src/HOL/Orderings.thy
changeset 23212 82881b1ae9c6
parent 23182 01fa88b79ddc
child 23247 b99dce43d252
     1.1 --- a/src/HOL/Orderings.thy	Sun Jun 03 15:44:35 2007 +0200
     1.2 +++ b/src/HOL/Orderings.thy	Sun Jun 03 16:57:51 2007 +0200
     1.3 @@ -99,94 +99,94 @@
     1.4  
     1.5  lemma eq_refl: "x = y \<Longrightarrow> x \<^loc>\<le> y"
     1.6      -- {* This form is useful with the classical reasoner. *}
     1.7 -  by (erule ssubst) (rule order_refl)
     1.8 +by (erule ssubst) (rule order_refl)
     1.9  
    1.10  lemma less_irrefl [iff]: "\<not> x \<^loc>< x"
    1.11 -  by (simp add: less_le)
    1.12 +by (simp add: less_le)
    1.13  
    1.14  lemma le_less: "x \<^loc>\<le> y \<longleftrightarrow> x \<^loc>< y \<or> x = y"
    1.15      -- {* NOT suitable for iff, since it can cause PROOF FAILED. *}
    1.16 -  by (simp add: less_le) blast
    1.17 +by (simp add: less_le) blast
    1.18  
    1.19  lemma le_imp_less_or_eq: "x \<^loc>\<le> y \<Longrightarrow> x \<^loc>< y \<or> x = y"
    1.20 -  unfolding less_le by blast
    1.21 +unfolding less_le by blast
    1.22  
    1.23  lemma less_imp_le: "x \<^loc>< y \<Longrightarrow> x \<^loc>\<le> y"
    1.24 -  unfolding less_le by blast
    1.25 +unfolding less_le by blast
    1.26  
    1.27  lemma less_imp_neq: "x \<^loc>< y \<Longrightarrow> x \<noteq> y"
    1.28 -  by (erule contrapos_pn, erule subst, rule less_irrefl)
    1.29 +by (erule contrapos_pn, erule subst, rule less_irrefl)
    1.30  
    1.31  
    1.32  text {* Useful for simplification, but too risky to include by default. *}
    1.33  
    1.34  lemma less_imp_not_eq: "x \<^loc>< y \<Longrightarrow> (x = y) \<longleftrightarrow> False"
    1.35 -  by auto
    1.36 +by auto
    1.37  
    1.38  lemma less_imp_not_eq2: "x \<^loc>< y \<Longrightarrow> (y = x) \<longleftrightarrow> False"
    1.39 -  by auto
    1.40 +by auto
    1.41  
    1.42  
    1.43  text {* Transitivity rules for calculational reasoning *}
    1.44  
    1.45  lemma neq_le_trans: "a \<noteq> b \<Longrightarrow> a \<^loc>\<le> b \<Longrightarrow> a \<^loc>< b"
    1.46 -  by (simp add: less_le)
    1.47 +by (simp add: less_le)
    1.48  
    1.49  lemma le_neq_trans: "a \<^loc>\<le> b \<Longrightarrow> a \<noteq> b \<Longrightarrow> a \<^loc>< b"
    1.50 -  by (simp add: less_le)
    1.51 +by (simp add: less_le)
    1.52  
    1.53  
    1.54  text {* Asymmetry. *}
    1.55  
    1.56  lemma less_not_sym: "x \<^loc>< y \<Longrightarrow> \<not> (y \<^loc>< x)"
    1.57 -  by (simp add: less_le antisym)
    1.58 +by (simp add: less_le antisym)
    1.59  
    1.60  lemma less_asym: "x \<^loc>< y \<Longrightarrow> (\<not> P \<Longrightarrow> y \<^loc>< x) \<Longrightarrow> P"
    1.61 -  by (drule less_not_sym, erule contrapos_np) simp
    1.62 +by (drule less_not_sym, erule contrapos_np) simp
    1.63  
    1.64  lemma eq_iff: "x = y \<longleftrightarrow> x \<^loc>\<le> y \<and> y \<^loc>\<le> x"
    1.65 -  by (blast intro: antisym)
    1.66 +by (blast intro: antisym)
    1.67  
    1.68  lemma antisym_conv: "y \<^loc>\<le> x \<Longrightarrow> x \<^loc>\<le> y \<longleftrightarrow> x = y"
    1.69 -  by (blast intro: antisym)
    1.70 +by (blast intro: antisym)
    1.71  
    1.72  lemma less_imp_neq: "x \<^loc>< y \<Longrightarrow> x \<noteq> y"
    1.73 -  by (erule contrapos_pn, erule subst, rule less_irrefl)
    1.74 +by (erule contrapos_pn, erule subst, rule less_irrefl)
    1.75  
    1.76  
    1.77  text {* Transitivity. *}
    1.78  
    1.79  lemma less_trans: "x \<^loc>< y \<Longrightarrow> y \<^loc>< z \<Longrightarrow> x \<^loc>< z"
    1.80 -  by (simp add: less_le) (blast intro: order_trans antisym)
    1.81 +by (simp add: less_le) (blast intro: order_trans antisym)
    1.82  
    1.83  lemma le_less_trans: "x \<^loc>\<le> y \<Longrightarrow> y \<^loc>< z \<Longrightarrow> x \<^loc>< z"
    1.84 -  by (simp add: less_le) (blast intro: order_trans antisym)
    1.85 +by (simp add: less_le) (blast intro: order_trans antisym)
    1.86  
    1.87  lemma less_le_trans: "x \<^loc>< y \<Longrightarrow> y \<^loc>\<le> z \<Longrightarrow> x \<^loc>< z"
    1.88 -  by (simp add: less_le) (blast intro: order_trans antisym)
    1.89 +by (simp add: less_le) (blast intro: order_trans antisym)
    1.90  
    1.91  
    1.92  text {* Useful for simplification, but too risky to include by default. *}
    1.93  
    1.94  lemma less_imp_not_less: "x \<^loc>< y \<Longrightarrow> (\<not> y \<^loc>< x) \<longleftrightarrow> True"
    1.95 -  by (blast elim: less_asym)
    1.96 +by (blast elim: less_asym)
    1.97  
    1.98  lemma less_imp_triv: "x \<^loc>< y \<Longrightarrow> (y \<^loc>< x \<longrightarrow> P) \<longleftrightarrow> True"
    1.99 -  by (blast elim: less_asym)
   1.100 +by (blast elim: less_asym)
   1.101  
   1.102  
   1.103  text {* Transitivity rules for calculational reasoning *}
   1.104  
   1.105  lemma less_asym': "a \<^loc>< b \<Longrightarrow> b \<^loc>< a \<Longrightarrow> P"
   1.106 -  by (rule less_asym)
   1.107 +by (rule less_asym)
   1.108  
   1.109  
   1.110  text {* Reverse order *}
   1.111  
   1.112  lemma order_reverse:
   1.113    "order (\<lambda>x y. y \<^loc>\<le> x) (\<lambda>x y. y \<^loc>< x)"
   1.114 -  by unfold_locales
   1.115 -    (simp add: less_le, auto intro: antisym order_trans)
   1.116 +by unfold_locales
   1.117 +   (simp add: less_le, auto intro: antisym order_trans)
   1.118  
   1.119  end
   1.120  
   1.121 @@ -198,97 +198,103 @@
   1.122  begin
   1.123  
   1.124  lemma less_linear: "x \<^loc>< y \<or> x = y \<or> y \<^loc>< x"
   1.125 -  unfolding less_le using less_le linear by blast 
   1.126 +unfolding less_le using less_le linear by blast
   1.127  
   1.128  lemma le_less_linear: "x \<^loc>\<le> y \<or> y \<^loc>< x"
   1.129 -  by (simp add: le_less less_linear)
   1.130 +by (simp add: le_less less_linear)
   1.131  
   1.132  lemma le_cases [case_names le ge]:
   1.133    "(x \<^loc>\<le> y \<Longrightarrow> P) \<Longrightarrow> (y \<^loc>\<le> x \<Longrightarrow> P) \<Longrightarrow> P"
   1.134 -  using linear by blast
   1.135 +using linear by blast
   1.136  
   1.137  lemma linorder_cases [case_names less equal greater]:
   1.138 -    "(x \<^loc>< y \<Longrightarrow> P) \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> (y \<^loc>< x \<Longrightarrow> P) \<Longrightarrow> P"
   1.139 -  using less_linear by blast
   1.140 +  "(x \<^loc>< y \<Longrightarrow> P) \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> (y \<^loc>< x \<Longrightarrow> P) \<Longrightarrow> P"
   1.141 +using less_linear by blast
   1.142  
   1.143  lemma not_less: "\<not> x \<^loc>< y \<longleftrightarrow> y \<^loc>\<le> x"
   1.144 -  apply (simp add: less_le)
   1.145 -  using linear apply (blast intro: antisym)
   1.146 -  done
   1.147 +apply (simp add: less_le)
   1.148 +using linear apply (blast intro: antisym)
   1.149 +done
   1.150 +
   1.151 +lemma not_less_iff_gr_or_eq:
   1.152 + "\<not>(x \<^loc>< y) \<longleftrightarrow> (x \<^loc>> y | x = y)"
   1.153 +apply(simp add:not_less le_less)
   1.154 +apply blast
   1.155 +done
   1.156  
   1.157  lemma not_le: "\<not> x \<^loc>\<le> y \<longleftrightarrow> y \<^loc>< x"
   1.158 -  apply (simp add: less_le)
   1.159 -  using linear apply (blast intro: antisym)
   1.160 -  done
   1.161 +apply (simp add: less_le)
   1.162 +using linear apply (blast intro: antisym)
   1.163 +done
   1.164  
   1.165  lemma neq_iff: "x \<noteq> y \<longleftrightarrow> x \<^loc>< y \<or> y \<^loc>< x"
   1.166 -  by (cut_tac x = x and y = y in less_linear, auto)
   1.167 +by (cut_tac x = x and y = y in less_linear, auto)
   1.168  
   1.169  lemma neqE: "x \<noteq> y \<Longrightarrow> (x \<^loc>< y \<Longrightarrow> R) \<Longrightarrow> (y \<^loc>< x \<Longrightarrow> R) \<Longrightarrow> R"
   1.170 -  by (simp add: neq_iff) blast
   1.171 +by (simp add: neq_iff) blast
   1.172  
   1.173  lemma antisym_conv1: "\<not> x \<^loc>< y \<Longrightarrow> x \<^loc>\<le> y \<longleftrightarrow> x = y"
   1.174 -  by (blast intro: antisym dest: not_less [THEN iffD1])
   1.175 +by (blast intro: antisym dest: not_less [THEN iffD1])
   1.176  
   1.177  lemma antisym_conv2: "x \<^loc>\<le> y \<Longrightarrow> \<not> x \<^loc>< y \<longleftrightarrow> x = y"
   1.178 -  by (blast intro: antisym dest: not_less [THEN iffD1])
   1.179 +by (blast intro: antisym dest: not_less [THEN iffD1])
   1.180  
   1.181  lemma antisym_conv3: "\<not> y \<^loc>< x \<Longrightarrow> \<not> x \<^loc>< y \<longleftrightarrow> x = y"
   1.182 -  by (blast intro: antisym dest: not_less [THEN iffD1])
   1.183 +by (blast intro: antisym dest: not_less [THEN iffD1])
   1.184  
   1.185  text{*Replacing the old Nat.leI*}
   1.186  lemma leI: "\<not> x \<^loc>< y \<Longrightarrow> y \<^loc>\<le> x"
   1.187 -  unfolding not_less .
   1.188 +unfolding not_less .
   1.189  
   1.190  lemma leD: "y \<^loc>\<le> x \<Longrightarrow> \<not> x \<^loc>< y"
   1.191 -  unfolding not_less .
   1.192 +unfolding not_less .
   1.193  
   1.194  (*FIXME inappropriate name (or delete altogether)*)
   1.195  lemma not_leE: "\<not> y \<^loc>\<le> x \<Longrightarrow> x \<^loc>< y"
   1.196 -  unfolding not_le .
   1.197 +unfolding not_le .
   1.198  
   1.199  
   1.200  text {* Reverse order *}
   1.201  
   1.202  lemma linorder_reverse:
   1.203    "linorder (\<lambda>x y. y \<^loc>\<le> x) (\<lambda>x y. y \<^loc>< x)"
   1.204 -  by unfold_locales
   1.205 -    (simp add: less_le, auto intro: antisym order_trans simp add: linear)
   1.206 +by unfold_locales
   1.207 +  (simp add: less_le, auto intro: antisym order_trans simp add: linear)
   1.208  
   1.209  
   1.210  text {* min/max properties *}
   1.211  
   1.212  lemma min_le_iff_disj:
   1.213    "min x y \<^loc>\<le> z \<longleftrightarrow> x \<^loc>\<le> z \<or> y \<^loc>\<le> z"
   1.214 -  unfolding min_def using linear by (auto intro: order_trans)
   1.215 +unfolding min_def using linear by (auto intro: order_trans)
   1.216  
   1.217  lemma le_max_iff_disj:
   1.218    "z \<^loc>\<le> max x y \<longleftrightarrow> z \<^loc>\<le> x \<or> z \<^loc>\<le> y"
   1.219 -  unfolding max_def using linear by (auto intro: order_trans)
   1.220 +unfolding max_def using linear by (auto intro: order_trans)
   1.221  
   1.222  lemma min_less_iff_disj:
   1.223    "min x y \<^loc>< z \<longleftrightarrow> x \<^loc>< z \<or> y \<^loc>< z"
   1.224 -  unfolding min_def le_less using less_linear by (auto intro: less_trans)
   1.225 +unfolding min_def le_less using less_linear by (auto intro: less_trans)
   1.226  
   1.227  lemma less_max_iff_disj:
   1.228    "z \<^loc>< max x y \<longleftrightarrow> z \<^loc>< x \<or> z \<^loc>< y"
   1.229 -  unfolding max_def le_less using less_linear by (auto intro: less_trans)
   1.230 +unfolding max_def le_less using less_linear by (auto intro: less_trans)
   1.231  
   1.232  lemma min_less_iff_conj [simp]:
   1.233    "z \<^loc>< min x y \<longleftrightarrow> z \<^loc>< x \<and> z \<^loc>< y"
   1.234 -  unfolding min_def le_less using less_linear by (auto intro: less_trans)
   1.235 +unfolding min_def le_less using less_linear by (auto intro: less_trans)
   1.236  
   1.237  lemma max_less_iff_conj [simp]:
   1.238    "max x y \<^loc>< z \<longleftrightarrow> x \<^loc>< z \<and> y \<^loc>< z"
   1.239 -  unfolding max_def le_less using less_linear by (auto intro: less_trans)
   1.240 +unfolding max_def le_less using less_linear by (auto intro: less_trans)
   1.241  
   1.242  lemma split_min:
   1.243    "P (min i j) \<longleftrightarrow> (i \<^loc>\<le> j \<longrightarrow> P i) \<and> (\<not> i \<^loc>\<le> j \<longrightarrow> P j)"
   1.244 -  by (simp add: min_def)
   1.245 +by (simp add: min_def)
   1.246  
   1.247  lemma split_max:
   1.248    "P (max i j) \<longleftrightarrow> (i \<^loc>\<le> j \<longrightarrow> P j) \<and> (\<not> i \<^loc>\<le> j \<longrightarrow> P i)"
   1.249 -  by (simp add: max_def)
   1.250 +by (simp add: max_def)
   1.251  
   1.252  end
   1.253  
   1.254 @@ -564,16 +570,16 @@
   1.255  subsection {* Transitivity reasoning *}
   1.256  
   1.257  lemma ord_le_eq_trans: "a <= b ==> b = c ==> a <= c"
   1.258 -  by (rule subst)
   1.259 +by (rule subst)
   1.260  
   1.261  lemma ord_eq_le_trans: "a = b ==> b <= c ==> a <= c"
   1.262 -  by (rule ssubst)
   1.263 +by (rule ssubst)
   1.264  
   1.265  lemma ord_less_eq_trans: "a < b ==> b = c ==> a < c"
   1.266 -  by (rule subst)
   1.267 +by (rule subst)
   1.268  
   1.269  lemma ord_eq_less_trans: "a = b ==> b < c ==> a < c"
   1.270 -  by (rule ssubst)
   1.271 +by (rule ssubst)
   1.272  
   1.273  lemma order_less_subst2: "(a::'a::order) < b ==> f b < (c::'c::order) ==>
   1.274    (!!x y. x < y ==> f x < f y) ==> f a < c"
   1.275 @@ -812,16 +818,16 @@
   1.276    by intro_classes (auto simp add: le_bool_def less_bool_def)
   1.277  
   1.278  lemma le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<le> Q"
   1.279 -  by (simp add: le_bool_def)
   1.280 +by (simp add: le_bool_def)
   1.281  
   1.282  lemma le_boolI': "P \<longrightarrow> Q \<Longrightarrow> P \<le> Q"
   1.283 -  by (simp add: le_bool_def)
   1.284 +by (simp add: le_bool_def)
   1.285  
   1.286  lemma le_boolE: "P \<le> Q \<Longrightarrow> P \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
   1.287 -  by (simp add: le_bool_def)
   1.288 +by (simp add: le_bool_def)
   1.289  
   1.290  lemma le_boolD: "P \<le> Q \<Longrightarrow> P \<longrightarrow> Q"
   1.291 -  by (simp add: le_bool_def)
   1.292 +by (simp add: le_bool_def)
   1.293  
   1.294  lemma [code func]:
   1.295    "False \<le> b \<longleftrightarrow> True"
   1.296 @@ -850,41 +856,41 @@
   1.297        !!y. P y ==> x <= y;
   1.298        !!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |]
   1.299     ==> Q (Least P)"
   1.300 -  apply (unfold Least_def)
   1.301 -  apply (rule theI2)
   1.302 -    apply (blast intro: order_antisym)+
   1.303 -  done
   1.304 +apply (unfold Least_def)
   1.305 +apply (rule theI2)
   1.306 +  apply (blast intro: order_antisym)+
   1.307 +done
   1.308  
   1.309  lemma Least_equality:
   1.310 -    "[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k"
   1.311 -  apply (simp add: Least_def)
   1.312 -  apply (rule the_equality)
   1.313 -  apply (auto intro!: order_antisym)
   1.314 -  done
   1.315 +  "[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k"
   1.316 +apply (simp add: Least_def)
   1.317 +apply (rule the_equality)
   1.318 +apply (auto intro!: order_antisym)
   1.319 +done
   1.320  
   1.321  lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
   1.322 -  by (simp add: min_def)
   1.323 +by (simp add: min_def)
   1.324  
   1.325  lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
   1.326 -  by (simp add: max_def)
   1.327 +by (simp add: max_def)
   1.328  
   1.329  lemma min_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> min x least = least"
   1.330 -  apply (simp add: min_def)
   1.331 -  apply (blast intro: order_antisym)
   1.332 -  done
   1.333 +apply (simp add: min_def)
   1.334 +apply (blast intro: order_antisym)
   1.335 +done
   1.336  
   1.337  lemma max_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> max x least = x"
   1.338 -  apply (simp add: max_def)
   1.339 -  apply (blast intro: order_antisym)
   1.340 -  done
   1.341 +apply (simp add: max_def)
   1.342 +apply (blast intro: order_antisym)
   1.343 +done
   1.344  
   1.345  lemma min_of_mono:
   1.346 -    "(!!x y. (f x <= f y) = (x <= y)) ==> min (f m) (f n) = f (min m n)"
   1.347 -  by (simp add: min_def)
   1.348 +  "(!!x y. (f x <= f y) = (x <= y)) ==> min (f m) (f n) = f (min m n)"
   1.349 +by (simp add: min_def)
   1.350  
   1.351  lemma max_of_mono:
   1.352 -    "(!!x y. (f x <= f y) = (x <= y)) ==> max (f m) (f n) = f (max m n)"
   1.353 -  by (simp add: max_def)
   1.354 +  "(!!x y. (f x <= f y) = (x <= y)) ==> max (f m) (f n) = f (max m n)"
   1.355 +by (simp add: max_def)
   1.356  
   1.357  
   1.358  subsection {* legacy ML bindings *}