ... ...
 ... ... @@ -189,7 +189,7 @@ qed lemma ldistinct_lmerge: "\ lsorted xs; lsorted ys; ldistinct xs; ldistinct ys \ \ ldistinct (lmerge xs ys)" by(coinduction arbitrary: xs ys)(auto 4 3 simp add: lhd_lmerge ltl_lmerge not_lnull_conv lsorted_LCons not_less dest!: in_lset_lmergeD dest: antisym) by(coinduction arbitrary: xs ys)(auto 4 3 simp add: lhd_lmerge ltl_lmerge not_lnull_conv lsorted_LCons not_less dest!: in_lset_lmergeD dest: order.antisym) end ... ...
 ... ... @@ -50,7 +50,7 @@ lemma (in ccpo) Sup_Un_less: assumes chain: "Complete_Partial_Order.chain (\) (A \ B)" and AB: "\x\A. \y\B. x \ y" shows "Sup (A \ B) = Sup B" proof(rule antisym) proof(rule order.antisym) from chain have chain': "Complete_Partial_Order.chain (\) B" by(blast intro: chain_subset) show "Sup (A \ B) \ Sup B" using chain ... ...
 ... ... @@ -59,7 +59,7 @@ proof (rule order_class.order.antisym) by (simp add: complete_lattice_class.INF_lower complete_lattice_class.le_INF_iff) next have "\k. \i j. f (i + j) \ f k" by (metis add.left_neutral order_class.eq_iff) by (metis Nat.add_0_right order_refl) then have "\k. \i. (\j. f (i + j)) \ f k" by (meson UNIV_I complete_lattice_class.INF_lower2) then show "(\i j. f (i + j)) \ (\k. f k)" ... ...
 ... ... @@ -157,7 +157,7 @@ proof - have "linorder.sorted (\) (map fst (map (\(c, d). ((a, c), f a b c d)) ys))" using ys by(induct ys) auto thus ?case using x Cons by(fastforce simp add: set_alist_product a.not_less dest: bspec a.antisym intro: rev_image_eqI) by(fastforce simp add: set_alist_product a.not_less dest: bspec a.order_antisym intro: rev_image_eqI) qed qed ... ...
 ... ... @@ -105,7 +105,7 @@ next by fastforce ultimately show ?case by(simp add: "4.IH") next case 5 thus ?case by(simp add: not_less eq_iff) case 5 thus ?case by(simp add: not_less order_eq_iff) qed lemma sorted_quicksort [simp]: "sorted (quicksort xs)" ... ... @@ -1222,10 +1222,10 @@ context linorder begin lemma sorted_list_subset_correct: "\ sorted xs; distinct xs; sorted ys; distinct ys \ \ sorted_list_subset (=) xs ys \ set xs \ set ys" apply(induct "(=) :: 'a \ 'a \ bool" xs ys rule: sorted_list_subset.induct) apply(auto 6 2) apply auto by (metis eq_iff insert_iff subsetD) apply(induct "(=) :: 'a \ 'a \ bool" xs ys rule: sorted_list_subset.induct) apply(auto 6 2) using order_antisym apply auto done end ... ...
 ... ... @@ -1060,8 +1060,8 @@ next case Cons show ?thesis using assms unfolding Cons exhaustive.simps apply(subst exhaustive_above_iff) apply(auto simp add: less_le proper_interval_simps not_less) by (metis List.set_simps(2) UNIV_I eq_iff set_ConsD) apply(auto simp add: less_le proper_interval_simps not_less intro: order_antisym) done qed theorem proper_interval_set_aux: ... ...
 ... ... @@ -1144,7 +1144,7 @@ proof(intro mcontI contI) moreover have "ord' (lub' (f ` Y)) (f (lub Y))" using chain' by(rule ccpo'.ccpo_Sup_least)(blast intro: monotoneD[OF mono] ccpo.ccpo_Sup_upper[OF ccpo chain]) ultimately show "f (lub Y) = lub' (f ` Y)" by(rule ccpo'.antisym) ultimately show "f (lub Y) = lub' (f ` Y)" by(rule ccpo'.order.antisym) qed(fact mono) lemma rel_fun_curry: includes lifting_syntax shows ... ... @@ -1248,7 +1248,7 @@ proof - interpret c1: ccpo lub1 leq1 "mk_less leq1" by fact interpret c2: ccpo lub2 leq2 "mk_less leq2" by fact show ?thesis proof(rule c1.antisym) proof(rule c1.order.antisym) have fg: "monotone leq2 leq2 (\x. f (g x))" using f g by(rule monotone2monotone) simp_all have gf: "monotone leq1 leq1 (\x. g (f x))" using g f by(rule monotone2monotone) simp_all show "leq1 (c1.fixp (\x. g (f x))) (g (c2.fixp (\x. f (g x))))" using gf ... ...
 ... ... @@ -56,7 +56,7 @@ locale comparator = fixes comp :: "'a comparator" assumes sym: "invert_order (comp x y) = comp y x" and weak_eq: "comp x y = Eq \ x = y" and trans: "comp x y = Lt \ comp y z = Lt \ comp x z = Lt" and comp_trans: "comp x y = Lt \ comp y z = Lt \ comp x z = Lt" begin lemma eq: "(comp x y = Eq) = (x = y)" ... ... @@ -72,20 +72,33 @@ lemma comp_same [simp]: abbreviation "lt \ lt_of_comp comp" abbreviation "le \ le_of_comp comp" lemma linorder: "class.linorder le lt" sublocale ordering le lt proof note [simp] = lt_of_comp_def le_of_comp_def fix x y z :: 'a show "lt x y = (le x y \ \ le y x)" using sym [of x y] by (cases "comp x y") (simp_all) show "le x y \ le y x" using sym [of x y] by (cases "comp x y") (simp_all) show "le x x" using eq [of x x] by (simp) show "le x y \ le y x \ x = y" using sym [of x y] by (cases "comp x y") (simp_all add: eq) show "le x y \ le y z \ le x z" by (cases "comp x y" "comp y z" rule: order.exhaust [case_product order.exhaust]) (auto dest: trans simp: eq) (auto dest: comp_trans simp: eq) show "le x y \ le y x \ x = y" using sym [of x y] by (cases "comp x y") (simp_all add: eq) show "lt x y \ le x y \ x \ y" using eq [of x y] by (cases "comp x y") (simp_all) qed lemma linorder: "class.linorder le lt" proof (rule class.linorder.intro) interpret order le lt using ordering_axioms by (rule ordering_orderI) show \class.order le lt\ by (fact order_axioms) show \class.linorder_axioms le\ proof note [simp] = lt_of_comp_def le_of_comp_def fix x y :: 'a show "le x y \ le y x" using sym [of x y] by (cases "comp x y") (simp_all) qed qed sublocale linorder le lt ... ...
 ... ... @@ -205,7 +205,7 @@ proof - fix x y z assume "acomp x y \ Gt" "acomp y z \ Gt" thus "acomp x z \ Gt \ (acomp x y = Lt \ acomp y z = Lt \ acomp x z = Lt)" using trans [of x y z] and eq [of x y] and eq [of y z] using comp_trans [of x y z] and eq [of x y] and eq [of y z] by (cases "acomp x y" "acomp y z" rule: order.exhaust [case_product order.exhaust]) auto qed qed ... ...
 ... ... @@ -36,7 +36,7 @@ qed lemma add_nonneg_eq_0_iff: assumes x: "0 \ x" and y: "0 \ y" shows "x + y = 0 \ x = 0 \ y = 0" by (metis add.comm_neutral add.left_neutral add_left_mono antisym x y) by (metis local.add_0_left local.add_0_right local.add_left_mono local.antisym_conv x y) lemma add_incr: "0\b \ a \ a+b" by (metis add.comm_neutral add_left_mono) ... ...
 ... ... @@ -199,7 +199,7 @@ begin using assms proof induction case (liveSimple' n v) from liveSimple'.prems have "vs' = [v]" by (metis append_Nil butlast.simps(2) suffixI suffix_order.antisym suffix_unsnoc) by (metis append_Nil butlast.simps(2) suffixI suffix_order.order_antisym suffix_unsnoc) with liveSimple'.hyps show ?case by (auto intro: liveVal'.liveSimple') next case (livePhi' v vs v') ... ...
 ... ... @@ -55,7 +55,7 @@ lemma strict_prefix_butlast: using assms unfolding append_butlast_last_id[symmetric] by (auto simp add:less_le butlast_strict_prefix prefix_order.le_less_trans) lemma prefix_tl_subset: "prefix xs ys \ set (tl xs) \ set (tl ys)" by (metis Nil_tl prefix_bot.bot.extremum prefix_def set_mono_prefix tl_append2) by (metis Nil_tl prefix_bot.extremum prefix_def set_mono_prefix tl_append2) lemma suffix_tl_subset: "suffix xs ys \ set (tl xs) \ set (tl ys)" by (metis append_Nil suffix_def set_mono_suffix suffix_tl suffix_order.order_trans tl_append2) ... ...
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