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(* Title: HOL/Orderings.thy
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ID: $Id$
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Author: Tobias Nipkow, Markus Wenzel, and Larry Paulson
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FIXME: derive more of the min/max laws generically via semilattices
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*)
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header {* Type classes for $\le$ *}
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theory Orderings
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imports Lattice_Locales
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files ("antisym_setup.ML")
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begin
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subsection {* Order signatures and orders *}
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axclass
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ord < type
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syntax
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"op <" :: "['a::ord, 'a] => bool" ("op <")
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"op <=" :: "['a::ord, 'a] => bool" ("op <=")
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global
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consts
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"op <" :: "['a::ord, 'a] => bool" ("(_/ < _)" [50, 51] 50)
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"op <=" :: "['a::ord, 'a] => bool" ("(_/ <= _)" [50, 51] 50)
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local
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syntax (xsymbols)
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"op <=" :: "['a::ord, 'a] => bool" ("op \<le>")
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"op <=" :: "['a::ord, 'a] => bool" ("(_/ \<le> _)" [50, 51] 50)
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syntax (HTML output)
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"op <=" :: "['a::ord, 'a] => bool" ("op \<le>")
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"op <=" :: "['a::ord, 'a] => bool" ("(_/ \<le> _)" [50, 51] 50)
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text{* Syntactic sugar: *}
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consts
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"_gt" :: "'a::ord => 'a => bool" (infixl ">" 50)
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"_ge" :: "'a::ord => 'a => bool" (infixl ">=" 50)
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translations
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"x > y" => "y < x"
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"x >= y" => "y <= x"
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syntax (xsymbols)
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"_ge" :: "'a::ord => 'a => bool" (infixl "\<ge>" 50)
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syntax (HTML output)
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"_ge" :: "['a::ord, 'a] => bool" (infixl "\<ge>" 50)
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subsection {* Monotonicity *}
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locale mono =
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fixes f
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assumes mono: "A <= B ==> f A <= f B"
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lemmas monoI [intro?] = mono.intro
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and monoD [dest?] = mono.mono
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constdefs
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min :: "['a::ord, 'a] => 'a"
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"min a b == (if a <= b then a else b)"
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max :: "['a::ord, 'a] => 'a"
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"max a b == (if a <= b then b else a)"
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lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
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by (simp add: min_def)
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lemma min_of_mono:
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"ALL x y. (f x <= f y) = (x <= y) ==> min (f m) (f n) = f (min m n)"
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by (simp add: min_def)
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lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
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by (simp add: max_def)
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lemma max_of_mono:
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"ALL x y. (f x <= f y) = (x <= y) ==> max (f m) (f n) = f (max m n)"
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by (simp add: max_def)
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subsection "Orders"
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axclass order < ord
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order_refl [iff]: "x <= x"
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order_trans: "x <= y ==> y <= z ==> x <= z"
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order_antisym: "x <= y ==> y <= x ==> x = y"
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order_less_le: "(x < y) = (x <= y & x ~= y)"
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text{* Connection to locale: *}
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lemma partial_order_order:
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"partial_order (op \<le> :: 'a::order \<Rightarrow> 'a \<Rightarrow> bool)"
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apply(rule partial_order.intro)
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apply(rule order_refl, erule (1) order_trans, erule (1) order_antisym)
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done
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text {* Reflexivity. *}
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lemma order_eq_refl: "!!x::'a::order. x = y ==> x <= y"
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-- {* This form is useful with the classical reasoner. *}
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apply (erule ssubst)
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apply (rule order_refl)
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done
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lemma order_less_irrefl [iff]: "~ x < (x::'a::order)"
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by (simp add: order_less_le)
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lemma order_le_less: "((x::'a::order) <= y) = (x < y | x = y)"
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-- {* NOT suitable for iff, since it can cause PROOF FAILED. *}
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apply (simp add: order_less_le, blast)
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done
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lemmas order_le_imp_less_or_eq = order_le_less [THEN iffD1, standard]
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lemma order_less_imp_le: "!!x::'a::order. x < y ==> x <= y"
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by (simp add: order_less_le)
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text {* Asymmetry. *}
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lemma order_less_not_sym: "(x::'a::order) < y ==> ~ (y < x)"
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by (simp add: order_less_le order_antisym)
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lemma order_less_asym: "x < (y::'a::order) ==> (~P ==> y < x) ==> P"
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apply (drule order_less_not_sym)
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apply (erule contrapos_np, simp)
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done
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lemma order_eq_iff: "!!x::'a::order. (x = y) = (x \<le> y & y \<le> x)"
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by (blast intro: order_antisym)
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lemma order_antisym_conv: "(y::'a::order) <= x ==> (x <= y) = (x = y)"
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by(blast intro:order_antisym)
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text {* Transitivity. *}
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lemma order_less_trans: "!!x::'a::order. [| x < y; y < z |] ==> x < z"
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apply (simp add: order_less_le)
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apply (blast intro: order_trans order_antisym)
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done
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lemma order_le_less_trans: "!!x::'a::order. [| x <= y; y < z |] ==> x < z"
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apply (simp add: order_less_le)
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apply (blast intro: order_trans order_antisym)
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done
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lemma order_less_le_trans: "!!x::'a::order. [| x < y; y <= z |] ==> x < z"
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apply (simp add: order_less_le)
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apply (blast intro: order_trans order_antisym)
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done
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text {* Useful for simplification, but too risky to include by default. *}
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lemma order_less_imp_not_less: "(x::'a::order) < y ==> (~ y < x) = True"
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by (blast elim: order_less_asym)
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lemma order_less_imp_triv: "(x::'a::order) < y ==> (y < x --> P) = True"
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by (blast elim: order_less_asym)
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lemma order_less_imp_not_eq: "(x::'a::order) < y ==> (x = y) = False"
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by auto
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lemma order_less_imp_not_eq2: "(x::'a::order) < y ==> (y = x) = False"
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by auto
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text {* Other operators. *}
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lemma min_leastR: "(!!x::'a::order. least <= x) ==> min x least = least"
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apply (simp add: min_def)
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apply (blast intro: order_antisym)
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done
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lemma max_leastR: "(!!x::'a::order. least <= x) ==> max x least = x"
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apply (simp add: max_def)
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apply (blast intro: order_antisym)
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done
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subsection {* Transitivity rules for calculational reasoning *}
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lemma order_neq_le_trans: "a ~= b ==> (a::'a::order) <= b ==> a < b"
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by (simp add: order_less_le)
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lemma order_le_neq_trans: "(a::'a::order) <= b ==> a ~= b ==> a < b"
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by (simp add: order_less_le)
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lemma order_less_asym': "(a::'a::order) < b ==> b < a ==> P"
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by (rule order_less_asym)
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subsection {* Least value operator *}
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constdefs
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Least :: "('a::ord => bool) => 'a" (binder "LEAST " 10)
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"Least P == THE x. P x & (ALL y. P y --> x <= y)"
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-- {* We can no longer use LeastM because the latter requires Hilbert-AC. *}
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lemma LeastI2:
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"[| P (x::'a::order);
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!!y. P y ==> x <= y;
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!!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |]
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==> Q (Least P)"
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apply (unfold Least_def)
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apply (rule theI2)
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apply (blast intro: order_antisym)+
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done
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lemma Least_equality:
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"[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k"
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apply (simp add: Least_def)
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apply (rule the_equality)
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apply (auto intro!: order_antisym)
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done
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subsection "Linear / total orders"
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axclass linorder < order
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linorder_linear: "x <= y | y <= x"
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lemma linorder_less_linear: "!!x::'a::linorder. x<y | x=y | y<x"
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apply (simp add: order_less_le)
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apply (insert linorder_linear, blast)
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done
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lemma linorder_le_less_linear: "!!x::'a::linorder. x\<le>y | y<x"
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by (simp add: order_le_less linorder_less_linear)
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lemma linorder_le_cases [case_names le ge]:
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"((x::'a::linorder) \<le> y ==> P) ==> (y \<le> x ==> P) ==> P"
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by (insert linorder_linear, blast)
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lemma linorder_cases [case_names less equal greater]:
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"((x::'a::linorder) < y ==> P) ==> (x = y ==> P) ==> (y < x ==> P) ==> P"
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by (insert linorder_less_linear, blast)
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lemma linorder_not_less: "!!x::'a::linorder. (~ x < y) = (y <= x)"
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apply (simp add: order_less_le)
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apply (insert linorder_linear)
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apply (blast intro: order_antisym)
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done
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lemma linorder_not_le: "!!x::'a::linorder. (~ x <= y) = (y < x)"
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apply (simp add: order_less_le)
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apply (insert linorder_linear)
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apply (blast intro: order_antisym)
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done
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lemma linorder_neq_iff: "!!x::'a::linorder. (x ~= y) = (x<y | y<x)"
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by (cut_tac x = x and y = y in linorder_less_linear, auto)
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lemma linorder_neqE: "x ~= (y::'a::linorder) ==> (x < y ==> R) ==> (y < x ==> R) ==> R"
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by (simp add: linorder_neq_iff, blast)
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lemma linorder_antisym_conv1: "~ (x::'a::linorder) < y ==> (x <= y) = (x = y)"
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by(blast intro:order_antisym dest:linorder_not_less[THEN iffD1])
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lemma linorder_antisym_conv2: "(x::'a::linorder) <= y ==> (~ x < y) = (x = y)"
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by(blast intro:order_antisym dest:linorder_not_less[THEN iffD1])
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lemma linorder_antisym_conv3: "~ (y::'a::linorder) < x ==> (~ x < y) = (x = y)"
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by(blast intro:order_antisym dest:linorder_not_less[THEN iffD1])
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use "antisym_setup.ML";
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setup antisym_setup
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subsection {* Setup of transitivity reasoner as Solver *}
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lemma less_imp_neq: "[| (x::'a::order) < y |] ==> x ~= y"
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by (erule contrapos_pn, erule subst, rule order_less_irrefl)
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lemma eq_neq_eq_imp_neq: "[| x = a ; a ~= b; b = y |] ==> x ~= y"
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by (erule subst, erule ssubst, assumption)
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ML_setup {*
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(* The setting up of Quasi_Tac serves as a demo. Since there is no
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class for quasi orders, the tactics Quasi_Tac.trans_tac and
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Quasi_Tac.quasi_tac are not of much use. *)
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fun decomp_gen sort sign (Trueprop $ t) =
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let fun of_sort t = Sign.of_sort sign (type_of t, sort)
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fun dec (Const ("Not", _) $ t) = (
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case dec t of
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None => None
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| Some (t1, rel, t2) => Some (t1, "~" ^ rel, t2))
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| dec (Const ("op =", _) $ t1 $ t2) =
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if of_sort t1
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then Some (t1, "=", t2)
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else None
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| dec (Const ("op <=", _) $ t1 $ t2) =
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if of_sort t1
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then Some (t1, "<=", t2)
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else None
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| dec (Const ("op <", _) $ t1 $ t2) =
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if of_sort t1
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then Some (t1, "<", t2)
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else None
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| dec _ = None
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in dec t end;
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structure Quasi_Tac = Quasi_Tac_Fun (
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struct
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val le_trans = thm "order_trans";
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val le_refl = thm "order_refl";
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val eqD1 = thm "order_eq_refl";
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val eqD2 = thm "sym" RS thm "order_eq_refl";
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val less_reflE = thm "order_less_irrefl" RS thm "notE";
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val less_imp_le = thm "order_less_imp_le";
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val le_neq_trans = thm "order_le_neq_trans";
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val neq_le_trans = thm "order_neq_le_trans";
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val less_imp_neq = thm "less_imp_neq";
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val decomp_trans = decomp_gen ["Orderings.order"];
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val decomp_quasi = decomp_gen ["Orderings.order"];
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end); (* struct *)
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structure Order_Tac = Order_Tac_Fun (
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struct
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val less_reflE = thm "order_less_irrefl" RS thm "notE";
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val le_refl = thm "order_refl";
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val less_imp_le = thm "order_less_imp_le";
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val not_lessI = thm "linorder_not_less" RS thm "iffD2";
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val not_leI = thm "linorder_not_le" RS thm "iffD2";
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val not_lessD = thm "linorder_not_less" RS thm "iffD1";
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val not_leD = thm "linorder_not_le" RS thm "iffD1";
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val eqI = thm "order_antisym";
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val eqD1 = thm "order_eq_refl";
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val eqD2 = thm "sym" RS thm "order_eq_refl";
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val less_trans = thm "order_less_trans";
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val less_le_trans = thm "order_less_le_trans";
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val le_less_trans = thm "order_le_less_trans";
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val le_trans = thm "order_trans";
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val le_neq_trans = thm "order_le_neq_trans";
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val neq_le_trans = thm "order_neq_le_trans";
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val less_imp_neq = thm "less_imp_neq";
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val eq_neq_eq_imp_neq = thm "eq_neq_eq_imp_neq";
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val decomp_part = decomp_gen ["Orderings.order"];
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val decomp_lin = decomp_gen ["Orderings.linorder"];
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end); (* struct *)
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simpset_ref() := simpset ()
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addSolver (mk_solver "Trans_linear" (fn _ => Order_Tac.linear_tac))
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addSolver (mk_solver "Trans_partial" (fn _ => Order_Tac.partial_tac));
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(* Adding the transitivity reasoners also as safe solvers showed a slight
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speed up, but the reasoning strength appears to be not higher (at least
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no breaking of additional proofs in the entire HOL distribution, as
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of 5 March 2004, was observed). *)
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*}
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(* Optional setup of methods *)
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(*
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method_setup trans_partial =
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{* Method.no_args (Method.SIMPLE_METHOD' HEADGOAL (Order_Tac.partial_tac)) *}
|
|
365 |
{* transitivity reasoner for partial orders *}
|
|
366 |
method_setup trans_linear =
|
|
367 |
{* Method.no_args (Method.SIMPLE_METHOD' HEADGOAL (Order_Tac.linear_tac)) *}
|
|
368 |
{* transitivity reasoner for linear orders *}
|
|
369 |
*)
|
|
370 |
|
|
371 |
(*
|
|
372 |
declare order.order_refl [simp del] order_less_irrefl [simp del]
|
|
373 |
|
|
374 |
can currently not be removed, abel_cancel relies on it.
|
|
375 |
*)
|
|
376 |
|
|
377 |
|
|
378 |
subsection "Min and max on (linear) orders"
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|
379 |
|
|
380 |
lemma min_same [simp]: "min (x::'a::order) x = x"
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|
381 |
by (simp add: min_def)
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|
382 |
|
|
383 |
lemma max_same [simp]: "max (x::'a::order) x = x"
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|
384 |
by (simp add: max_def)
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|
385 |
|
|
386 |
text{* Instantiate locales: *}
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|
387 |
|
|
388 |
lemma lower_semilattice_lin_min:
|
|
389 |
"lower_semilattice(op \<le>) (min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a)"
|
|
390 |
apply(rule lower_semilattice.intro)
|
|
391 |
apply(rule partial_order_order)
|
|
392 |
apply(rule lower_semilattice_axioms.intro)
|
|
393 |
apply(simp add:min_def linorder_not_le order_less_imp_le)
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|
394 |
apply(simp add:min_def linorder_not_le order_less_imp_le)
|
|
395 |
apply(simp add:min_def linorder_not_le order_less_imp_le)
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|
396 |
done
|
|
397 |
|
|
398 |
lemma upper_semilattice_lin_max:
|
|
399 |
"upper_semilattice(op \<le>) (max :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a)"
|
|
400 |
apply(rule upper_semilattice.intro)
|
|
401 |
apply(rule partial_order_order)
|
|
402 |
apply(rule upper_semilattice_axioms.intro)
|
|
403 |
apply(simp add: max_def linorder_not_le order_less_imp_le)
|
|
404 |
apply(simp add: max_def linorder_not_le order_less_imp_le)
|
|
405 |
apply(simp add: max_def linorder_not_le order_less_imp_le)
|
|
406 |
done
|
|
407 |
|
|
408 |
lemma lattice_min_max: "lattice (op \<le>) (min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a) max"
|
|
409 |
apply(rule lattice.intro)
|
|
410 |
apply(rule partial_order_order)
|
|
411 |
apply(rule lower_semilattice.axioms[OF lower_semilattice_lin_min])
|
|
412 |
apply(rule upper_semilattice.axioms[OF upper_semilattice_lin_max])
|
|
413 |
done
|
|
414 |
|
|
415 |
lemma distrib_lattice_min_max:
|
|
416 |
"distrib_lattice (op \<le>) (min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a) max"
|
|
417 |
apply(rule distrib_lattice.intro)
|
|
418 |
apply(rule partial_order_order)
|
|
419 |
apply(rule lower_semilattice.axioms[OF lower_semilattice_lin_min])
|
|
420 |
apply(rule upper_semilattice.axioms[OF upper_semilattice_lin_max])
|
|
421 |
apply(rule distrib_lattice_axioms.intro)
|
|
422 |
apply(rule_tac x=x and y=y in linorder_le_cases)
|
|
423 |
apply(rule_tac x=x and y=z in linorder_le_cases)
|
|
424 |
apply(rule_tac x=y and y=z in linorder_le_cases)
|
|
425 |
apply(simp add:min_def max_def)
|
|
426 |
apply(simp add:min_def max_def)
|
|
427 |
apply(rule_tac x=y and y=z in linorder_le_cases)
|
|
428 |
apply(simp add:min_def max_def)
|
|
429 |
apply(simp add:min_def max_def)
|
|
430 |
apply(rule_tac x=x and y=z in linorder_le_cases)
|
|
431 |
apply(rule_tac x=y and y=z in linorder_le_cases)
|
|
432 |
apply(simp add:min_def max_def)
|
|
433 |
apply(simp add:min_def max_def)
|
|
434 |
apply(rule_tac x=y and y=z in linorder_le_cases)
|
|
435 |
apply(simp add:min_def max_def)
|
|
436 |
apply(simp add:min_def max_def)
|
|
437 |
done
|
|
438 |
|
|
439 |
lemma le_max_iff_disj: "!!z::'a::linorder. (z <= max x y) = (z <= x | z <= y)"
|
|
440 |
apply(simp add:max_def)
|
|
441 |
apply (insert linorder_linear)
|
|
442 |
apply (blast intro: order_trans)
|
|
443 |
done
|
|
444 |
|
|
445 |
lemma le_maxI1: "(x::'a::linorder) <= max x y"
|
|
446 |
by(rule upper_semilattice.sup_ge1[OF upper_semilattice_lin_max])
|
|
447 |
|
|
448 |
lemma le_maxI2: "(y::'a::linorder) <= max x y"
|
|
449 |
-- {* CANNOT use with @{text "[intro!]"} because blast will give PROOF FAILED. *}
|
|
450 |
by(rule upper_semilattice.sup_ge2[OF upper_semilattice_lin_max])
|
|
451 |
|
|
452 |
lemma less_max_iff_disj: "!!z::'a::linorder. (z < max x y) = (z < x | z < y)"
|
|
453 |
apply (simp add: max_def order_le_less)
|
|
454 |
apply (insert linorder_less_linear)
|
|
455 |
apply (blast intro: order_less_trans)
|
|
456 |
done
|
|
457 |
|
|
458 |
lemma max_le_iff_conj [simp]:
|
|
459 |
"!!z::'a::linorder. (max x y <= z) = (x <= z & y <= z)"
|
|
460 |
by (rule upper_semilattice.above_sup_conv[OF upper_semilattice_lin_max])
|
|
461 |
|
|
462 |
lemma max_less_iff_conj [simp]:
|
|
463 |
"!!z::'a::linorder. (max x y < z) = (x < z & y < z)"
|
|
464 |
apply (simp add: order_le_less max_def)
|
|
465 |
apply (insert linorder_less_linear)
|
|
466 |
apply (blast intro: order_less_trans)
|
|
467 |
done
|
|
468 |
|
|
469 |
lemma le_min_iff_conj [simp]:
|
|
470 |
"!!z::'a::linorder. (z <= min x y) = (z <= x & z <= y)"
|
|
471 |
-- {* @{text "[iff]"} screws up a @{text blast} in MiniML *}
|
|
472 |
by (rule lower_semilattice.below_inf_conv[OF lower_semilattice_lin_min])
|
|
473 |
|
|
474 |
lemma min_less_iff_conj [simp]:
|
|
475 |
"!!z::'a::linorder. (z < min x y) = (z < x & z < y)"
|
|
476 |
apply (simp add: order_le_less min_def)
|
|
477 |
apply (insert linorder_less_linear)
|
|
478 |
apply (blast intro: order_less_trans)
|
|
479 |
done
|
|
480 |
|
|
481 |
lemma min_le_iff_disj: "!!z::'a::linorder. (min x y <= z) = (x <= z | y <= z)"
|
|
482 |
apply (simp add: min_def)
|
|
483 |
apply (insert linorder_linear)
|
|
484 |
apply (blast intro: order_trans)
|
|
485 |
done
|
|
486 |
|
|
487 |
lemma min_less_iff_disj: "!!z::'a::linorder. (min x y < z) = (x < z | y < z)"
|
|
488 |
apply (simp add: min_def order_le_less)
|
|
489 |
apply (insert linorder_less_linear)
|
|
490 |
apply (blast intro: order_less_trans)
|
|
491 |
done
|
|
492 |
|
|
493 |
lemma max_assoc: "!!x::'a::linorder. max (max x y) z = max x (max y z)"
|
|
494 |
by (rule upper_semilattice.sup_assoc[OF upper_semilattice_lin_max])
|
|
495 |
|
|
496 |
lemma max_commute: "!!x::'a::linorder. max x y = max y x"
|
|
497 |
by (rule upper_semilattice.sup_commute[OF upper_semilattice_lin_max])
|
|
498 |
|
|
499 |
lemmas max_ac = max_assoc max_commute
|
|
500 |
mk_left_commute[of max,OF max_assoc max_commute]
|
|
501 |
|
|
502 |
lemma min_assoc: "!!x::'a::linorder. min (min x y) z = min x (min y z)"
|
|
503 |
by (rule lower_semilattice.inf_assoc[OF lower_semilattice_lin_min])
|
|
504 |
|
|
505 |
lemma min_commute: "!!x::'a::linorder. min x y = min y x"
|
|
506 |
by (rule lower_semilattice.inf_commute[OF lower_semilattice_lin_min])
|
|
507 |
|
|
508 |
lemmas min_ac = min_assoc min_commute
|
|
509 |
mk_left_commute[of min,OF min_assoc min_commute]
|
|
510 |
|
|
511 |
lemma split_min:
|
|
512 |
"P (min (i::'a::linorder) j) = ((i <= j --> P(i)) & (~ i <= j --> P(j)))"
|
|
513 |
by (simp add: min_def)
|
|
514 |
|
|
515 |
lemma split_max:
|
|
516 |
"P (max (i::'a::linorder) j) = ((i <= j --> P(j)) & (~ i <= j --> P(i)))"
|
|
517 |
by (simp add: max_def)
|
|
518 |
|
|
519 |
|
|
520 |
subsection "Bounded quantifiers"
|
|
521 |
|
|
522 |
syntax
|
|
523 |
"_lessAll" :: "[idt, 'a, bool] => bool" ("(3ALL _<_./ _)" [0, 0, 10] 10)
|
|
524 |
"_lessEx" :: "[idt, 'a, bool] => bool" ("(3EX _<_./ _)" [0, 0, 10] 10)
|
|
525 |
"_leAll" :: "[idt, 'a, bool] => bool" ("(3ALL _<=_./ _)" [0, 0, 10] 10)
|
|
526 |
"_leEx" :: "[idt, 'a, bool] => bool" ("(3EX _<=_./ _)" [0, 0, 10] 10)
|
|
527 |
|
|
528 |
"_gtAll" :: "[idt, 'a, bool] => bool" ("(3ALL _>_./ _)" [0, 0, 10] 10)
|
|
529 |
"_gtEx" :: "[idt, 'a, bool] => bool" ("(3EX _>_./ _)" [0, 0, 10] 10)
|
|
530 |
"_geAll" :: "[idt, 'a, bool] => bool" ("(3ALL _>=_./ _)" [0, 0, 10] 10)
|
|
531 |
"_geEx" :: "[idt, 'a, bool] => bool" ("(3EX _>=_./ _)" [0, 0, 10] 10)
|
|
532 |
|
|
533 |
syntax (xsymbols)
|
|
534 |
"_lessAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_<_./ _)" [0, 0, 10] 10)
|
|
535 |
"_lessEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_<_./ _)" [0, 0, 10] 10)
|
|
536 |
"_leAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
|
|
537 |
"_leEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
|
|
538 |
|
|
539 |
"_gtAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_>_./ _)" [0, 0, 10] 10)
|
|
540 |
"_gtEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_>_./ _)" [0, 0, 10] 10)
|
|
541 |
"_geAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
|
|
542 |
"_geEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
|
|
543 |
|
|
544 |
syntax (HOL)
|
|
545 |
"_lessAll" :: "[idt, 'a, bool] => bool" ("(3! _<_./ _)" [0, 0, 10] 10)
|
|
546 |
"_lessEx" :: "[idt, 'a, bool] => bool" ("(3? _<_./ _)" [0, 0, 10] 10)
|
|
547 |
"_leAll" :: "[idt, 'a, bool] => bool" ("(3! _<=_./ _)" [0, 0, 10] 10)
|
|
548 |
"_leEx" :: "[idt, 'a, bool] => bool" ("(3? _<=_./ _)" [0, 0, 10] 10)
|
|
549 |
|
|
550 |
syntax (HTML output)
|
|
551 |
"_lessAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_<_./ _)" [0, 0, 10] 10)
|
|
552 |
"_lessEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_<_./ _)" [0, 0, 10] 10)
|
|
553 |
"_leAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
|
|
554 |
"_leEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
|
|
555 |
|
|
556 |
"_gtAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_>_./ _)" [0, 0, 10] 10)
|
|
557 |
"_gtEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_>_./ _)" [0, 0, 10] 10)
|
|
558 |
"_geAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
|
|
559 |
"_geEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
|
|
560 |
|
|
561 |
translations
|
|
562 |
"ALL x<y. P" => "ALL x. x < y --> P"
|
|
563 |
"EX x<y. P" => "EX x. x < y & P"
|
|
564 |
"ALL x<=y. P" => "ALL x. x <= y --> P"
|
|
565 |
"EX x<=y. P" => "EX x. x <= y & P"
|
|
566 |
"ALL x>y. P" => "ALL x. x > y --> P"
|
|
567 |
"EX x>y. P" => "EX x. x > y & P"
|
|
568 |
"ALL x>=y. P" => "ALL x. x >= y --> P"
|
|
569 |
"EX x>=y. P" => "EX x. x >= y & P"
|
|
570 |
|
|
571 |
print_translation {*
|
|
572 |
let
|
|
573 |
fun mk v v' q n P =
|
|
574 |
if v=v' andalso not(v mem (map fst (Term.add_frees([],n))))
|
|
575 |
then Syntax.const q $ Syntax.mark_bound v' $ n $ P else raise Match;
|
|
576 |
fun all_tr' [Const ("_bound",_) $ Free (v,_),
|
|
577 |
Const("op -->",_) $ (Const ("op <",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] =
|
|
578 |
mk v v' "_lessAll" n P
|
|
579 |
|
|
580 |
| all_tr' [Const ("_bound",_) $ Free (v,_),
|
|
581 |
Const("op -->",_) $ (Const ("op <=",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] =
|
|
582 |
mk v v' "_leAll" n P
|
|
583 |
|
|
584 |
| all_tr' [Const ("_bound",_) $ Free (v,_),
|
|
585 |
Const("op -->",_) $ (Const ("op <",_) $ n $ (Const ("_bound",_) $ Free (v',_))) $ P] =
|
|
586 |
mk v v' "_gtAll" n P
|
|
587 |
|
|
588 |
| all_tr' [Const ("_bound",_) $ Free (v,_),
|
|
589 |
Const("op -->",_) $ (Const ("op <=",_) $ n $ (Const ("_bound",_) $ Free (v',_))) $ P] =
|
|
590 |
mk v v' "_geAll" n P;
|
|
591 |
|
|
592 |
fun ex_tr' [Const ("_bound",_) $ Free (v,_),
|
|
593 |
Const("op &",_) $ (Const ("op <",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] =
|
|
594 |
mk v v' "_lessEx" n P
|
|
595 |
|
|
596 |
| ex_tr' [Const ("_bound",_) $ Free (v,_),
|
|
597 |
Const("op &",_) $ (Const ("op <=",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] =
|
|
598 |
mk v v' "_leEx" n P
|
|
599 |
|
|
600 |
| ex_tr' [Const ("_bound",_) $ Free (v,_),
|
|
601 |
Const("op &",_) $ (Const ("op <",_) $ n $ (Const ("_bound",_) $ Free (v',_))) $ P] =
|
|
602 |
mk v v' "_gtEx" n P
|
|
603 |
|
|
604 |
| ex_tr' [Const ("_bound",_) $ Free (v,_),
|
|
605 |
Const("op &",_) $ (Const ("op <=",_) $ n $ (Const ("_bound",_) $ Free (v',_))) $ P] =
|
|
606 |
mk v v' "_geEx" n P
|
|
607 |
in
|
|
608 |
[("ALL ", all_tr'), ("EX ", ex_tr')]
|
|
609 |
end
|
|
610 |
*}
|
|
611 |
|
|
612 |
end
|