author  hoelzl 
Mon, 10 Mar 2014 20:04:40 +0100  
changeset 56020  f92479477c52 
parent 54868  bab6cade3cc5 
child 56508  af08160c5a4c 
permissions  rwrr 
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(* Title: HOL/Orderings.thy 
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Author: Tobias Nipkow, Markus Wenzel, and Larry Paulson 
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*) 

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header {* Abstract orderings *} 
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theory Orderings 

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distributed theory Algebras to theories Groups and Lattices
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imports HOL 
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declare command keywords via theory header, including strict checking outside Pure;
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keywords "print_orders" :: diag 
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begin 
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ML_file "~~/src/Provers/order.ML" 
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ML_file "~~/src/Provers/quasi.ML" (* FIXME unused? *) 

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subsection {* Abstract ordering *} 
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locale ordering = 

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fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<preceq>" 50) 

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and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<prec>" 50) 

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assumes strict_iff_order: "a \<prec> b \<longleftrightarrow> a \<preceq> b \<and> a \<noteq> b" 

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assumes refl: "a \<preceq> a"  {* not @{text iff}: makes problems due to multiple (dual) interpretations *} 

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and antisym: "a \<preceq> b \<Longrightarrow> b \<preceq> a \<Longrightarrow> a = b" 

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and trans: "a \<preceq> b \<Longrightarrow> b \<preceq> c \<Longrightarrow> a \<preceq> c" 

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begin 

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lemma strict_implies_order: 

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"a \<prec> b \<Longrightarrow> a \<preceq> b" 

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by (simp add: strict_iff_order) 

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lemma strict_implies_not_eq: 

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"a \<prec> b \<Longrightarrow> a \<noteq> b" 

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by (simp add: strict_iff_order) 

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lemma not_eq_order_implies_strict: 

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"a \<noteq> b \<Longrightarrow> a \<preceq> b \<Longrightarrow> a \<prec> b" 

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by (simp add: strict_iff_order) 

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lemma order_iff_strict: 

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"a \<preceq> b \<longleftrightarrow> a \<prec> b \<or> a = b" 

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by (auto simp add: strict_iff_order refl) 

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lemma irrefl:  {* not @{text iff}: makes problems due to multiple (dual) interpretations *} 

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"\<not> a \<prec> a" 

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by (simp add: strict_iff_order) 

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lemma asym: 

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"a \<prec> b \<Longrightarrow> b \<prec> a \<Longrightarrow> False" 

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by (auto simp add: strict_iff_order intro: antisym) 

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lemma strict_trans1: 

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"a \<preceq> b \<Longrightarrow> b \<prec> c \<Longrightarrow> a \<prec> c" 

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by (auto simp add: strict_iff_order intro: trans antisym) 

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lemma strict_trans2: 

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"a \<prec> b \<Longrightarrow> b \<preceq> c \<Longrightarrow> a \<prec> c" 

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by (auto simp add: strict_iff_order intro: trans antisym) 

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lemma strict_trans: 

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"a \<prec> b \<Longrightarrow> b \<prec> c \<Longrightarrow> a \<prec> c" 

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by (auto intro: strict_trans1 strict_implies_order) 

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end 

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locale ordering_top = ordering + 

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fixes top :: "'a" 

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assumes extremum [simp]: "a \<preceq> top" 

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begin 

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lemma extremum_uniqueI: 

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"top \<preceq> a \<Longrightarrow> a = top" 

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by (rule antisym) auto 

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lemma extremum_unique: 

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"top \<preceq> a \<longleftrightarrow> a = top" 

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by (auto intro: antisym) 

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lemma extremum_strict [simp]: 

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"\<not> (top \<prec> a)" 

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using extremum [of a] by (auto simp add: order_iff_strict intro: asym irrefl) 

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lemma not_eq_extremum: 

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"a \<noteq> top \<longleftrightarrow> a \<prec> top" 

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by (auto simp add: order_iff_strict intro: not_eq_order_implies_strict extremum) 

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end 

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subsection {* Syntactic orders *} 
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class ord = 
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fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" 
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and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" 
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begin 
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notation 
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less_eq ("op <=") and 
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less_eq ("(_/ <= _)" [51, 51] 50) and 
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less ("op <") and 
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less ("(_/ < _)" [51, 51] 50) 
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notation (xsymbols) 
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less_eq ("op \<le>") and 
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less_eq ("(_/ \<le> _)" [51, 51] 50) 
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notation (HTML output) 
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less_eq ("op \<le>") and 
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less_eq ("(_/ \<le> _)" [51, 51] 50) 
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abbreviation (input) 
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greater_eq (infix ">=" 50) where 
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"x >= y \<equiv> y <= x" 
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notation (input) 
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greater_eq (infix "\<ge>" 50) 
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abbreviation (input) 
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greater (infix ">" 50) where 
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"x > y \<equiv> y < x" 
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end 
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subsection {* Quasi orders *} 
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class preorder = ord + 
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assumes less_le_not_le: "x < y \<longleftrightarrow> x \<le> y \<and> \<not> (y \<le> x)" 

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and order_refl [iff]: "x \<le> x" 
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and order_trans: "x \<le> y \<Longrightarrow> y \<le> z \<Longrightarrow> x \<le> z" 

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begin 
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text {* Reflexivity. *} 
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lemma eq_refl: "x = y \<Longrightarrow> x \<le> y" 
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 {* This form is useful with the classical reasoner. *} 
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by (erule ssubst) (rule order_refl) 
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lemma less_irrefl [iff]: "\<not> x < x" 
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by (simp add: less_le_not_le) 
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lemma less_imp_le: "x < y \<Longrightarrow> x \<le> y" 

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unfolding less_le_not_le by blast 

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text {* Asymmetry. *} 

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lemma less_not_sym: "x < y \<Longrightarrow> \<not> (y < x)" 

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by (simp add: less_le_not_le) 

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lemma less_asym: "x < y \<Longrightarrow> (\<not> P \<Longrightarrow> y < x) \<Longrightarrow> P" 

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by (drule less_not_sym, erule contrapos_np) simp 

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text {* Transitivity. *} 

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lemma less_trans: "x < y \<Longrightarrow> y < z \<Longrightarrow> x < z" 

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by (auto simp add: less_le_not_le intro: order_trans) 

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lemma le_less_trans: "x \<le> y \<Longrightarrow> y < z \<Longrightarrow> x < z" 

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by (auto simp add: less_le_not_le intro: order_trans) 

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lemma less_le_trans: "x < y \<Longrightarrow> y \<le> z \<Longrightarrow> x < z" 

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by (auto simp add: less_le_not_le intro: order_trans) 

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text {* Useful for simplification, but too risky to include by default. *} 

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lemma less_imp_not_less: "x < y \<Longrightarrow> (\<not> y < x) \<longleftrightarrow> True" 

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by (blast elim: less_asym) 

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lemma less_imp_triv: "x < y \<Longrightarrow> (y < x \<longrightarrow> P) \<longleftrightarrow> True" 

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by (blast elim: less_asym) 

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text {* Transitivity rules for calculational reasoning *} 

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lemma less_asym': "a < b \<Longrightarrow> b < a \<Longrightarrow> P" 

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by (rule less_asym) 

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text {* Dual order *} 

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lemma dual_preorder: 

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"class.preorder (op \<ge>) (op >)" 
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proof qed (auto simp add: less_le_not_le intro: order_trans) 
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end 

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subsection {* Partial orders *} 

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class order = preorder + 

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assumes antisym: "x \<le> y \<Longrightarrow> y \<le> x \<Longrightarrow> x = y" 

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begin 

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lemma less_le: "x < y \<longleftrightarrow> x \<le> y \<and> x \<noteq> y" 
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by (auto simp add: less_le_not_le intro: antisym) 

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sublocale order!: ordering less_eq less + dual_order!: ordering greater_eq greater 
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by default (auto intro: antisym order_trans simp add: less_le) 
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text {* Reflexivity. *} 

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lemma le_less: "x \<le> y \<longleftrightarrow> x < y \<or> x = y" 
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 {* NOT suitable for iff, since it can cause PROOF FAILED. *} 
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by (fact order.order_iff_strict) 
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lemma le_imp_less_or_eq: "x \<le> y \<Longrightarrow> x < y \<or> x = y" 
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unfolding less_le by blast 
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text {* Useful for simplification, but too risky to include by default. *} 

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lemma less_imp_not_eq: "x < y \<Longrightarrow> (x = y) \<longleftrightarrow> False" 
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by auto 
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lemma less_imp_not_eq2: "x < y \<Longrightarrow> (y = x) \<longleftrightarrow> False" 
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by auto 
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text {* Transitivity rules for calculational reasoning *} 

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lemma neq_le_trans: "a \<noteq> b \<Longrightarrow> a \<le> b \<Longrightarrow> a < b" 
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by (fact order.not_eq_order_implies_strict) 
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lemma le_neq_trans: "a \<le> b \<Longrightarrow> a \<noteq> b \<Longrightarrow> a < b" 
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by (rule order.not_eq_order_implies_strict) 
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text {* Asymmetry. *} 

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lemma eq_iff: "x = y \<longleftrightarrow> x \<le> y \<and> y \<le> x" 
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by (blast intro: antisym) 
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lemma antisym_conv: "y \<le> x \<Longrightarrow> x \<le> y \<longleftrightarrow> x = y" 
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by (blast intro: antisym) 
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lemma less_imp_neq: "x < y \<Longrightarrow> x \<noteq> y" 
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by (fact order.strict_implies_not_eq) 
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text {* Least value operator *} 
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definition (in ord) 
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Least :: "('a \<Rightarrow> bool) \<Rightarrow> 'a" (binder "LEAST " 10) where 
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"Least P = (THE x. P x \<and> (\<forall>y. P y \<longrightarrow> x \<le> y))" 

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lemma Least_equality: 

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assumes "P x" 

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and "\<And>y. P y \<Longrightarrow> x \<le> y" 

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shows "Least P = x" 

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unfolding Least_def by (rule the_equality) 

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(blast intro: assms antisym)+ 

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lemma LeastI2_order: 

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assumes "P x" 

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and "\<And>y. P y \<Longrightarrow> x \<le> y" 

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and "\<And>x. P x \<Longrightarrow> \<forall>y. P y \<longrightarrow> x \<le> y \<Longrightarrow> Q x" 

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shows "Q (Least P)" 

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unfolding Least_def by (rule theI2) 

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(blast intro: assms antisym)+ 

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text {* Dual order *} 
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lemma dual_order: 
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"class.order (op \<ge>) (op >)" 
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by (intro_locales, rule dual_preorder) (unfold_locales, rule antisym) 
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end 
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21329  272 

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subsection {* Linear (total) orders *} 

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class linorder = order + 
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assumes linear: "x \<le> y \<or> y \<le> x" 
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begin 
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lemma less_linear: "x < y \<or> x = y \<or> y < x" 
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unfolding less_le using less_le linear by blast 
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lemma le_less_linear: "x \<le> y \<or> y < x" 
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by (simp add: le_less less_linear) 
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lemma le_cases [case_names le ge]: 

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"(x \<le> y \<Longrightarrow> P) \<Longrightarrow> (y \<le> x \<Longrightarrow> P) \<Longrightarrow> P" 
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using linear by blast 
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lemma linorder_cases [case_names less equal greater]: 
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"(x < y \<Longrightarrow> P) \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> (y < x \<Longrightarrow> P) \<Longrightarrow> P" 
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using less_linear by blast 
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lemma not_less: "\<not> x < y \<longleftrightarrow> y \<le> x" 
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apply (simp add: less_le) 
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using linear apply (blast intro: antisym) 

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done 

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lemma not_less_iff_gr_or_eq: 

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"\<not>(x < y) \<longleftrightarrow> (x > y  x = y)" 
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apply(simp add:not_less le_less) 
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apply blast 

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done 

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lemma not_le: "\<not> x \<le> y \<longleftrightarrow> y < x" 
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apply (simp add: less_le) 
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using linear apply (blast intro: antisym) 

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done 

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lemma neq_iff: "x \<noteq> y \<longleftrightarrow> x < y \<or> y < x" 
23212  310 
by (cut_tac x = x and y = y in less_linear, auto) 
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25062  312 
lemma neqE: "x \<noteq> y \<Longrightarrow> (x < y \<Longrightarrow> R) \<Longrightarrow> (y < x \<Longrightarrow> R) \<Longrightarrow> R" 
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by (simp add: neq_iff) blast 
15524  314 

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lemma antisym_conv1: "\<not> x < y \<Longrightarrow> x \<le> y \<longleftrightarrow> x = y" 
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by (blast intro: antisym dest: not_less [THEN iffD1]) 
15524  317 

25062  318 
lemma antisym_conv2: "x \<le> y \<Longrightarrow> \<not> x < y \<longleftrightarrow> x = y" 
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by (blast intro: antisym dest: not_less [THEN iffD1]) 
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25062  321 
lemma antisym_conv3: "\<not> y < x \<Longrightarrow> \<not> x < y \<longleftrightarrow> x = y" 
23212  322 
by (blast intro: antisym dest: not_less [THEN iffD1]) 
15524  323 

25062  324 
lemma leI: "\<not> x < y \<Longrightarrow> y \<le> x" 
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unfolding not_less . 
16796  326 

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lemma leD: "y \<le> x \<Longrightarrow> \<not> x < y" 
23212  328 
unfolding not_less . 
16796  329 

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(*FIXME inappropriate name (or delete altogether)*) 

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lemma not_leE: "\<not> y \<le> x \<Longrightarrow> x < y" 
23212  332 
unfolding not_le . 
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22916  334 

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text {* Dual order *} 
22916  336 

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lemma dual_linorder: 
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"class.linorder (op \<ge>) (op >)" 
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by (rule class.linorder.intro, rule dual_order) (unfold_locales, rule linear) 
22916  340 

21248  341 
end 
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subsection {* Reasoning tools setup *} 
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ML {* 
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signature ORDERS = 
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sig 
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val print_structures: Proof.context > unit 
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val attrib_setup: theory > theory 
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val order_tac: Proof.context > thm list > int > tactic 
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end; 
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structure Orders: ORDERS = 
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struct 
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(** Theory and context data **) 
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360 
fun struct_eq ((s1: string, ts1), (s2, ts2)) = 
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361 
(s1 = s2) andalso eq_list (op aconv) (ts1, ts2); 
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362 

33519  363 
structure Data = Generic_Data 
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364 
( 
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365 
type T = ((string * term list) * Order_Tac.less_arith) list; 
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366 
(* Order structures: 
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367 
identifier of the structure, list of operations and record of theorems 
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368 
needed to set up the transitivity reasoner, 
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369 
identifier and operations identify the structure uniquely. *) 
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370 
val empty = []; 
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371 
val extend = I; 
33519  372 
fun merge data = AList.join struct_eq (K fst) data; 
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373 
); 
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374 

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375 
fun print_structures ctxt = 
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376 
let 
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377 
val structs = Data.get (Context.Proof ctxt); 
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378 
fun pretty_term t = Pretty.block 
24920  379 
[Pretty.quote (Syntax.pretty_term ctxt t), Pretty.brk 1, 
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380 
Pretty.str "::", Pretty.brk 1, 
24920  381 
Pretty.quote (Syntax.pretty_typ ctxt (type_of t))]; 
24641
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382 
fun pretty_struct ((s, ts), _) = Pretty.block 
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383 
[Pretty.str s, Pretty.str ":", Pretty.brk 1, 
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384 
Pretty.enclose "(" ")" (Pretty.breaks (map pretty_term ts))]; 
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385 
in 
51579  386 
Pretty.writeln (Pretty.big_list "order structures:" (map pretty_struct structs)) 
24641
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387 
end; 
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388 

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389 

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390 
(** Method **) 
21091  391 

32215  392 
fun struct_tac ((s, [eq, le, less]), thms) ctxt prems = 
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393 
let 
30107
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394 
fun decomp thy (@{const Trueprop} $ t) = 
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395 
let 
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396 
fun excluded t = 
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397 
(* exclude numeric types: linear arithmetic subsumes transitivity *) 
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398 
let val T = type_of t 
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399 
in 
32960
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400 
T = HOLogic.natT orelse T = HOLogic.intT orelse T = HOLogic.realT 
24641
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401 
end; 
32960
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402 
fun rel (bin_op $ t1 $ t2) = 
24641
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403 
if excluded t1 then NONE 
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404 
else if Pattern.matches thy (eq, bin_op) then SOME (t1, "=", t2) 
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405 
else if Pattern.matches thy (le, bin_op) then SOME (t1, "<=", t2) 
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406 
else if Pattern.matches thy (less, bin_op) then SOME (t1, "<", t2) 
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407 
else NONE 
32960
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408 
 rel _ = NONE; 
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409 
fun dec (Const (@{const_name Not}, _) $ t) = (case rel t 
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410 
of NONE => NONE 
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changeset

411 
 SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2)) 
24741
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changeset

412 
 dec x = rel x; 
30107
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parents:
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diff
changeset

413 
in dec t end 
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changeset

414 
 decomp thy _ = NONE; 
24641
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415 
in 
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416 
case s of 
32215  417 
"order" => Order_Tac.partial_tac decomp thms ctxt prems 
418 
 "linorder" => Order_Tac.linear_tac decomp thms ctxt prems 

24641
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419 
 _ => error ("Unknown kind of order `" ^ s ^ "' encountered in transitivity reasoner.") 
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420 
end 
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421 

32215  422 
fun order_tac ctxt prems = 
423 
FIRST' (map (fn s => CHANGED o struct_tac s ctxt prems) (Data.get (Context.Proof ctxt))); 

24641
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424 

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425 

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426 
(** Attribute **) 
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427 

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428 
fun add_struct_thm s tag = 
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429 
Thm.declaration_attribute 
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430 
(fn thm => Data.map (AList.map_default struct_eq (s, Order_Tac.empty TrueI) (Order_Tac.update tag thm))); 
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431 
fun del_struct s = 
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432 
Thm.declaration_attribute 
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433 
(fn _ => Data.map (AList.delete struct_eq s)); 
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changeset

434 

30722
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parents:
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diff
changeset

435 
val attrib_setup = 
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diff
changeset

436 
Attrib.setup @{binding order} 
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changeset

437 
(Scan.lift ((Args.add  Args.name >> (fn (_, s) => SOME s)  Args.del >> K NONE)  
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changeset

438 
Args.colon (* FIXME  Scan.succeed true *) )  Scan.lift Args.name  
623d4831c8cf
simplified attribute and method setup: eliminating bottomup styles makes it easier to keep things in one place, and also SML/NJ happy;
wenzelm
parents:
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diff
changeset

439 
Scan.repeat Args.term 
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wenzelm
parents:
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diff
changeset

440 
>> (fn ((SOME tag, n), ts) => add_struct_thm (n, ts) tag 
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simplified attribute and method setup: eliminating bottomup styles makes it easier to keep things in one place, and also SML/NJ happy;
wenzelm
parents:
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diff
changeset

441 
 ((NONE, n), ts) => del_struct (n, ts))) 
623d4831c8cf
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parents:
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changeset

442 
"theorems controlling transitivity reasoner"; 
24641
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443 

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444 

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445 
(** Diagnostic command **) 
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446 

24867  447 
val _ = 
46961
5c6955f487e5
outer syntax command definitions based on formal command_spec derived from theory header declarations;
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parents:
46950
diff
changeset

448 
Outer_Syntax.improper_command @{command_spec "print_orders"} 
5c6955f487e5
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diff
changeset

449 
"print order structures available to transitivity reasoner" 
51658
21c10672633b
discontinued Toplevel.no_timing complication  also recovers timing of diagnostic commands, e.g. 'find_theorems';
wenzelm
parents:
51579
diff
changeset

450 
(Scan.succeed (Toplevel.unknown_context o 
21c10672633b
discontinued Toplevel.no_timing complication  also recovers timing of diagnostic commands, e.g. 'find_theorems';
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diff
changeset

451 
Toplevel.keep (print_structures o Toplevel.context_of))); 
24641
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452 

21091  453 
end; 
24641
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changeset

454 

21091  455 
*} 
456 

47432  457 
setup Orders.attrib_setup 
458 

459 
method_setup order = {* 

460 
Scan.succeed (fn ctxt => SIMPLE_METHOD' (Orders.order_tac ctxt [])) 

461 
*} "transitivity reasoner" 

24641
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462 

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463 

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464 
text {* Declarations to set up transitivity reasoner of partial and linear orders. *} 
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465 

25076  466 
context order 
467 
begin 

468 

24641
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469 
(* The type constraint on @{term op =} below is necessary since the operation 
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470 
is not a parameter of the locale. *) 
25076  471 

27689  472 
declare less_irrefl [THEN notE, order add less_reflE: order "op = :: 'a \<Rightarrow> 'a \<Rightarrow> bool" "op <=" "op <"] 
473 

474 
declare order_refl [order add le_refl: order "op = :: 'a => 'a => bool" "op <=" "op <"] 

475 

476 
declare less_imp_le [order add less_imp_le: order "op = :: 'a => 'a => bool" "op <=" "op <"] 

477 

478 
declare antisym [order add eqI: order "op = :: 'a => 'a => bool" "op <=" "op <"] 

479 

480 
declare eq_refl [order add eqD1: order "op = :: 'a => 'a => bool" "op <=" "op <"] 

481 

482 
declare sym [THEN eq_refl, order add eqD2: order "op = :: 'a => 'a => bool" "op <=" "op <"] 

483 

484 
declare less_trans [order add less_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] 

485 

486 
declare less_le_trans [order add less_le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] 

487 

488 
declare le_less_trans [order add le_less_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] 

489 

490 
declare order_trans [order add le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] 

491 

492 
declare le_neq_trans [order add le_neq_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] 

493 

494 
declare neq_le_trans [order add neq_le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] 

495 

496 
declare less_imp_neq [order add less_imp_neq: order "op = :: 'a => 'a => bool" "op <=" "op <"] 

497 

498 
declare eq_neq_eq_imp_neq [order add eq_neq_eq_imp_neq: order "op = :: 'a => 'a => bool" "op <=" "op <"] 

499 

500 
declare not_sym [order add not_sym: order "op = :: 'a => 'a => bool" "op <=" "op <"] 

24641
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changeset

501 

25076  502 
end 
503 

504 
context linorder 

505 
begin 

24641
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diff
changeset

506 

27689  507 
declare [[order del: order "op = :: 'a => 'a => bool" "op <=" "op <"]] 
508 

509 
declare less_irrefl [THEN notE, order add less_reflE: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] 

510 

511 
declare order_refl [order add le_refl: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] 

512 

513 
declare less_imp_le [order add less_imp_le: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] 

514 

515 
declare not_less [THEN iffD2, order add not_lessI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] 

516 

517 
declare not_le [THEN iffD2, order add not_leI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] 

518 

519 
declare not_less [THEN iffD1, order add not_lessD: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] 

520 

521 
declare not_le [THEN iffD1, order add not_leD: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] 

522 

523 
declare antisym [order add eqI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] 

524 

525 
declare eq_refl [order add eqD1: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] 

25076  526 

27689  527 
declare sym [THEN eq_refl, order add eqD2: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] 
528 

529 
declare less_trans [order add less_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] 

530 

531 
declare less_le_trans [order add less_le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] 

532 

533 
declare le_less_trans [order add le_less_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] 

534 

535 
declare order_trans [order add le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] 

536 

537 
declare le_neq_trans [order add le_neq_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] 

538 

539 
declare neq_le_trans [order add neq_le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] 

540 

541 
declare less_imp_neq [order add less_imp_neq: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] 

542 

543 
declare eq_neq_eq_imp_neq [order add eq_neq_eq_imp_neq: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] 

544 

545 
declare not_sym [order add not_sym: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] 

24641
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changeset

546 

25076  547 
end 
548 

24641
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diff
changeset

549 

21083  550 
setup {* 
551 
let 

552 

44058  553 
fun prp t thm = Thm.prop_of thm = t; (* FIXME aconv!? *) 
15524  554 

51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
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parents:
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diff
changeset

555 
fun prove_antisym_le ctxt ((le as Const(_,T)) $ r $ s) = 
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51658
diff
changeset

556 
let val prems = Simplifier.prems_of ctxt; 
22916  557 
val less = Const (@{const_name less}, T); 
21083  558 
val t = HOLogic.mk_Trueprop(le $ s $ r); 
559 
in case find_first (prp t) prems of 

560 
NONE => 

561 
let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s)) 

562 
in case find_first (prp t) prems of 

563 
NONE => NONE 

24422  564 
 SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv1})) 
21083  565 
end 
24422  566 
 SOME thm => SOME(mk_meta_eq(thm RS @{thm order_class.antisym_conv})) 
21083  567 
end 
568 
handle THM _ => NONE; 

15524  569 

51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
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parents:
51658
diff
changeset

570 
fun prove_antisym_less ctxt (NotC $ ((less as Const(_,T)) $ r $ s)) = 
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51658
diff
changeset

571 
let val prems = Simplifier.prems_of ctxt; 
22916  572 
val le = Const (@{const_name less_eq}, T); 
21083  573 
val t = HOLogic.mk_Trueprop(le $ r $ s); 
574 
in case find_first (prp t) prems of 

575 
NONE => 

576 
let val t = HOLogic.mk_Trueprop(NotC $ (less $ s $ r)) 

577 
in case find_first (prp t) prems of 

578 
NONE => NONE 

24422  579 
 SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv3})) 
21083  580 
end 
24422  581 
 SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv2})) 
21083  582 
end 
583 
handle THM _ => NONE; 

15524  584 

21248  585 
fun add_simprocs procs thy = 
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51658
diff
changeset

586 
map_theory_simpset (fn ctxt => ctxt 
21248  587 
addsimprocs (map (fn (name, raw_ts, proc) => 
38715
6513ea67d95d
renamed Simplifier.simproc(_i) to Simplifier.simproc_global(_i) to emphasize that this is not the real thing;
wenzelm
parents:
38705
diff
changeset

588 
Simplifier.simproc_global thy name raw_ts proc) procs)) thy; 
42795
66fcc9882784
clarified map_simpset versus Simplifier.map_simpset_global;
wenzelm
parents:
42287
diff
changeset

589 

26496
49ae9456eba9
purely functional setup of claset/simpset/clasimpset;
wenzelm
parents:
26324
diff
changeset

590 
fun add_solver name tac = 
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51658
diff
changeset

591 
map_theory_simpset (fn ctxt0 => ctxt0 addSolver 
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51658
diff
changeset

592 
mk_solver name (fn ctxt => tac ctxt (Simplifier.prems_of ctxt))); 
21083  593 

594 
in 

21248  595 
add_simprocs [ 
596 
("antisym le", ["(x::'a::order) <= y"], prove_antisym_le), 

597 
("antisym less", ["~ (x::'a::linorder) < y"], prove_antisym_less) 

598 
] 

24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset

599 
#> add_solver "Transitivity" Orders.order_tac 
21248  600 
(* Adding the transitivity reasoners also as safe solvers showed a slight 
601 
speed up, but the reasoning strength appears to be not higher (at least 

602 
no breaking of additional proofs in the entire HOL distribution, as 

603 
of 5 March 2004, was observed). *) 

21083  604 
end 
605 
*} 

15524  606 

607 

21083  608 
subsection {* Bounded quantifiers *} 
609 

610 
syntax 

21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

611 
"_All_less" :: "[idt, 'a, bool] => bool" ("(3ALL _<_./ _)" [0, 0, 10] 10) 
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

612 
"_Ex_less" :: "[idt, 'a, bool] => bool" ("(3EX _<_./ _)" [0, 0, 10] 10) 
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

613 
"_All_less_eq" :: "[idt, 'a, bool] => bool" ("(3ALL _<=_./ _)" [0, 0, 10] 10) 
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

614 
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3EX _<=_./ _)" [0, 0, 10] 10) 
21083  615 

21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

616 
"_All_greater" :: "[idt, 'a, bool] => bool" ("(3ALL _>_./ _)" [0, 0, 10] 10) 
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

617 
"_Ex_greater" :: "[idt, 'a, bool] => bool" ("(3EX _>_./ _)" [0, 0, 10] 10) 
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

618 
"_All_greater_eq" :: "[idt, 'a, bool] => bool" ("(3ALL _>=_./ _)" [0, 0, 10] 10) 
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

619 
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3EX _>=_./ _)" [0, 0, 10] 10) 
21083  620 

621 
syntax (xsymbols) 

21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

622 
"_All_less" :: "[idt, 'a, bool] => bool" ("(3\<forall>_<_./ _)" [0, 0, 10] 10) 
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

623 
"_Ex_less" :: "[idt, 'a, bool] => bool" ("(3\<exists>_<_./ _)" [0, 0, 10] 10) 
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

624 
"_All_less_eq" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10) 
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

625 
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10) 
21083  626 

21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

627 
"_All_greater" :: "[idt, 'a, bool] => bool" ("(3\<forall>_>_./ _)" [0, 0, 10] 10) 
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

628 
"_Ex_greater" :: "[idt, 'a, bool] => bool" ("(3\<exists>_>_./ _)" [0, 0, 10] 10) 
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

629 
"_All_greater_eq" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10) 
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

630 
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10) 
21083  631 

632 
syntax (HOL) 

21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

633 
"_All_less" :: "[idt, 'a, bool] => bool" ("(3! _<_./ _)" [0, 0, 10] 10) 
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

634 
"_Ex_less" :: "[idt, 'a, bool] => bool" ("(3? _<_./ _)" [0, 0, 10] 10) 
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

635 
"_All_less_eq" :: "[idt, 'a, bool] => bool" ("(3! _<=_./ _)" [0, 0, 10] 10) 
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

636 
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3? _<=_./ _)" [0, 0, 10] 10) 
21083  637 

638 
syntax (HTML output) 

21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

639 
"_All_less" :: "[idt, 'a, bool] => bool" ("(3\<forall>_<_./ _)" [0, 0, 10] 10) 
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

640 
"_Ex_less" :: "[idt, 'a, bool] => bool" ("(3\<exists>_<_./ _)" [0, 0, 10] 10) 
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

641 
"_All_less_eq" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10) 
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

642 
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10) 
21083  643 

21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

644 
"_All_greater" :: "[idt, 'a, bool] => bool" ("(3\<forall>_>_./ _)" [0, 0, 10] 10) 
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

645 
"_Ex_greater" :: "[idt, 'a, bool] => bool" ("(3\<exists>_>_./ _)" [0, 0, 10] 10) 
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

646 
"_All_greater_eq" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10) 
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

647 
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10) 
21083  648 

649 
translations 

650 
"ALL x<y. P" => "ALL x. x < y \<longrightarrow> P" 

651 
"EX x<y. P" => "EX x. x < y \<and> P" 

652 
"ALL x<=y. P" => "ALL x. x <= y \<longrightarrow> P" 

653 
"EX x<=y. P" => "EX x. x <= y \<and> P" 

654 
"ALL x>y. P" => "ALL x. x > y \<longrightarrow> P" 

655 
"EX x>y. P" => "EX x. x > y \<and> P" 

656 
"ALL x>=y. P" => "ALL x. x >= y \<longrightarrow> P" 

657 
"EX x>=y. P" => "EX x. x >= y \<and> P" 

658 

659 
print_translation {* 

660 
let 

42287
d98eb048a2e4
discontinued special treatment of structure Mixfix;
wenzelm
parents:
42284
diff
changeset

661 
val All_binder = Mixfix.binder_name @{const_syntax All}; 
d98eb048a2e4
discontinued special treatment of structure Mixfix;
wenzelm
parents:
42284
diff
changeset

662 
val Ex_binder = Mixfix.binder_name @{const_syntax Ex}; 
38786
e46e7a9cb622
formerly unnamed infix impliciation now named HOL.implies
haftmann
parents:
38715
diff
changeset

663 
val impl = @{const_syntax HOL.implies}; 
38795
848be46708dc
formerly unnamed infix conjunction and disjunction now named HOL.conj and HOL.disj
haftmann
parents:
38786
diff
changeset

664 
val conj = @{const_syntax HOL.conj}; 
22916  665 
val less = @{const_syntax less}; 
666 
val less_eq = @{const_syntax less_eq}; 

21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

667 

f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

668 
val trans = 
35115  669 
[((All_binder, impl, less), 
670 
(@{syntax_const "_All_less"}, @{syntax_const "_All_greater"})), 

671 
((All_binder, impl, less_eq), 

672 
(@{syntax_const "_All_less_eq"}, @{syntax_const "_All_greater_eq"})), 

673 
((Ex_binder, conj, less), 

674 
(@{syntax_const "_Ex_less"}, @{syntax_const "_Ex_greater"})), 

675 
((Ex_binder, conj, less_eq), 

676 
(@{syntax_const "_Ex_less_eq"}, @{syntax_const "_Ex_greater_eq"}))]; 

21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

677 

35115  678 
fun matches_bound v t = 
679 
(case t of 

35364  680 
Const (@{syntax_const "_bound"}, _) $ Free (v', _) => v = v' 
35115  681 
 _ => false); 
682 
fun contains_var v = Term.exists_subterm (fn Free (x, _) => x = v  _ => false); 

49660
de49d9b4d7bc
more explicit Syntax_Trans.mark_bound_abs/mark_bound_body: preserve type information for show_markup;
wenzelm
parents:
48891
diff
changeset

683 
fun mk x c n P = Syntax.const c $ Syntax_Trans.mark_bound_body x $ n $ P; 
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

684 

52143  685 
fun tr' q = (q, fn _ => 
686 
(fn [Const (@{syntax_const "_bound"}, _) $ Free (v, T), 

35364  687 
Const (c, _) $ (Const (d, _) $ t $ u) $ P] => 
35115  688 
(case AList.lookup (op =) trans (q, c, d) of 
689 
NONE => raise Match 

690 
 SOME (l, g) => 

49660
de49d9b4d7bc
more explicit Syntax_Trans.mark_bound_abs/mark_bound_body: preserve type information for show_markup;
wenzelm
parents:
48891
diff
changeset

691 
if matches_bound v t andalso not (contains_var v u) then mk (v, T) l u P 
de49d9b4d7bc
more explicit Syntax_Trans.mark_bound_abs/mark_bound_body: preserve type information for show_markup;
wenzelm
parents:
48891
diff
changeset

692 
else if matches_bound v u andalso not (contains_var v t) then mk (v, T) g t P 
35115  693 
else raise Match) 
52143  694 
 _ => raise Match)); 
21524  695 
in [tr' All_binder, tr' Ex_binder] end 
21083  696 
*} 
697 

698 

21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

699 
subsection {* Transitivity reasoning *} 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

700 

25193  701 
context ord 
702 
begin 

21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

703 

25193  704 
lemma ord_le_eq_trans: "a \<le> b \<Longrightarrow> b = c \<Longrightarrow> a \<le> c" 
705 
by (rule subst) 

21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

706 

25193  707 
lemma ord_eq_le_trans: "a = b \<Longrightarrow> b \<le> c \<Longrightarrow> a \<le> c" 
708 
by (rule ssubst) 

21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

709 

25193  710 
lemma ord_less_eq_trans: "a < b \<Longrightarrow> b = c \<Longrightarrow> a < c" 
711 
by (rule subst) 

712 

713 
lemma ord_eq_less_trans: "a = b \<Longrightarrow> b < c \<Longrightarrow> a < c" 

714 
by (rule ssubst) 

715 

716 
end 

21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

717 

17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

718 
lemma order_less_subst2: "(a::'a::order) < b ==> f b < (c::'c::order) ==> 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

719 
(!!x y. x < y ==> f x < f y) ==> f a < c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

720 
proof  
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

721 
assume r: "!!x y. x < y ==> f x < f y" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

722 
assume "a < b" hence "f a < f b" by (rule r) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

723 
also assume "f b < c" 
34250
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset

724 
finally (less_trans) show ?thesis . 
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

725 
qed 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

726 

17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

727 
lemma order_less_subst1: "(a::'a::order) < f b ==> (b::'b::order) < c ==> 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

728 
(!!x y. x < y ==> f x < f y) ==> a < f c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

729 
proof  
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

730 
assume r: "!!x y. x < y ==> f x < f y" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

731 
assume "a < f b" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

732 
also assume "b < c" hence "f b < f c" by (rule r) 
34250
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset

733 
finally (less_trans) show ?thesis . 
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

734 
qed 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

735 

17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

736 
lemma order_le_less_subst2: "(a::'a::order) <= b ==> f b < (c::'c::order) ==> 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

737 
(!!x y. x <= y ==> f x <= f y) ==> f a < c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

738 
proof  
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

739 
assume r: "!!x y. x <= y ==> f x <= f y" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

740 
assume "a <= b" hence "f a <= f b" by (rule r) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

741 
also assume "f b < c" 
34250
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset

742 
finally (le_less_trans) show ?thesis . 
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

743 
qed 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

744 

17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

745 
lemma order_le_less_subst1: "(a::'a::order) <= f b ==> (b::'b::order) < c ==> 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

746 
(!!x y. x < y ==> f x < f y) ==> a < f c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

747 
proof  
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

748 
assume r: "!!x y. x < y ==> f x < f y" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

749 
assume "a <= f b" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

750 
also assume "b < c" hence "f b < f c" by (rule r) 
34250
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset

751 
finally (le_less_trans) show ?thesis . 
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

752 
qed 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

753 

17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

754 
lemma order_less_le_subst2: "(a::'a::order) < b ==> f b <= (c::'c::order) ==> 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

755 
(!!x y. x < y ==> f x < f y) ==> f a < c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

756 
proof  
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

757 
assume r: "!!x y. x < y ==> f x < f y" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

758 
assume "a < b" hence "f a < f b" by (rule r) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

759 
also assume "f b <= c" 
34250
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset

760 
finally (less_le_trans) show ?thesis . 
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

761 
qed 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

762 

17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

763 
lemma order_less_le_subst1: "(a::'a::order) < f b ==> (b::'b::order) <= c ==> 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

764 
(!!x y. x <= y ==> f x <= f y) ==> a < f c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

765 
proof  
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

766 
assume r: "!!x y. x <= y ==> f x <= f y" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

767 
assume "a < f b" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

768 
also assume "b <= c" hence "f b <= f c" by (rule r) 
34250
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset

769 
finally (less_le_trans) show ?thesis . 
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

770 
qed 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

771 

17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

772 
lemma order_subst1: "(a::'a::order) <= f b ==> (b::'b::order) <= c ==> 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

773 
(!!x y. x <= y ==> f x <= f y) ==> a <= f c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

774 
proof  
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

775 
assume r: "!!x y. x <= y ==> f x <= f y" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

776 
assume "a <= f b" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

777 
also assume "b <= c" hence "f b <= f c" by (rule r) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

778 
finally (order_trans) show ?thesis . 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

779 
qed 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

780 

17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

781 
lemma order_subst2: "(a::'a::order) <= b ==> f b <= (c::'c::order) ==> 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

782 
(!!x y. x <= y ==> f x <= f y) ==> f a <= c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

783 
proof  
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

784 
assume r: "!!x y. x <= y ==> f x <= f y" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

785 
assume "a <= b" hence "f a <= f b" by (rule r) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

786 
also assume "f b <= c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

787 
finally (order_trans) show ?thesis . 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

788 
qed 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

789 

17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

790 
lemma ord_le_eq_subst: "a <= b ==> f b = c ==> 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

791 
(!!x y. x <= y ==> f x <= f y) ==> f a <= c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

792 
proof  
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

793 
assume r: "!!x y. x <= y ==> f x <= f y" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

794 
assume "a <= b" hence "f a <= f b" by (rule r) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

795 
also assume "f b = c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

796 
finally (ord_le_eq_trans) show ?thesis . 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

797 
qed 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

798 

17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

799 
lemma ord_eq_le_subst: "a = f b ==> b <= c ==> 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

800 
(!!x y. x <= y ==> f x <= f y) ==> a <= f c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

801 
proof  
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

802 
assume r: "!!x y. x <= y ==> f x <= f y" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

803 
assume "a = f b" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

804 
also assume "b <= c" hence "f b <= f c" by (rule r) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

805 
finally (ord_eq_le_trans) show ?thesis . 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

806 
qed 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

807 

17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

808 
lemma ord_less_eq_subst: "a < b ==> f b = c ==> 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

809 
(!!x y. x < y ==> f x < f y) ==> f a < c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

810 
proof  
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

811 
assume r: "!!x y. x < y ==> f x < f y" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

812 
assume "a < b" hence "f a < f b" by (rule r) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

813 
also assume "f b = c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

814 
finally (ord_less_eq_trans) show ?thesis . 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

815 
qed 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

816 

17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

817 
lemma ord_eq_less_subst: "a = f b ==> b < c ==> 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

818 
(!!x y. x < y ==> f x < f y) ==> a < f c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

819 
proof  
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

820 
assume r: "!!x y. x < y ==> f x < f y" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

821 
assume "a = f b" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

822 
also assume "b < c" hence "f b < f c" by (rule r) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

823 
finally (ord_eq_less_trans) show ?thesis . 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

824 
qed 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

825 

17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

826 
text {* 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

827 
Note that this list of rules is in reverse order of priorities. 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

828 
*} 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

829 

27682  830 
lemmas [trans] = 
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

831 
order_less_subst2 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

832 
order_less_subst1 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

833 
order_le_less_subst2 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

834 
order_le_less_subst1 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

835 
order_less_le_subst2 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

836 
order_less_le_subst1 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

837 
order_subst2 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

838 
order_subst1 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

839 
ord_le_eq_subst 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

840 
ord_eq_le_subst 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

841 
ord_less_eq_subst 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

842 
ord_eq_less_subst 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

843 
forw_subst 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

844 
back_subst 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

845 
rev_mp 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

846 
mp 
27682  847 

848 
lemmas (in order) [trans] = 

849 
neq_le_trans 

850 
le_neq_trans 

851 

852 
lemmas (in preorder) [trans] = 

853 
less_trans 

854 
less_asym' 

855 
le_less_trans 

856 
less_le_trans 

21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

857 
order_trans 
27682  858 

859 
lemmas (in order) [trans] = 

860 
antisym 

861 

862 
lemmas (in ord) [trans] = 

863 
ord_le_eq_trans 

864 
ord_eq_le_trans 

865 
ord_less_eq_trans 

866 
ord_eq_less_trans 

867 

868 
lemmas [trans] = 

869 
trans 

870 

871 
lemmas order_trans_rules = 

872 
order_less_subst2 

873 
order_less_subst1 

874 
order_le_less_subst2 

875 
order_le_less_subst1 

876 
order_less_le_subst2 

877 
order_less_le_subst1 

878 
order_subst2 

879 
order_subst1 

880 
ord_le_eq_subst 

881 
ord_eq_le_subst 

882 
ord_less_eq_subst 

883 
ord_eq_less_subst 

884 
forw_subst 

885 
back_subst 

886 
rev_mp 

887 
mp 

888 
neq_le_trans 

889 
le_neq_trans 

890 
less_trans 

891 
less_asym' 

892 
le_less_trans 

893 
less_le_trans 

894 
order_trans 

895 
antisym 

21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

896 
ord_le_eq_trans 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

897 
ord_eq_le_trans 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

898 
ord_less_eq_trans 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

899 
ord_eq_less_trans 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

900 
trans 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

901 

21083  902 
text {* These support proving chains of decreasing inequalities 
903 
a >= b >= c ... in Isar proofs. *} 

904 

45221
3eadb9b6a055
mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents:
44921
diff
changeset

905 
lemma xt1 [no_atp]: 
21083  906 
"a = b ==> b > c ==> a > c" 
907 
"a > b ==> b = c ==> a > c" 

908 
"a = b ==> b >= c ==> a >= c" 

909 
"a >= b ==> b = c ==> a >= c" 

910 
"(x::'a::order) >= y ==> y >= x ==> x = y" 

911 
"(x::'a::order) >= y ==> y >= z ==> x >= z" 

912 
"(x::'a::order) > y ==> y >= z ==> x > z" 

913 
"(x::'a::order) >= y ==> y > z ==> x > z" 

23417  914 
"(a::'a::order) > b ==> b > a ==> P" 
21083  915 
"(x::'a::order) > y ==> y > z ==> x > z" 
916 
"(a::'a::order) >= b ==> a ~= b ==> a > b" 

917 
"(a::'a::order) ~= b ==> a >= b ==> a > b" 

918 
"a = f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c" 

919 
"a > b ==> f b = c ==> (!!x y. x > y ==> f x > f y) ==> f a > c" 

920 
"a = f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c" 

921 
"a >= b ==> f b = c ==> (!! x y. x >= y ==> f x >= f y) ==> f a >= c" 

25076  922 
by auto 
21083  923 

45221
3eadb9b6a055
mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents:
44921
diff
changeset

924 
lemma xt2 [no_atp]: 
21083  925 
"(a::'a::order) >= f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c" 
926 
by (subgoal_tac "f b >= f c", force, force) 

927 

45221
3eadb9b6a055
mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents:
44921
diff
changeset

928 
lemma xt3 [no_atp]: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==> 
21083  929 
(!!x y. x >= y ==> f x >= f y) ==> f a >= c" 
930 
by (subgoal_tac "f a >= f b", force, force) 

931 

45221
3eadb9b6a055
mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents:
44921
diff
changeset

932 
lemma xt4 [no_atp]: "(a::'a::order) > f b ==> (b::'b::order) >= c ==> 
21083  933 
(!!x y. x >= y ==> f x >= f y) ==> a > f c" 
934 
by (subgoal_tac "f b >= f c", force, force) 

935 

45221
3eadb9b6a055
mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents:
44921
diff
changeset

936 
lemma xt5 [no_atp]: "(a::'a::order) > b ==> (f b::'b::order) >= c==> 
21083  937 
(!!x y. x > y ==> f x > f y) ==> f a > c" 
938 
by (subgoal_tac "f a > f b", force, force) 

939 

45221
3eadb9b6a055
mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents:
44921
diff
changeset

940 
lemma xt6 [no_atp]: "(a::'a::order) >= f b ==> b > c ==> 
21083  941 
(!!x y. x > y ==> f x > f y) ==> a > f c" 
942 
by (subgoal_tac "f b > f c", force, force) 

943 

45221
3eadb9b6a055
mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents:
44921
diff
changeset

944 
lemma xt7 [no_atp]: "(a::'a::order) >= b ==> (f b::'b::order) > c ==> 
21083  945 
(!!x y. x >= y ==> f x >= f y) ==> f a > c" 
946 
by (subgoal_tac "f a >= f b", force, force) 

947 

45221
3eadb9b6a055
mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents:
44921
diff
changeset

948 
lemma xt8 [no_atp]: "(a::'a::order) > f b ==> (b::'b::order) > c ==> 
21083  949 
(!!x y. x > y ==> f x > f y) ==> a > f c" 
950 
by (subgoal_tac "f b > f c", force, force) 

951 

45221
3eadb9b6a055
mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents:
44921
diff
changeset

952 
lemma xt9 [no_atp]: "(a::'a::order) > b ==> (f b::'b::order) > c ==> 
21083  953 
(!!x y. x > y ==> f x > f y) ==> f a > c" 
954 
by (subgoal_tac "f a > f b", force, force) 

955 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53216
diff
changeset

956 
lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9 
21083  957 

958 
(* 

959 
Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands 

960 
for the wrong thing in an Isar proof. 

961 

962 
The extra transitivity rules can be used as follows: 

963 

964 
lemma "(a::'a::order) > z" 

965 
proof  

966 
have "a >= b" (is "_ >= ?rhs") 

967 
sorry 

968 
also have "?rhs >= c" (is "_ >= ?rhs") 

969 
sorry 

970 
also (xtrans) have "?rhs = d" (is "_ = ?rhs") 

971 
sorry 

972 
also (xtrans) have "?rhs >= e" (is "_ >= ?rhs") 

973 
sorry 

974 
also (xtrans) have "?rhs > f" (is "_ > ?rhs") 

975 
sorry 

976 
also (xtrans) have "?rhs > z" 

977 
sorry 

978 
finally (xtrans) show ?thesis . 

979 
qed 

980 

981 
Alternatively, one can use "declare xtrans [trans]" and then 

982 
leave out the "(xtrans)" above. 

983 
*) 

984 

23881  985 

54860  986 
subsection {* Monotonicity *} 
21083  987 

25076  988 
context order 
989 
begin 

990 

30298  991 
definition mono :: "('a \<Rightarrow> 'b\<Colon>order) \<Rightarrow> bool" where 
25076  992 
"mono f \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> f x \<le> f y)" 
993 

994 
lemma monoI [intro?]: 

995 
fixes f :: "'a \<Rightarrow> 'b\<Colon>order" 

996 
shows "(\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y) \<Longrightarrow> mono f" 

997 
unfolding mono_def by iprover 

21216
1c8580913738
made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents:
21204
diff
changeset

998 

25076  999 
lemma monoD [dest?]: 
1000 
fixes f :: "'a \<Rightarrow> 'b\<Colon>order" 

1001 
shows "mono f \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y" 

1002 
unfolding mono_def by iprover 

1003 

51263
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset

1004 
lemma monoE: 
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset

1005 
fixes f :: "'a \<Rightarrow> 'b\<Colon>order" 
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset

1006 
assumes "mono f" 
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset

1007 
assumes "x \<le> y" 
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset

1008 
obtains "f x \<le> f y" 
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset

1009 
proof 
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset

1010 
from assms show "f x \<le> f y" by (simp add: mono_def) 
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset

1011 
qed 
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset

1012 

56020
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset

1013 
definition antimono :: "('a \<Rightarrow> 'b\<Colon>order) \<Rightarrow> bool" where 
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset

1014 
"antimono f \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> f x \<ge> f y)" 
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset

1015 

f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset

1016 
lemma antimonoI [intro?]: 
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset

1017 
fixes f :: "'a \<Rightarrow> 'b\<Colon>order" 
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset

1018 
shows "(\<And>x y. x \<le> y \<Longrightarrow> f x \<ge> f y) \<Longrightarrow> antimono f" 
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset

1019 
unfolding antimono_def by iprover 
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset

1020 

f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset

1021 
lemma antimonoD [dest?]: 
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset

1022 
fixes f :: "'a \<Rightarrow> 'b\<Colon>order" 
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset

1023 
shows "antimono f \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<ge> f y" 
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset

1024 
unfolding antimono_def by iprover 
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset

1025 

f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset

1026 
lemma antimonoE: 
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset

1027 
fixes f :: "'a \<Rightarrow> 'b\<Colon>order" 
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset

1028 
assumes "antimono f" 
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset

1029 
assumes "x \<le> y" 
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset

1030 
obtains "f x \<ge> f y" 
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset

1031 
proof 
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset

1032 
from assms show "f x \<ge> f y" by (simp add: antimono_def) 
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset

1033 
qed 
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset

1034 

30298  1035 
definition strict_mono :: "('a \<Rightarrow> 'b\<Colon>order) \<Rightarrow> bool" where 
1036 
"strict_mono f \<longleftrightarrow> (\<forall>x y. x < y \<longrightarrow> f x < f y)" 

1037 

1038 
lemma strict_monoI [intro?]: 

1039 
assumes "\<And>x y. x < y \<Longrightarrow> f x < f y" 

1040 
shows "strict_mono f" 

1041 
using assms unfolding strict_mono_def by auto 

1042 

1043 
lemma strict_monoD [dest?]: 

1044 
"strict_mono f \<Longrightarrow> x < y \<Longrightarrow> f x < f y" 

1045 
unfolding strict_mono_def by auto 

1046 

1047 
lemma strict_mono_mono [dest?]: 

1048 
assumes "strict_mono f" 

1049 
shows "mono f" 

1050 
proof (rule monoI) 

1051 
fix x y 

1052 
assume "x \<le> y" 

1053 
show "f x \<le> f y" 

1054 
proof (cases "x = y") 

1055 
case True then show ?thesis by simp 

1056 
next 

1057 
case False with `x \<le> y` have "x < y" by simp 

1058 
with assms strict_monoD have "f x < f y" by auto 

1059 
then show ?thesis by simp 

1060 
qed 

1061 
qed 

1062 

25076  1063 
end 
1064 

1065 
context linorder 

1066 
begin 

1067 

51263
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset

1068 
lemma mono_invE: 
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset

1069 
fixes f :: "'a \<Rightarrow> 'b\<Colon>order" 
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset

1070 
assumes "mono f" 
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset

1071 
assumes "f x < f y" 
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset

1072 
obtains "x \<le> y" 
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset

1073 
proof 
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset

1074 
show "x \<le> y" 
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset

1075 
proof (rule ccontr) 
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset

1076 
assume "\<not> x \<le> y" 
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset

1077 
then have "y \<le> x" by simp 
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset

1078 
with `mono f` obtain "f y \<le> f x" by (rule monoE) 
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset

1079 
with `f x < f y` show False by simp 
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset

1080 
qed 
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset

1081 
qed 
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset

1082 

30298  1083 
lemma strict_mono_eq: 
1084 
assumes "strict_mono f" 

1085 
shows "f x = f y \<longleftrightarrow> x = y" 

1086 
proof 

1087 
assume "f x = f y" 

1088 
show "x = y" proof (cases x y rule: linorder_cases) 

1089 
case less with assms strict_monoD have "f x < f y" by auto 

1090 
with `f x = f y` show ?thesis by simp 

1091 
next 

1092 
case equal then show ?thesis . 

1093 
next 

1094 
case greater with assms strict_monoD have "f y < f x" by auto 

1095 
with `f x = f y` show ?thesis by simp 

1096 
qed 

1097 
qed simp 

1098 

1099 
lemma strict_mono_less_eq: 

1100 
assumes "strict_mono f" 

1101 
shows "f x \<le> f y \<longleftrightarrow> x \<le> y" 

1102 
proof 

1103 
assume "x \<le> y" 

1104 
with assms strict_mono_mono monoD show "f x \<le> f y" by auto 

1105 
next 

1106 
assume "f x \<le> f y" 

1107 
show "x \<le> y" proof (rule ccontr) 

1108 
assume "\<not> x \<le> y" then have "y < x" by simp 

1109 
with assms strict_monoD have "f y < f x" by auto 

1110 
with `f x \<le> f y` show False by simp 

1111 
qed 

1112 
qed 

1113 

1114 
lemma strict_mono_less: 

1115 
assumes "strict_mono f" 

1116 
shows "f x < f y \<longleftrightarrow> x < y" 

1117 
using assms 

1118 
by (auto simp add: less_le Orderings.less_le strict_mono_eq strict_mono_less_eq) 

1119 

54860  1120 
end 
1121 

1122 

54861
00d551179872
postponed min/max lemmas until abstract lattice is available
haftmann
parents:
54860
diff
changeset

1123 
subsection {* min and max  fundamental *} 
54860  1124 

1125 
definition (in ord) min :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where 

1126 
"min a b = (if a \<le> b then a else b)" 

1127 

1128 
definition (in ord) max :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where 

1129 
"max a b = (if a \<le> b then b else a)" 

1130 

45931  1131 
lemma min_absorb1: "x \<le> y \<Longrightarrow> min x y = x" 
54861
00d551179872
postponed min/max lemmas until abstract lattice is available
haftmann
parents:
54860
diff
changeset

1132 
by (simp add: min_def) 
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

1133 

54857  1134 
lemma max_absorb2: "x \<le> y \<Longrightarrow> max x y = y" 
54861
00d551179872
postponed min/max lemmas until abstract lattice is available
haftmann
parents:
54860
diff
changeset

1135 
by (simp add: max_def) 
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

1136 

45931  1137 
lemma min_absorb2: "(y\<Colon>'a\<Colon>order) \<le> x \<Longrightarrow> min x y = y" 
54861
00d551179872
postponed min/max lemmas until abstract lattice is available
haftmann
parents:
54860
diff
changeset

1138 
by (simp add:min_def) 
45893  1139 

45931  1140 
lemma max_absorb1: "(y\<Colon>'a\<Colon>order) \<le> x \<Longrightarrow> max x y = x" 
54861
00d551179872
postponed min/max lemmas until abstract lattice is available
haftmann
parents:
54860
diff
changeset

1141 
by (simp add: max_def) 
45893  1142 

1143 

43813
07f0650146f2
tightened specification of classes bot and top: uniqueness of top and bot elements
haftmann
parents:
43597
diff
changeset

1144 
subsection {* (Unique) top and bottom elements *} 
28685  1145 

52729
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset

1146 
class bot = 
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset

1147 
fixes bot :: 'a ("\<bottom>") 
52729
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset

1148 

412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset

1149 
class order_bot = order + bot + 
51487  1150 
assumes bot_least: "\<bottom> \<le> a" 
54868  1151 
begin 
51487  1152 

54868  1153 
sublocale bot!: ordering_top greater_eq greater bot 
51546
2e26df807dc7
more uniform style for interpretation and sublocale declarations
haftmann
parents:
51487
diff
changeset

1154 
by default (fact bot_least) 
51487  1155 

43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset

1156 
lemma le_bot: 
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset

1157 
"a \<le> \<bottom> \<Longrightarrow> a = \<bottom>" 
51487  1158 
by (fact bot.extremum_uniqueI) 
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset

1159 

43816  1160 
lemma bot_unique: 
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset

1161 
"a \<le> \<bottom> \<longleftrightarrow> a = \<bottom>" 
51487  1162 
by (fact bot.extremum_unique) 
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset

1163 

51487  1164 
lemma not_less_bot: 
1165 
"\<not> a < \<bottom>" 

1166 
by (fact bot.extremum_strict) 

43816  1167 

43814
58791b75cf1f
moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents:
43813
diff
changeset

1168 
lemma bot_less: 
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset

1169 
"a \<noteq> \<bottom> \<longleftrightarrow> \<bottom> < a" 
51487  1170 
by (fact bot.not_eq_extremum) 
43814
58791b75cf1f
moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents:
43813
diff
changeset

1171 

58791b75cf1f
moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents:
43813
diff
changeset

1172 
end 
41082  1173 

52729
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset

1174 
class top = 
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset

1175 
fixes top :: 'a ("\<top>") 
52729
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset

1176 

412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset

1177 
class order_top = order + top + 
51487  1178 
assumes top_greatest: "a \<le> \<top>" 
54868  1179 
begin 
51487  1180 

54868  1181 
sublocale top!: ordering_top less_eq less top 
51546
2e26df807dc7
more uniform style for interpretation and sublocale declarations
haftmann
parents:
51487
diff
changeset

1182 
by default (fact top_greatest) 
51487  1183 

43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset

1184 
lemma top_le: 
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset

1185 
"\<top> \<le> a \<Longrightarrow> a = \<top>" 
51487  1186 
by (fact top.extremum_uniqueI) 
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset

1187 

43816  1188 
lemma top_unique: 
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset

1189 
"\<top> \<le> a \<longleftrightarrow> a = \<top>" 
51487  1190 
by (fact top.extremum_unique) 
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset

1191 

51487  1192 
lemma not_top_less: 
1193 
"\<not> \<top> < a" 

1194 
by (fact top.extremum_strict) 

43816  1195 

43814
58791b75cf1f
moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents:
43813
diff
changeset

1196 
lemma less_top: 
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset

1197 
"a \<noteq> \<top> \<longleftrightarrow> a < \<top>" 
51487  1198 
by (fact top.not_eq_extremum) 
43814
58791b75cf1f
moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents:
43813
diff
changeset

1199 

58791b75cf1f
moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents:
43813
diff
changeset

1200 
end 
28685  1201 

1202 

27823  1203 
subsection {* Dense orders *} 
1204 

53216  1205 
class dense_order = order + 
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1206 
assumes dense: "x < y \<Longrightarrow> (\<exists>z. x < z \<and> z < y)" 
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1207 

53216  1208 
class dense_linorder = linorder + dense_order 
35579
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset

1209 
begin 
27823  1210 

35579
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset

1211 
lemma dense_le: 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset

1212 
fixes y z :: 'a 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset

1213 
assumes "\<And>x. x < y \<Longrightarrow> x \<le> z" 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset

1214 
shows "y \<le> z" 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset

1215 
proof (rule ccontr) 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset

1216 
assume "\<not> ?thesis" 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset

1217 
hence "z < y" by simp 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset

1218 
from dense[OF this] 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset

1219 
obtain x where "x < y" and "z < x" by safe 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset

1220 
moreover have "x \<le> z" using assms[OF `x < y`] . 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset

1221 
ultimately show False by auto 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset

1222 
qed 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset

1223 

cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset

1224 
lemma dense_le_bounded: 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset

1225 
fixes x y z :: 'a 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset

1226 
assumes "x < y" 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset

1227 
assumes *: "\<And>w. \<lbrakk> x < w ; w < y \<rbrakk> \<Longrightarrow> w \<le> z" 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset

1228 
shows "y \<le> z" 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset

1229 
proof (rule dense_le) 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset

1230 
fix w assume "w < y" 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset

1231 
from dense[OF `x < y`] obtain u where "x < u" "u < y" by safe 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset

1232 
from linear[of u w] 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset

1233 
show "w \<le> z" 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset

1234 
proof (rule disjE) 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset

1235 
assume "u \<le> w" 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset

1236 
from less_le_trans[OF `x < u` `u \<le> w`] `w < y` 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset

1237 
show "w \<le> z" by (rule *) 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset

1238 
next 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset

1239 
assume "w \<le> u" 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset

1240 
from `w \<le> u` *[OF `x < u` `u < y`] 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset

1241 
show "w \<le> z" by (rule order_trans) 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset

1242 
qed 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset

1243 
qed 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset

1244 

51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1245 
lemma dense_ge: 
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1246 
fixes y z :: 'a 
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1247 
assumes "\<And>x. z < x \<Longrightarrow> y \<le> x" 
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1248 
shows "y \<le> z" 
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1249 
proof (rule ccontr) 
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1250 
assume "\<not> ?thesis" 
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1251 
hence "z < y" by simp 
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1252 
from dense[OF this] 
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1253 
obtain x where "x < y" and "z < x" by safe 
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1254 
moreover have "y \<le> x" using assms[OF `z < x`] . 
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1255 
ultimately show False by auto 
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1256 
qed 
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1257 

4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1258 
lemma dense_ge_bounded: 
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1259 
fixes x y z :: 'a 
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1260 
assumes "z < x" 
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1261 
assumes *: "\<And>w. \<lbrakk> z < w ; w < x \<rbrakk> \<Longrightarrow> y \<le> w" 
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1262 
shows "y \<le> z" 
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1263 
proof (rule dense_ge) 
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1264 
fix w assume "z < w" 
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1265 
from dense[OF `z < x`] obtain u where "z < u" "u < x" by safe 
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1266 
from linear[of u w] 
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1267 
show "y \<le> w" 
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1268 
proof (rule disjE) 
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1269 
assume "w \<le> u" 
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1270 
from `z < w` le_less_trans[OF `w \<le> u` `u < x`] 
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1271 
show "y \<le> w" by (rule *) 
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1272 
next 
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1273 
assume "u \<le> w" 
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1274 
from *[OF `z < u` `u < x`] `u \<le> w` 
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1275 
show "y \<le> w" by (rule order_trans) 
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1276 
qed 
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1277 
qed 
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1278 

35579
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset

1279 
end 
27823  1280 

51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1281 
class no_top = order + 
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1282 
assumes gt_ex: "\<exists>y. x < y" 
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1283 

4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1284 
class no_bot = order + 
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1285 
assumes lt_ex: "\<exists>y. y < x" 
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1286 

53216  1287 
class unbounded_dense_linorder = dense_linorder + no_top + no_bot 
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

1288 

51546
2e26df807dc7
more uniform style for interpretation and sublocale declarations
haftmann
parents:
51487
diff
changeset

1289 

27823  1290 
subsection {* Wellorders *} 
1291 

1292 
class wellorder = linorder + 

1293 
assumes less_induct [case_names less]: "(\<And>x. (\<And>y. y < x \<Longrightarrow> P y) \<Longrightarrow> P x) \<Longrightarrow> P a" 

1294 
begin 

1295 

1296 
lemma wellorder_Least_lemma: 

1297 
fixes k :: 'a 

1298 
assumes "P k" 

34250
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset

1299 
shows LeastI: "P (LEAST x. P x)" and Least_le: "(LEAST x. P x) \<le> k" 
27823  1300 
proof  
1301 
have "P (LEAST x. P x) \<and> (LEAST x. P x) \<le> k" 

1302 
using assms proof (induct k rule: less_induct) 

1303 
case (less x) then have "P x" by simp 

1304 
show ?case proof (rule classical) 

1305 
assume assm: "\<not> (P (LEAST a. P a) \<and> (LEAST a. P a) \<le> x)" 

1306 
have "\<And>y. P y \<Longrightarrow> x \<le> y" 

1307 
proof (rule classical) 

1308 
fix y 

38705 