src/HOL/HOL.thy

 (*  Title:      HOL/HOL.thy
     ID:         $Id$
-    Author:     Tobias Nipkow
-    Copyright   1993  University of Cambridge
-
-Higher-Order Logic.
+    Author:     Tobias Nipkow, Markus Wenzel, and Larry Paulson
 *)
+
+header {* The basis of Higher-Order Logic *}
 
 theory HOL = CPure
 files ("HOL_lemmas.ML") ("cladata.ML") ("blastdata.ML") ("simpdata.ML"):
 
 
-(** Core syntax **)
+subsection {* Primitive logic *}
+
+subsubsection {* Core syntax *}
 
 global
 
 classes "term" < logic
 defaultsort "term"

 
 arities
   bool :: "term"
   fun :: ("term", "term") "term"
 
+judgment
+  Trueprop      :: "bool => prop"                   ("(_)" 5)
+
 consts
-
-  (* Constants *)
-
-  Trueprop      :: "bool => prop"                   ("(_)" 5)
   Not           :: "bool => bool"                   ("~ _" [40] 40)
   True          :: bool
   False         :: bool
   If            :: "[bool, 'a, 'a] => 'a"           ("(if (_)/ then (_)/ else (_))" 10)
   arbitrary     :: 'a
 
-  (* Binders *)
-
   The           :: "('a => bool) => 'a"
   All           :: "('a => bool) => bool"           (binder "ALL " 10)
   Ex            :: "('a => bool) => bool"           (binder "EX " 10)
   Ex1           :: "('a => bool) => bool"           (binder "EX! " 10)
   Let           :: "['a, 'a => 'b] => 'b"
 
-  (* Infixes *)
-
   "="           :: "['a, 'a] => bool"               (infixl 50)
   &             :: "[bool, bool] => bool"           (infixr 35)
   "|"           :: "[bool, bool] => bool"           (infixr 30)
   -->           :: "[bool, bool] => bool"           (infixr 25)
 
 local
 
 
-(* Overloaded Constants *)
-
-axclass zero < "term"
-axclass one < "term"
-axclass plus < "term"
-axclass minus < "term"
-axclass times < "term"
-axclass inverse < "term"
-
-global
-
-consts
-  "0"           :: "'a::zero"                       ("0")
-  "1"           :: "'a::one"                        ("1")
-  "+"           :: "['a::plus, 'a]  => 'a"          (infixl 65)
-  -             :: "['a::minus, 'a] => 'a"          (infixl 65)
-  uminus        :: "['a::minus] => 'a"              ("- _" [81] 80)
-  *             :: "['a::times, 'a] => 'a"          (infixl 70)
-
-typed_print_translation {*
-  let
-    fun tr' c = (c, fn show_sorts => fn T => fn ts =>
-      if T = dummyT orelse not (! show_types) andalso can Term.dest_Type T then raise Match
-      else Syntax.const Syntax.constrainC $ Syntax.const c $ Syntax.term_of_typ show_sorts T);
-  in [tr' "0", tr' "1"] end;
-*}
-
-local
-
-consts
-  abs           :: "'a::minus => 'a"
-  inverse       :: "'a::inverse => 'a"
-  divide        :: "['a::inverse, 'a] => 'a"        (infixl "'/" 70)
-
-syntax (xsymbols)
-  abs :: "'a::minus => 'a"    ("\<bar>_\<bar>")
-syntax (HTML output)
-  abs :: "'a::minus => 'a"    ("\<bar>_\<bar>")
-
-axclass plus_ac0 < plus, zero
-  commute: "x + y = y + x"
-  assoc:   "(x + y) + z = x + (y + z)"
-  zero:    "0 + x = x"
-
-
-(** Additional concrete syntax **)
+subsubsection {* Additional concrete syntax *}
 
 nonterminals
   letbinds  letbind
   case_syn  cases_syn
 
 syntax
   ~=            :: "['a, 'a] => bool"                    (infixl 50)
   "_The"        :: "[pttrn, bool] => 'a"                 ("(3THE _./ _)" [0, 10] 10)
-
-  (* Let expressions *)
 
   "_bind"       :: "[pttrn, 'a] => letbind"              ("(2_ =/ _)" 10)
   ""            :: "letbind => letbinds"                 ("_")
   "_binds"      :: "[letbind, letbinds] => letbinds"     ("_;/ _")
   "_Let"        :: "[letbinds, 'a] => 'a"                ("(let (_)/ in (_))" 10)
-
-  (* Case expressions *)
 
   "_case_syntax":: "['a, cases_syn] => 'b"               ("(case _ of/ _)" 10)
   "_case1"      :: "['a, 'b] => case_syn"                ("(2_ =>/ _)" 10)
   ""            :: "case_syn => cases_syn"               ("_")
   "_case2"      :: "[case_syn, cases_syn] => cases_syn"  ("_/ | _")

   "ALL "        :: "[idts, bool] => bool"                ("(3! _./ _)" [0, 10] 10)
   "EX "         :: "[idts, bool] => bool"                ("(3? _./ _)" [0, 10] 10)
   "EX! "        :: "[idts, bool] => bool"                ("(3?! _./ _)" [0, 10] 10)
 
 
-
-(** Rules and definitions **)
+subsubsection {* Axioms and basic definitions *}
 
 axioms
-
   eq_reflection: "(x=y) ==> (x==y)"
-
-  (* Basic Rules *)
 
   refl:         "t = (t::'a)"
   subst:        "[| s = t; P(s) |] ==> P(t::'a)"
 
-  (*Extensionality is built into the meta-logic, and this rule expresses
-    a related property.  It is an eta-expanded version of the traditional
-    rule, and similar to the ABS rule of HOL.*)
   ext:          "(!!x::'a. (f x ::'b) = g x) ==> (%x. f x) = (%x. g x)"
+    -- {* Extensionality is built into the meta-logic, and this rule expresses *}
+    -- {* a related property.  It is an eta-expanded version of the traditional *}
+    -- {* rule, and similar to the ABS rule of HOL *}
 
   the_eq_trivial: "(THE x. x = a) = (a::'a)"
 
   impI:         "(P ==> Q) ==> P-->Q"
   mp:           "[| P-->Q;  P |] ==> Q"
 
 defs
-
   True_def:     "True      == ((%x::bool. x) = (%x. x))"
   All_def:      "All(P)    == (P = (%x. True))"
   Ex_def:       "Ex(P)     == !Q. (!x. P x --> Q) --> Q"
   False_def:    "False     == (!P. P)"
   not_def:      "~ P       == P-->False"
   and_def:      "P & Q     == !R. (P-->Q-->R) --> R"
   or_def:       "P | Q     == !R. (P-->R) --> (Q-->R) --> R"
   Ex1_def:      "Ex1(P)    == ? x. P(x) & (! y. P(y) --> y=x)"
 
 axioms
-  (* Axioms *)
-
   iff:          "(P-->Q) --> (Q-->P) --> (P=Q)"
   True_or_False:  "(P=True) | (P=False)"
 
 defs
-  (*misc definitions*)
   Let_def:      "Let s f == f(s)"
   if_def:       "If P x y == THE z::'a. (P=True --> z=x) & (P=False --> z=y)"
 
-  (*arbitrary is completely unspecified, but is made to appear as a
-    definition syntactically*)
   arbitrary_def:  "False ==> arbitrary == (THE x. False)"
-
-
-
-(* theory and package setup *)
+    -- {* @{term arbitrary} is completely unspecified, but is made to appear as a
+    definition syntactically *}
+
+
+subsubsection {* Generic algebraic operations *}
+
+axclass zero < "term"
+axclass one < "term"
+axclass plus < "term"
+axclass minus < "term"
+axclass times < "term"
+axclass inverse < "term"
+
+global
+
+consts
+  "0"           :: "'a::zero"                       ("0")
+  "1"           :: "'a::one"                        ("1")
+  "+"           :: "['a::plus, 'a]  => 'a"          (infixl 65)
+  -             :: "['a::minus, 'a] => 'a"          (infixl 65)
+  uminus        :: "['a::minus] => 'a"              ("- _" [81] 80)
+  *             :: "['a::times, 'a] => 'a"          (infixl 70)
+
+local
+
+typed_print_translation {*
+  let
+    fun tr' c = (c, fn show_sorts => fn T => fn ts =>
+      if T = dummyT orelse not (! show_types) andalso can Term.dest_Type T then raise Match
+      else Syntax.const Syntax.constrainC $ Syntax.const c $ Syntax.term_of_typ show_sorts T);
+  in [tr' "0", tr' "1"] end;
+*} -- {* show types that are presumably too general *}
+
+
+consts
+  abs           :: "'a::minus => 'a"
+  inverse       :: "'a::inverse => 'a"
+  divide        :: "['a::inverse, 'a] => 'a"        (infixl "'/" 70)
+
+syntax (xsymbols)
+  abs :: "'a::minus => 'a"    ("\<bar>_\<bar>")
+syntax (HTML output)
+  abs :: "'a::minus => 'a"    ("\<bar>_\<bar>")
+
+axclass plus_ac0 < plus, zero
+  commute: "x + y = y + x"
+  assoc:   "(x + y) + z = x + (y + z)"
+  zero:    "0 + x = x"
+
+
+subsection {* Theory and package setup *}
+
+subsubsection {* Basic lemmas *}
 
 use "HOL_lemmas.ML"
 theorems case_split = case_split_thm [case_names True False]
 
-declare trans [trans]  (*overridden in theory Calculation*)
-
-lemma atomize_all: "(!!x. P x) == Trueprop (ALL x. P x)"
+declare trans [trans]
+declare impE [CPure.elim]  iffD1 [CPure.elim]  iffD2 [CPure.elim]
+
+
+subsubsection {* Atomizing meta-level connectives *}
+
+lemma atomize_all [atomize]: "(!!x. P x) == Trueprop (ALL x. P x)"
 proof (rule equal_intr_rule)
   assume "!!x. P x"
   show "ALL x. P x" by (rule allI)
 next
   assume "ALL x. P x"
   thus "!!x. P x" by (rule allE)
 qed
 
-lemma atomize_imp: "(A ==> B) == Trueprop (A --> B)"
+lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
 proof (rule equal_intr_rule)
   assume r: "A ==> B"
   show "A --> B" by (rule impI) (rule r)
 next
   assume "A --> B" and A
   thus B by (rule mp)
 qed
 
-lemma atomize_eq: "(x == y) == Trueprop (x = y)"
+lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)"
 proof (rule equal_intr_rule)
   assume "x == y"
   show "x = y" by (unfold prems) (rule refl)
 next
   assume "x = y"
   thus "x == y" by (rule eq_reflection)
 qed
 
-lemmas atomize = atomize_all atomize_imp
-lemmas atomize' = atomize atomize_eq
+
+subsubsection {* Classical Reasoner setup *}
 
 use "cladata.ML"
 setup hypsubst_setup
 setup Classical.setup
 setup clasetup
 
-declare impE [CPure.elim]  iffD1 [CPure.elim]  iffD2 [CPure.elim]
-
 use "blastdata.ML"
 setup Blast.setup
+
+
+subsubsection {* Simplifier setup *}
 
 use "simpdata.ML"
 setup Simplifier.setup
 setup "Simplifier.method_setup Splitter.split_modifiers" setup simpsetup
 setup Splitter.setup setup Clasimp.setup
 
+
+subsection {* Order signatures and orders *}
+
+axclass
+  ord < "term"
+
+syntax
+  "op <"        :: "['a::ord, 'a] => bool"             ("op <")
+  "op <="       :: "['a::ord, 'a] => bool"             ("op <=")
+
+global
+
+consts
+  "op <"        :: "['a::ord, 'a] => bool"             ("(_/ < _)"  [50, 51] 50)
+  "op <="       :: "['a::ord, 'a] => bool"             ("(_/ <= _)" [50, 51] 50)
+
+local
+
+syntax (symbols)
+  "op <="       :: "['a::ord, 'a] => bool"             ("op \<le>")
+  "op <="       :: "['a::ord, 'a] => bool"             ("(_/ \<le> _)"  [50, 51] 50)
+
+(*Tell blast about overloading of < and <= to reduce the risk of
+  its applying a rule for the wrong type*)
+ML {*
+Blast.overloaded ("op <" , domain_type);
+Blast.overloaded ("op <=", domain_type);
+*}
+
+
+subsubsection {* Monotonicity *}
+
+constdefs
+  mono :: "['a::ord => 'b::ord] => bool"
+  "mono f == ALL A B. A <= B --> f A <= f B"
+
+lemma monoI [intro?]: "(!!A B. A <= B ==> f A <= f B) ==> mono f"
+  by (unfold mono_def) blast
+
+lemma monoD [dest?]: "mono f ==> A <= B ==> f A <= f B"
+  by (unfold mono_def) blast
+
+constdefs
+  min :: "['a::ord, 'a] => 'a"
+  "min a b == (if a <= b then a else b)"
+  max :: "['a::ord, 'a] => 'a"
+  "max a b == (if a <= b then b else a)"
+
+lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
+  by (simp add: min_def)
+
+lemma min_of_mono:
+    "ALL x y. (f x <= f y) = (x <= y) ==> min (f m) (f n) = f (min m n)"
+  by (simp add: min_def)
+
+lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
+  by (simp add: max_def)
+
+lemma max_of_mono:
+    "ALL x y. (f x <= f y) = (x <= y) ==> max (f m) (f n) = f (max m n)"
+  by (simp add: max_def)
+
+
+subsubsection "Orders"
+
+axclass order < ord
+  order_refl [iff]: "x <= x"
+  order_trans: "x <= y ==> y <= z ==> x <= z"
+  order_antisym: "x <= y ==> y <= x ==> x = y"
+  order_less_le: "(x < y) = (x <= y & x ~= y)"
+
+
+text {* Reflexivity. *}
+
+lemma order_eq_refl: "!!x::'a::order. x = y ==> x <= y"
+    -- {* This form is useful with the classical reasoner. *}
+  apply (erule ssubst)
+  apply (rule order_refl)
+  done
+
+lemma order_less_irrefl [simp]: "~ x < (x::'a::order)"
+  by (simp add: order_less_le)
+
+lemma order_le_less: "((x::'a::order) <= y) = (x < y | x = y)"
+    -- {* NOT suitable for iff, since it can cause PROOF FAILED. *}
+  apply (simp add: order_less_le)
+  apply (blast intro!: order_refl)
+  done
+
+lemmas order_le_imp_less_or_eq = order_le_less [THEN iffD1, standard]
+
+lemma order_less_imp_le: "!!x::'a::order. x < y ==> x <= y"
+  by (simp add: order_less_le)
+
+
+text {* Asymmetry. *}
+
+lemma order_less_not_sym: "(x::'a::order) < y ==> ~ (y < x)"
+  by (simp add: order_less_le order_antisym)
+
+lemma order_less_asym: "x < (y::'a::order) ==> (~P ==> y < x) ==> P"
+  apply (drule order_less_not_sym)
+  apply (erule contrapos_np)
+  apply simp
+  done
+
+
+text {* Transitivity. *}
+
+lemma order_less_trans: "!!x::'a::order. [| x < y; y < z |] ==> x < z"
+  apply (simp add: order_less_le)
+  apply (blast intro: order_trans order_antisym)
+  done
+
+lemma order_le_less_trans: "!!x::'a::order. [| x <= y; y < z |] ==> x < z"
+  apply (simp add: order_less_le)
+  apply (blast intro: order_trans order_antisym)
+  done
+
+lemma order_less_le_trans: "!!x::'a::order. [| x < y; y <= z |] ==> x < z"
+  apply (simp add: order_less_le)
+  apply (blast intro: order_trans order_antisym)
+  done
+
+
+text {* Useful for simplification, but too risky to include by default. *}
+
+lemma order_less_imp_not_less: "(x::'a::order) < y ==>  (~ y < x) = True"
+  by (blast elim: order_less_asym)
+
+lemma order_less_imp_triv: "(x::'a::order) < y ==>  (y < x --> P) = True"
+  by (blast elim: order_less_asym)
+
+lemma order_less_imp_not_eq: "(x::'a::order) < y ==>  (x = y) = False"
+  by auto
+
+lemma order_less_imp_not_eq2: "(x::'a::order) < y ==>  (y = x) = False"
+  by auto
+
+
+text {* Other operators. *}
+
+lemma min_leastR: "(!!x::'a::order. least <= x) ==> min x least = least"
+  apply (simp add: min_def)
+  apply (blast intro: order_antisym)
+  done
+
+lemma max_leastR: "(!!x::'a::order. least <= x) ==> max x least = x"
+  apply (simp add: max_def)
+  apply (blast intro: order_antisym)
+  done
+
+
+subsubsection {* Least value operator *}
+
+constdefs
+  Least :: "('a::ord => bool) => 'a"               (binder "LEAST " 10)
+  "Least P == THE x. P x & (ALL y. P y --> x <= y)"
+    -- {* We can no longer use LeastM because the latter requires Hilbert-AC. *}
+
+lemma LeastI2:
+  "[| P (x::'a::order);
+      !!y. P y ==> x <= y;
+      !!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |]
+   ==> Q (Least P)";
+  apply (unfold Least_def)
+  apply (rule theI2)
+    apply (blast intro: order_antisym)+
+  done
+
+lemma Least_equality:
+    "[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k";
+  apply (simp add: Least_def)
+  apply (rule the_equality)
+  apply (auto intro!: order_antisym)
+  done
+
+
+subsubsection "Linear / total orders"
+
+axclass linorder < order
+  linorder_linear: "x <= y | y <= x"
+
+lemma linorder_less_linear: "!!x::'a::linorder. x<y | x=y | y<x"
+  apply (simp add: order_less_le)
+  apply (insert linorder_linear)
+  apply blast
+  done
+
+lemma linorder_cases [case_names less equal greater]:
+    "((x::'a::linorder) < y ==> P) ==> (x = y ==> P) ==> (y < x ==> P) ==> P"
+  apply (insert linorder_less_linear)
+  apply blast
+  done
+
+lemma linorder_not_less: "!!x::'a::linorder. (~ x < y) = (y <= x)"
+  apply (simp add: order_less_le)
+  apply (insert linorder_linear)
+  apply (blast intro: order_antisym)
+  done
+
+lemma linorder_not_le: "!!x::'a::linorder. (~ x <= y) = (y < x)"
+  apply (simp add: order_less_le)
+  apply (insert linorder_linear)
+  apply (blast intro: order_antisym)
+  done
+
+lemma linorder_neq_iff: "!!x::'a::linorder. (x ~= y) = (x<y | y<x)"
+  apply (cut_tac x = x and y = y in linorder_less_linear)
+  apply auto
+  done
+
+lemma linorder_neqE: "x ~= (y::'a::linorder) ==> (x < y ==> R) ==> (y < x ==> R) ==> R"
+  apply (simp add: linorder_neq_iff)
+  apply blast
+  done
+
+
+subsubsection "Min and max on (linear) orders"
+
+lemma min_same [simp]: "min (x::'a::order) x = x"
+  by (simp add: min_def)
+
+lemma max_same [simp]: "max (x::'a::order) x = x"
+  by (simp add: max_def)
+
+lemma le_max_iff_disj: "!!z::'a::linorder. (z <= max x y) = (z <= x | z <= y)"
+  apply (simp add: max_def)
+  apply (insert linorder_linear)
+  apply (blast intro: order_trans)
+  done
+
+lemma le_maxI1: "(x::'a::linorder) <= max x y"
+  by (simp add: le_max_iff_disj)
+
+lemma le_maxI2: "(y::'a::linorder) <= max x y"
+    -- {* CANNOT use with @{text "[intro!]"} because blast will give PROOF FAILED. *}
+  by (simp add: le_max_iff_disj)
+
+lemma less_max_iff_disj: "!!z::'a::linorder. (z < max x y) = (z < x | z < y)"
+  apply (simp add: max_def order_le_less)
+  apply (insert linorder_less_linear)
+  apply (blast intro: order_less_trans)
+  done
+
+lemma max_le_iff_conj [simp]:
+    "!!z::'a::linorder. (max x y <= z) = (x <= z & y <= z)"
+  apply (simp add: max_def)
+  apply (insert linorder_linear)
+  apply (blast intro: order_trans)
+  done
+
+lemma max_less_iff_conj [simp]:
+    "!!z::'a::linorder. (max x y < z) = (x < z & y < z)"
+  apply (simp add: order_le_less max_def)
+  apply (insert linorder_less_linear)
+  apply (blast intro: order_less_trans)
+  done
+
+lemma le_min_iff_conj [simp]:
+    "!!z::'a::linorder. (z <= min x y) = (z <= x & z <= y)"
+    -- {* @{text "[iff]"} screws up a Q{text blast} in MiniML *}
+  apply (simp add: min_def)
+  apply (insert linorder_linear)
+  apply (blast intro: order_trans)
+  done
+
+lemma min_less_iff_conj [simp]:
+    "!!z::'a::linorder. (z < min x y) = (z < x & z < y)"
+  apply (simp add: order_le_less min_def)
+  apply (insert linorder_less_linear)
+  apply (blast intro: order_less_trans)
+  done
+
+lemma min_le_iff_disj: "!!z::'a::linorder. (min x y <= z) = (x <= z | y <= z)"
+  apply (simp add: min_def)
+  apply (insert linorder_linear)
+  apply (blast intro: order_trans)
+  done
+
+lemma min_less_iff_disj: "!!z::'a::linorder. (min x y < z) = (x < z | y < z)"
+  apply (simp add: min_def order_le_less)
+  apply (insert linorder_less_linear)
+  apply (blast intro: order_less_trans)
+  done
+
+lemma split_min:
+    "P (min (i::'a::linorder) j) = ((i <= j --> P(i)) & (~ i <= j --> P(j)))"
+  by (simp add: min_def)
+
+lemma split_max:
+    "P (max (i::'a::linorder) j) = ((i <= j --> P(j)) & (~ i <= j --> P(i)))"
+  by (simp add: max_def)
+
+
+subsubsection "Bounded quantifiers"
+
+syntax
+  "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3ALL _<_./ _)"  [0, 0, 10] 10)
+  "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3EX _<_./ _)"  [0, 0, 10] 10)
+  "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3ALL _<=_./ _)" [0, 0, 10] 10)
+  "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3EX _<=_./ _)" [0, 0, 10] 10)
+
+syntax (symbols)
+  "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
+  "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
+  "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
+  "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
+
+syntax (HOL)
+  "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3! _<_./ _)"  [0, 0, 10] 10)
+  "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3? _<_./ _)"  [0, 0, 10] 10)
+  "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3! _<=_./ _)" [0, 0, 10] 10)
+  "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3? _<=_./ _)" [0, 0, 10] 10)
+
+translations
+ "ALL x<y. P"   =>  "ALL x. x < y --> P"
+ "EX x<y. P"    =>  "EX x. x < y  & P"
+ "ALL x<=y. P"  =>  "ALL x. x <= y --> P"
+ "EX x<=y. P"   =>  "EX x. x <= y & P"
+
 end