
nipkow@15524: (*  Title:      HOL/Orderings.thy
nipkow@15524:     ID:         $Id$
nipkow@15524:     Author:     Tobias Nipkow, Markus Wenzel, and Larry Paulson
nipkow@15524: *)
nipkow@15524: 
haftmann@21083: header {* Abstract orderings *}
nipkow@15524: 
nipkow@15524: theory Orderings
haftmann@21216: imports Code_Generator
nipkow@15524: begin
nipkow@15524: 
haftmann@21083: section {* Abstract orderings *}
haftmann@21083: 
haftmann@21083: subsection {* Order signatures *}
nipkow@15524: 
haftmann@21194: class ord =
haftmann@20588:   fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
wenzelm@21204:     and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
wenzelm@21204: begin
wenzelm@21204: 
wenzelm@21204: notation
wenzelm@21204:   less_eq  ("op \<^loc><=")
wenzelm@21204:   less_eq  ("(_/ \<^loc><= _)" [50, 51] 50)
wenzelm@21204:   less  ("op \<^loc><")
wenzelm@21204:   less  ("(_/ \<^loc>< _)"  [50, 51] 50)
wenzelm@21204: 
wenzelm@21204: notation (xsymbols)
wenzelm@21204:   less_eq  ("op \<^loc>\<le>")
wenzelm@21204:   less_eq  ("(_/ \<^loc>\<le> _)"  [50, 51] 50)
nipkow@15524: 
wenzelm@21204: notation (HTML output)
wenzelm@21204:   less_eq  ("op \<^loc>\<le>")
wenzelm@21204:   less_eq  ("(_/ \<^loc>\<le> _)"  [50, 51] 50)
wenzelm@21204: 
wenzelm@21204: abbreviation (input)
wenzelm@21204:   greater  (infix "\<^loc>>" 50)
wenzelm@21204:   "x \<^loc>> y \<equiv> y \<^loc>< x"
wenzelm@21204:   greater_eq  (infix "\<^loc>>=" 50)
wenzelm@21204:   "x \<^loc>>= y \<equiv> y \<^loc><= x"
wenzelm@21204: 
wenzelm@21204: notation (xsymbols)
wenzelm@21204:   greater_eq  (infixl "\<^loc>\<ge>" 50)
wenzelm@21204: 
wenzelm@21204: end
wenzelm@21204: 
wenzelm@21204: notation
wenzelm@19656:   less_eq  ("op <=")
wenzelm@19656:   less_eq  ("(_/ <= _)" [50, 51] 50)
wenzelm@21204:   less  ("op <")
wenzelm@21204:   less  ("(_/ < _)"  [50, 51] 50)
wenzelm@21204:   
wenzelm@21204: notation (xsymbols)
wenzelm@19656:   less_eq  ("op \<le>")
wenzelm@19656:   less_eq  ("(_/ \<le> _)"  [50, 51] 50)
nipkow@15524: 
wenzelm@21204: notation (HTML output)
wenzelm@21204:   less_eq  ("op \<le>")
wenzelm@21204:   less_eq  ("(_/ \<le> _)"  [50, 51] 50)
haftmann@20714: 
wenzelm@19536: abbreviation (input)
wenzelm@19536:   greater  (infixl ">" 50)
haftmann@20714:   "x > y \<equiv> y < x"
wenzelm@19536:   greater_eq  (infixl ">=" 50)
haftmann@20714:   "x >= y \<equiv> y <= x"
haftmann@20714:   
wenzelm@21204: notation (xsymbols)
wenzelm@19656:   greater_eq  (infixl "\<ge>" 50)
nipkow@15524: 
nipkow@15524: 
haftmann@21083: subsection {* Partial orderings *}
nipkow@15524: 
haftmann@21216: locale partial_order =
haftmann@21216:   fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "\<sqsubseteq>" 50)
haftmann@21216:   fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "\<sqsubset>" 50)
haftmann@21216:   assumes refl [iff]: "x \<sqsubseteq> x"
haftmann@21216:   and trans: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> z"
haftmann@21216:   and antisym: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> x \<Longrightarrow> x = y"
haftmann@21216:   and less_le: "(x \<sqsubset> y) = (x \<sqsubseteq> y \<and> x \<noteq> y)"
haftmann@21216: 
nipkow@15524: axclass order < ord
nipkow@15524:   order_refl [iff]: "x <= x"
nipkow@15524:   order_trans: "x <= y ==> y <= z ==> x <= z"
nipkow@15524:   order_antisym: "x <= y ==> y <= x ==> x = y"
nipkow@15524:   order_less_le: "(x < y) = (x <= y & x ~= y)"
nipkow@15524: 
ballarin@15837: interpretation order:
haftmann@21216:   partial_order ["op \<le> \<Colon> 'a\<Colon>order \<Rightarrow> 'a \<Rightarrow> bool" "op < \<Colon> 'a\<Colon>order \<Rightarrow> 'a \<Rightarrow> bool"]
nipkow@15524: apply(rule partial_order.intro)
haftmann@21216: apply(rule order_refl, erule (1) order_trans, erule (1) order_antisym, rule order_less_le)
nipkow@15524: done
nipkow@15524: 
nipkow@15524: text {* Reflexivity. *}
nipkow@15524: 
haftmann@21216: lemma order_eq_refl: "(x \<Colon> 'a\<Colon>order) = y \<Longrightarrow> x \<le> y"
nipkow@15524:     -- {* This form is useful with the classical reasoner. *}
nipkow@15524:   apply (erule ssubst)
nipkow@15524:   apply (rule order_refl)
nipkow@15524:   done
nipkow@15524: 
nipkow@15524: lemma order_less_irrefl [iff]: "~ x < (x::'a::order)"
nipkow@15524:   by (simp add: order_less_le)
nipkow@15524: 
nipkow@15524: lemma order_le_less: "((x::'a::order) <= y) = (x < y | x = y)"
nipkow@15524:     -- {* NOT suitable for iff, since it can cause PROOF FAILED. *}
nipkow@15524:   apply (simp add: order_less_le, blast)
nipkow@15524:   done
nipkow@15524: 
nipkow@15524: lemmas order_le_imp_less_or_eq = order_le_less [THEN iffD1, standard]
nipkow@15524: 
nipkow@15524: lemma order_less_imp_le: "!!x::'a::order. x < y ==> x <= y"
nipkow@15524:   by (simp add: order_less_le)
nipkow@15524: 
nipkow@15524: text {* Asymmetry. *}
nipkow@15524: 
nipkow@15524: lemma order_less_not_sym: "(x::'a::order) < y ==> ~ (y < x)"
nipkow@15524:   by (simp add: order_less_le order_antisym)
nipkow@15524: 
nipkow@15524: lemma order_less_asym: "x < (y::'a::order) ==> (~P ==> y < x) ==> P"
nipkow@15524:   apply (drule order_less_not_sym)
nipkow@15524:   apply (erule contrapos_np, simp)
nipkow@15524:   done
nipkow@15524: 
nipkow@15524: lemma order_eq_iff: "!!x::'a::order. (x = y) = (x \<le> y & y \<le> x)"
nipkow@15524: by (blast intro: order_antisym)
nipkow@15524: 
nipkow@15524: lemma order_antisym_conv: "(y::'a::order) <= x ==> (x <= y) = (x = y)"
nipkow@15524: by(blast intro:order_antisym)
nipkow@15524: 
haftmann@21083: lemma less_imp_neq: "[| (x::'a::order) < y |] ==> x ~= y"
haftmann@21083:   by (erule contrapos_pn, erule subst, rule order_less_irrefl)
haftmann@21083: 
nipkow@15524: text {* Transitivity. *}
nipkow@15524: 
nipkow@15524: lemma order_less_trans: "!!x::'a::order. [| x < y; y < z |] ==> x < z"
nipkow@15524:   apply (simp add: order_less_le)
nipkow@15524:   apply (blast intro: order_trans order_antisym)
nipkow@15524:   done
nipkow@15524: 
nipkow@15524: lemma order_le_less_trans: "!!x::'a::order. [| x <= y; y < z |] ==> x < z"
nipkow@15524:   apply (simp add: order_less_le)
nipkow@15524:   apply (blast intro: order_trans order_antisym)
nipkow@15524:   done
nipkow@15524: 
nipkow@15524: lemma order_less_le_trans: "!!x::'a::order. [| x < y; y <= z |] ==> x < z"
nipkow@15524:   apply (simp add: order_less_le)
nipkow@15524:   apply (blast intro: order_trans order_antisym)
nipkow@15524:   done
nipkow@15524: 
haftmann@21083: lemma eq_neq_eq_imp_neq: "[| x = a ; a ~= b; b = y |] ==> x ~= y"
haftmann@21083:   by (erule subst, erule ssubst, assumption)
nipkow@15524: 
nipkow@15524: text {* Useful for simplification, but too risky to include by default. *}
nipkow@15524: 
nipkow@15524: lemma order_less_imp_not_less: "(x::'a::order) < y ==>  (~ y < x) = True"
nipkow@15524:   by (blast elim: order_less_asym)
nipkow@15524: 
nipkow@15524: lemma order_less_imp_triv: "(x::'a::order) < y ==>  (y < x --> P) = True"
nipkow@15524:   by (blast elim: order_less_asym)
nipkow@15524: 
nipkow@15524: lemma order_less_imp_not_eq: "(x::'a::order) < y ==>  (x = y) = False"
nipkow@15524:   by auto
nipkow@15524: 
nipkow@15524: lemma order_less_imp_not_eq2: "(x::'a::order) < y ==>  (y = x) = False"
nipkow@15524:   by auto
nipkow@15524: 
haftmann@21083: text {* Transitivity rules for calculational reasoning *}
nipkow@15524: 
nipkow@15524: lemma order_neq_le_trans: "a ~= b ==> (a::'a::order) <= b ==> a < b"
nipkow@15524:   by (simp add: order_less_le)
nipkow@15524: 
nipkow@15524: lemma order_le_neq_trans: "(a::'a::order) <= b ==> a ~= b ==> a < b"
nipkow@15524:   by (simp add: order_less_le)
nipkow@15524: 
nipkow@15524: lemma order_less_asym': "(a::'a::order) < b ==> b < a ==> P"
nipkow@15524:   by (rule order_less_asym)
nipkow@15524: 
nipkow@15524: 
haftmann@21216: subsection {* Linear (total) orderings *}
haftmann@21216: 
haftmann@21216: locale linear_order = partial_order +
haftmann@21216:   assumes linear: "x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
nipkow@15524: 
nipkow@15524: axclass linorder < order
nipkow@15524:   linorder_linear: "x <= y | y <= x"
nipkow@15524: 
haftmann@21216: interpretation linorder:
haftmann@21216:   linear_order ["op \<le> \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> bool" "op < \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> bool"]
haftmann@21216:   by unfold_locales (rule linorder_linear)
haftmann@21216: 
nipkow@15524: lemma linorder_less_linear: "!!x::'a::linorder. x<y | x=y | y<x"
nipkow@15524:   apply (simp add: order_less_le)
nipkow@15524:   apply (insert linorder_linear, blast)
nipkow@15524:   done
nipkow@15524: 
nipkow@15524: lemma linorder_le_less_linear: "!!x::'a::linorder. x\<le>y | y<x"
nipkow@15524:   by (simp add: order_le_less linorder_less_linear)
nipkow@15524: 
nipkow@15524: lemma linorder_le_cases [case_names le ge]:
nipkow@15524:     "((x::'a::linorder) \<le> y ==> P) ==> (y \<le> x ==> P) ==> P"
nipkow@15524:   by (insert linorder_linear, blast)
nipkow@15524: 
nipkow@15524: lemma linorder_cases [case_names less equal greater]:
nipkow@15524:     "((x::'a::linorder) < y ==> P) ==> (x = y ==> P) ==> (y < x ==> P) ==> P"
nipkow@15524:   by (insert linorder_less_linear, blast)
nipkow@15524: 
nipkow@15524: lemma linorder_not_less: "!!x::'a::linorder. (~ x < y) = (y <= x)"
nipkow@15524:   apply (simp add: order_less_le)
nipkow@15524:   apply (insert linorder_linear)
nipkow@15524:   apply (blast intro: order_antisym)
nipkow@15524:   done
nipkow@15524: 
nipkow@15524: lemma linorder_not_le: "!!x::'a::linorder. (~ x <= y) = (y < x)"
nipkow@15524:   apply (simp add: order_less_le)
nipkow@15524:   apply (insert linorder_linear)
nipkow@15524:   apply (blast intro: order_antisym)
nipkow@15524:   done
nipkow@15524: 
nipkow@15524: lemma linorder_neq_iff: "!!x::'a::linorder. (x ~= y) = (x<y | y<x)"
nipkow@15524: by (cut_tac x = x and y = y in linorder_less_linear, auto)
nipkow@15524: 
nipkow@15524: lemma linorder_neqE: "x ~= (y::'a::linorder) ==> (x < y ==> R) ==> (y < x ==> R) ==> R"
nipkow@15524: by (simp add: linorder_neq_iff, blast)
nipkow@15524: 
nipkow@15524: lemma linorder_antisym_conv1: "~ (x::'a::linorder) < y ==> (x <= y) = (x = y)"
nipkow@15524: by(blast intro:order_antisym dest:linorder_not_less[THEN iffD1])
nipkow@15524: 
nipkow@15524: lemma linorder_antisym_conv2: "(x::'a::linorder) <= y ==> (~ x < y) = (x = y)"
nipkow@15524: by(blast intro:order_antisym dest:linorder_not_less[THEN iffD1])
nipkow@15524: 
nipkow@15524: lemma linorder_antisym_conv3: "~ (y::'a::linorder) < x ==> (~ x < y) = (x = y)"
nipkow@15524: by(blast intro:order_antisym dest:linorder_not_less[THEN iffD1])
nipkow@15524: 
paulson@16796: text{*Replacing the old Nat.leI*}
paulson@16796: lemma leI: "~ x < y ==> y <= (x::'a::linorder)"
paulson@16796:   by (simp only: linorder_not_less)
paulson@16796: 
paulson@16796: lemma leD: "y <= (x::'a::linorder) ==> ~ x < y"
paulson@16796:   by (simp only: linorder_not_less)
paulson@16796: 
paulson@16796: (*FIXME inappropriate name (or delete altogether)*)
paulson@16796: lemma not_leE: "~ y <= (x::'a::linorder) ==> x < y"
paulson@16796:   by (simp only: linorder_not_le)
paulson@16796: 
haftmann@21083: 
haftmann@21083: subsection {* Reasoning tools setup *}
haftmann@21083: 
haftmann@21091: ML {*
haftmann@21091: local
haftmann@21091: 
haftmann@21091: fun decomp_gen sort thy (Trueprop $ t) =
haftmann@21091:   let fun of_sort t = let val T = type_of t in
haftmann@21091:         (* exclude numeric types: linear arithmetic subsumes transitivity *)
haftmann@21091:         T <> HOLogic.natT andalso T <> HOLogic.intT andalso
haftmann@21091:         T <> HOLogic.realT andalso Sign.of_sort thy (T, sort) end
haftmann@21091:   fun dec (Const ("Not", _) $ t) = (
haftmann@21091: 	  case dec t of
haftmann@21091: 	    NONE => NONE
haftmann@21091: 	  | SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2))
haftmann@21091: 	| dec (Const ("op =",  _) $ t1 $ t2) =
haftmann@21091: 	    if of_sort t1
haftmann@21091: 	    then SOME (t1, "=", t2)
haftmann@21091: 	    else NONE
haftmann@21091: 	| dec (Const ("Orderings.less_eq",  _) $ t1 $ t2) =
haftmann@21091: 	    if of_sort t1
haftmann@21091: 	    then SOME (t1, "<=", t2)
haftmann@21091: 	    else NONE
haftmann@21091: 	| dec (Const ("Orderings.less",  _) $ t1 $ t2) =
haftmann@21091: 	    if of_sort t1
haftmann@21091: 	    then SOME (t1, "<", t2)
haftmann@21091: 	    else NONE
haftmann@21091: 	| dec _ = NONE
haftmann@21091:   in dec t end;
haftmann@21091: 
haftmann@21091: in
haftmann@21091: 
haftmann@21091: structure Quasi_Tac = Quasi_Tac_Fun (
haftmann@21091: (* The setting up of Quasi_Tac serves as a demo.  Since there is no
haftmann@21091:    class for quasi orders, the tactics Quasi_Tac.trans_tac and
haftmann@21091:    Quasi_Tac.quasi_tac are not of much use. *)
haftmann@21091:   struct
haftmann@21091:     val le_trans = thm "order_trans";
haftmann@21091:     val le_refl = thm "order_refl";
haftmann@21091:     val eqD1 = thm "order_eq_refl";
haftmann@21091:     val eqD2 = thm "sym" RS thm "order_eq_refl";
haftmann@21091:     val less_reflE = thm "order_less_irrefl" RS thm "notE";
haftmann@21091:     val less_imp_le = thm "order_less_imp_le";
haftmann@21091:     val le_neq_trans = thm "order_le_neq_trans";
haftmann@21091:     val neq_le_trans = thm "order_neq_le_trans";
haftmann@21091:     val less_imp_neq = thm "less_imp_neq";
haftmann@21091:     val decomp_trans = decomp_gen ["Orderings.order"];
haftmann@21091:     val decomp_quasi = decomp_gen ["Orderings.order"];
haftmann@21091: 
haftmann@21091:   end);
haftmann@21091: 
haftmann@21091: structure Order_Tac = Order_Tac_Fun (
haftmann@21091:   struct
haftmann@21091:     val less_reflE = thm "order_less_irrefl" RS thm "notE";
haftmann@21091:     val le_refl = thm "order_refl";
haftmann@21091:     val less_imp_le = thm "order_less_imp_le";
haftmann@21091:     val not_lessI = thm "linorder_not_less" RS thm "iffD2";
haftmann@21091:     val not_leI = thm "linorder_not_le" RS thm "iffD2";
haftmann@21091:     val not_lessD = thm "linorder_not_less" RS thm "iffD1";
haftmann@21091:     val not_leD = thm "linorder_not_le" RS thm "iffD1";
haftmann@21091:     val eqI = thm "order_antisym";
haftmann@21091:     val eqD1 = thm "order_eq_refl";
haftmann@21091:     val eqD2 = thm "sym" RS thm "order_eq_refl";
haftmann@21091:     val less_trans = thm "order_less_trans";
haftmann@21091:     val less_le_trans = thm "order_less_le_trans";
haftmann@21091:     val le_less_trans = thm "order_le_less_trans";
haftmann@21091:     val le_trans = thm "order_trans";
haftmann@21091:     val le_neq_trans = thm "order_le_neq_trans";
haftmann@21091:     val neq_le_trans = thm "order_neq_le_trans";
haftmann@21091:     val less_imp_neq = thm "less_imp_neq";
haftmann@21091:     val eq_neq_eq_imp_neq = thm "eq_neq_eq_imp_neq";
haftmann@21091:     val not_sym = thm "not_sym";
haftmann@21091:     val decomp_part = decomp_gen ["Orderings.order"];
haftmann@21091:     val decomp_lin = decomp_gen ["Orderings.linorder"];
haftmann@21091: 
haftmann@21091:   end);
haftmann@21091: 
haftmann@21091: end;
haftmann@21091: *}
haftmann@21091: 
haftmann@21083: setup {*
haftmann@21083: let
haftmann@21083: 
haftmann@21083: val order_antisym_conv = thm "order_antisym_conv"
haftmann@21083: val linorder_antisym_conv1 = thm "linorder_antisym_conv1"
haftmann@21083: val linorder_antisym_conv2 = thm "linorder_antisym_conv2"
haftmann@21083: val linorder_antisym_conv3 = thm "linorder_antisym_conv3"
haftmann@21083: 
haftmann@21083: fun prp t thm = (#prop (rep_thm thm) = t);
nipkow@15524: 
haftmann@21083: fun prove_antisym_le sg ss ((le as Const(_,T)) $ r $ s) =
haftmann@21083:   let val prems = prems_of_ss ss;
haftmann@21083:       val less = Const("Orderings.less",T);
haftmann@21083:       val t = HOLogic.mk_Trueprop(le $ s $ r);
haftmann@21083:   in case find_first (prp t) prems of
haftmann@21083:        NONE =>
haftmann@21083:          let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s))
haftmann@21083:          in case find_first (prp t) prems of
haftmann@21083:               NONE => NONE
haftmann@21083:             | SOME thm => SOME(mk_meta_eq(thm RS linorder_antisym_conv1))
haftmann@21083:          end
haftmann@21083:      | SOME thm => SOME(mk_meta_eq(thm RS order_antisym_conv))
haftmann@21083:   end
haftmann@21083:   handle THM _ => NONE;
nipkow@15524: 
haftmann@21083: fun prove_antisym_less sg ss (NotC $ ((less as Const(_,T)) $ r $ s)) =
haftmann@21083:   let val prems = prems_of_ss ss;
haftmann@21083:       val le = Const("Orderings.less_eq",T);
haftmann@21083:       val t = HOLogic.mk_Trueprop(le $ r $ s);
haftmann@21083:   in case find_first (prp t) prems of
haftmann@21083:        NONE =>
haftmann@21083:          let val t = HOLogic.mk_Trueprop(NotC $ (less $ s $ r))
haftmann@21083:          in case find_first (prp t) prems of
haftmann@21083:               NONE => NONE
haftmann@21083:             | SOME thm => SOME(mk_meta_eq(thm RS linorder_antisym_conv3))
haftmann@21083:          end
haftmann@21083:      | SOME thm => SOME(mk_meta_eq(thm RS linorder_antisym_conv2))
haftmann@21083:   end
haftmann@21083:   handle THM _ => NONE;
nipkow@15524: 
haftmann@21083: val antisym_le = Simplifier.simproc (the_context())
haftmann@21083:   "antisym le" ["(x::'a::order) <= y"] prove_antisym_le;
haftmann@21083: val antisym_less = Simplifier.simproc (the_context())
haftmann@21083:   "antisym less" ["~ (x::'a::linorder) < y"] prove_antisym_less;
haftmann@21083: 
haftmann@21083: in
haftmann@21091:   (fn thy => (Simplifier.change_simpset_of thy
haftmann@21091:     (fn ss => ss
haftmann@21091:        addsimprocs [antisym_le, antisym_less]
haftmann@21091:        addSolver (mk_solver "Trans_linear" (fn _ => Order_Tac.linear_tac))
haftmann@21091:        addSolver (mk_solver "Trans_partial" (fn _ => Order_Tac.partial_tac)))
haftmann@21091:        (* Adding the transitivity reasoners also as safe solvers showed a slight
haftmann@21091:           speed up, but the reasoning strength appears to be not higher (at least
haftmann@21091:           no breaking of additional proofs in the entire HOL distribution, as
haftmann@21091:           of 5 March 2004, was observed). *); thy))
haftmann@21083: end
haftmann@21083: *}
nipkow@15524: 
nipkow@15524: 
haftmann@21083: subsection {* Bounded quantifiers *}
haftmann@21083: 
haftmann@21083: syntax
wenzelm@21180:   "_All_less" :: "[idt, 'a, bool] => bool"    ("(3ALL _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180:   "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3EX _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180:   "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _<=_./ _)" [0, 0, 10] 10)
wenzelm@21180:   "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _<=_./ _)" [0, 0, 10] 10)
haftmann@21083: 
wenzelm@21180:   "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3ALL _>_./ _)"  [0, 0, 10] 10)
wenzelm@21180:   "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3EX _>_./ _)"  [0, 0, 10] 10)
wenzelm@21180:   "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _>=_./ _)" [0, 0, 10] 10)
wenzelm@21180:   "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _>=_./ _)" [0, 0, 10] 10)
haftmann@21083: 
haftmann@21083: syntax (xsymbols)
wenzelm@21180:   "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180:   "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180:   "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
wenzelm@21180:   "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
haftmann@21083: 
wenzelm@21180:   "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180:   "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180:   "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
wenzelm@21180:   "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
haftmann@21083: 
haftmann@21083: syntax (HOL)
wenzelm@21180:   "_All_less" :: "[idt, 'a, bool] => bool"    ("(3! _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180:   "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3? _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180:   "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3! _<=_./ _)" [0, 0, 10] 10)
wenzelm@21180:   "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3? _<=_./ _)" [0, 0, 10] 10)
haftmann@21083: 
haftmann@21083: syntax (HTML output)
wenzelm@21180:   "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180:   "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180:   "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
wenzelm@21180:   "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
haftmann@21083: 
wenzelm@21180:   "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180:   "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180:   "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
wenzelm@21180:   "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
haftmann@21083: 
haftmann@21083: translations
haftmann@21083:   "ALL x<y. P"   =>  "ALL x. x < y \<longrightarrow> P"
haftmann@21083:   "EX x<y. P"    =>  "EX x. x < y \<and> P"
haftmann@21083:   "ALL x<=y. P"  =>  "ALL x. x <= y \<longrightarrow> P"
haftmann@21083:   "EX x<=y. P"   =>  "EX x. x <= y \<and> P"
haftmann@21083:   "ALL x>y. P"   =>  "ALL x. x > y \<longrightarrow> P"
haftmann@21083:   "EX x>y. P"    =>  "EX x. x > y \<and> P"
haftmann@21083:   "ALL x>=y. P"  =>  "ALL x. x >= y \<longrightarrow> P"
haftmann@21083:   "EX x>=y. P"   =>  "EX x. x >= y \<and> P"
haftmann@21083: 
haftmann@21083: print_translation {*
haftmann@21083: let
wenzelm@21180:   val syntax_name = Sign.const_syntax_name (the_context ());
wenzelm@21180:   val impl = syntax_name "op -->";
wenzelm@21180:   val conj = syntax_name "op &";
wenzelm@21180:   val less = syntax_name "Orderings.less";
wenzelm@21180:   val less_eq = syntax_name "Orderings.less_eq";
wenzelm@21180: 
wenzelm@21180:   val trans =
wenzelm@21180:    [(("ALL ", impl, less), ("_All_less", "_All_greater")),
wenzelm@21180:     (("ALL ", impl, less_eq), ("_All_less_eq", "_All_greater_eq")),
wenzelm@21180:     (("EX ", conj, less), ("_Ex_less", "_Ex_greater")),
wenzelm@21180:     (("EX ", conj, less_eq), ("_Ex_less_eq", "_Ex_greater_eq"))];
wenzelm@21180: 
haftmann@21083:   fun mk v v' c n P =
wenzelm@21180:     if v = v' andalso not (Term.exists_subterm (fn Free (x, _) => x = v | _ => false) n)
haftmann@21083:     then Syntax.const c $ Syntax.mark_bound v' $ n $ P else raise Match;
wenzelm@21180: 
wenzelm@21180:   fun tr' q = (q,
wenzelm@21180:     fn [Const ("_bound", _) $ Free (v, _), Const (c, _) $ (Const (d, _) $ t $ u) $ P] =>
wenzelm@21180:       (case AList.lookup (op =) trans (q, c, d) of
wenzelm@21180:         NONE => raise Match
wenzelm@21180:       | SOME (l, g) =>
wenzelm@21180:           (case (t, u) of
wenzelm@21180:             (Const ("_bound", _) $ Free (v', _), n) => mk v v' l n P
wenzelm@21180:           | (n, Const ("_bound", _) $ Free (v', _)) => mk v v' g n P
wenzelm@21180:           | _ => raise Match))
wenzelm@21180:      | _ => raise Match);
wenzelm@21180: in [tr' "ALL ", tr' "EX "] end
haftmann@21083: *}
haftmann@21083: 
haftmann@21083: 
haftmann@21083: subsection {* Transitivity reasoning on decreasing inequalities *}
haftmann@21083: 
wenzelm@21180: (* FIXME cleanup *)
wenzelm@21180: 
haftmann@21083: text {* These support proving chains of decreasing inequalities
haftmann@21083:     a >= b >= c ... in Isar proofs. *}
haftmann@21083: 
haftmann@21083: lemma xt1:
haftmann@21083:   "a = b ==> b > c ==> a > c"
haftmann@21083:   "a > b ==> b = c ==> a > c"
haftmann@21083:   "a = b ==> b >= c ==> a >= c"
haftmann@21083:   "a >= b ==> b = c ==> a >= c"
haftmann@21083:   "(x::'a::order) >= y ==> y >= x ==> x = y"
haftmann@21083:   "(x::'a::order) >= y ==> y >= z ==> x >= z"
haftmann@21083:   "(x::'a::order) > y ==> y >= z ==> x > z"
haftmann@21083:   "(x::'a::order) >= y ==> y > z ==> x > z"
haftmann@21083:   "(a::'a::order) > b ==> b > a ==> ?P"
haftmann@21083:   "(x::'a::order) > y ==> y > z ==> x > z"
haftmann@21083:   "(a::'a::order) >= b ==> a ~= b ==> a > b"
haftmann@21083:   "(a::'a::order) ~= b ==> a >= b ==> a > b"
haftmann@21083:   "a = f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c" 
haftmann@21083:   "a > b ==> f b = c ==> (!!x y. x > y ==> f x > f y) ==> f a > c"
haftmann@21083:   "a = f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
haftmann@21083:   "a >= b ==> f b = c ==> (!! x y. x >= y ==> f x >= f y) ==> f a >= c"
haftmann@21083: by auto
haftmann@21083: 
haftmann@21083: lemma xt2:
haftmann@21083:   "(a::'a::order) >= f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
haftmann@21083: by (subgoal_tac "f b >= f c", force, force)
haftmann@21083: 
haftmann@21083: lemma xt3: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==> 
haftmann@21083:     (!!x y. x >= y ==> f x >= f y) ==> f a >= c"
haftmann@21083: by (subgoal_tac "f a >= f b", force, force)
haftmann@21083: 
haftmann@21083: lemma xt4: "(a::'a::order) > f b ==> (b::'b::order) >= c ==>
haftmann@21083:   (!!x y. x >= y ==> f x >= f y) ==> a > f c"
haftmann@21083: by (subgoal_tac "f b >= f c", force, force)
haftmann@21083: 
haftmann@21083: lemma xt5: "(a::'a::order) > b ==> (f b::'b::order) >= c==>
haftmann@21083:     (!!x y. x > y ==> f x > f y) ==> f a > c"
haftmann@21083: by (subgoal_tac "f a > f b", force, force)
haftmann@21083: 
haftmann@21083: lemma xt6: "(a::'a::order) >= f b ==> b > c ==>
haftmann@21083:     (!!x y. x > y ==> f x > f y) ==> a > f c"
haftmann@21083: by (subgoal_tac "f b > f c", force, force)
haftmann@21083: 
haftmann@21083: lemma xt7: "(a::'a::order) >= b ==> (f b::'b::order) > c ==>
haftmann@21083:     (!!x y. x >= y ==> f x >= f y) ==> f a > c"
haftmann@21083: by (subgoal_tac "f a >= f b", force, force)
haftmann@21083: 
haftmann@21083: lemma xt8: "(a::'a::order) > f b ==> (b::'b::order) > c ==>
haftmann@21083:     (!!x y. x > y ==> f x > f y) ==> a > f c"
haftmann@21083: by (subgoal_tac "f b > f c", force, force)
haftmann@21083: 
haftmann@21083: lemma xt9: "(a::'a::order) > b ==> (f b::'b::order) > c ==>
haftmann@21083:     (!!x y. x > y ==> f x > f y) ==> f a > c"
haftmann@21083: by (subgoal_tac "f a > f b", force, force)
haftmann@21083: 
haftmann@21083: lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9
haftmann@21083: 
haftmann@21083: (* 
haftmann@21083:   Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands
haftmann@21083:   for the wrong thing in an Isar proof.
haftmann@21083: 
haftmann@21083:   The extra transitivity rules can be used as follows: 
haftmann@21083: 
haftmann@21083: lemma "(a::'a::order) > z"
haftmann@21083: proof -
haftmann@21083:   have "a >= b" (is "_ >= ?rhs")
haftmann@21083:     sorry
haftmann@21083:   also have "?rhs >= c" (is "_ >= ?rhs")
haftmann@21083:     sorry
haftmann@21083:   also (xtrans) have "?rhs = d" (is "_ = ?rhs")
haftmann@21083:     sorry
haftmann@21083:   also (xtrans) have "?rhs >= e" (is "_ >= ?rhs")
haftmann@21083:     sorry
haftmann@21083:   also (xtrans) have "?rhs > f" (is "_ > ?rhs")
haftmann@21083:     sorry
haftmann@21083:   also (xtrans) have "?rhs > z"
haftmann@21083:     sorry
haftmann@21083:   finally (xtrans) show ?thesis .
haftmann@21083: qed
haftmann@21083: 
haftmann@21083:   Alternatively, one can use "declare xtrans [trans]" and then
haftmann@21083:   leave out the "(xtrans)" above.
haftmann@21083: *)
haftmann@21083: 
haftmann@21216: subsection {* Monotonicity, syntactic least value operator and syntactic min/max *}
haftmann@21083: 
haftmann@21216: locale mono =
haftmann@21216:   fixes f
haftmann@21216:   assumes mono: "A \<le> B \<Longrightarrow> f A \<le> f B"
haftmann@21216: 
haftmann@21216: lemmas monoI [intro?] = mono.intro
haftmann@21216:   and monoD [dest?] = mono.mono
haftmann@21083: 
haftmann@21083: constdefs
haftmann@21083:   Least :: "('a::ord => bool) => 'a"               (binder "LEAST " 10)
haftmann@21083:   "Least P == THE x. P x & (ALL y. P y --> x <= y)"
haftmann@21083:     -- {* We can no longer use LeastM because the latter requires Hilbert-AC. *}
haftmann@21083: 
haftmann@21083: constdefs
haftmann@21083:   min :: "['a::ord, 'a] => 'a"
haftmann@21083:   "min a b == (if a <= b then a else b)"
haftmann@21083:   max :: "['a::ord, 'a] => 'a"
haftmann@21083:   "max a b == (if a <= b then b else a)"
haftmann@21083: 
nipkow@15524: end




