--- a/src/HOL/HOL.thy Sun Oct 14 20:00:32 2001 +0200
+++ b/src/HOL/HOL.thy Sun Oct 14 20:01:42 2001 +0200
@@ -1,16 +1,17 @@
(* Title: HOL/HOL.thy
ID: $Id$
- Author: Tobias Nipkow
- Copyright 1993 University of Cambridge
+ Author: Tobias Nipkow, Markus Wenzel, and Larry Paulson
+*)
-Higher-Order Logic.
-*)
+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
@@ -23,27 +24,22 @@
bool :: "term"
fun :: ("term", "term") "term"
-consts
+judgment
+ Trueprop :: "bool => prop" ("(_)" 5)
- (* Constants *)
-
- Trueprop :: "bool => prop" ("(_)" 5)
+consts
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)
@@ -52,52 +48,7 @@
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
@@ -107,15 +58,11 @@
~= :: "['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" ("_")
@@ -158,22 +105,18 @@
"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)"
@@ -181,7 +124,6 @@
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"
@@ -192,30 +134,78 @@
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)"
+ -- {* @{term arbitrary} is completely unspecified, but is made to appear as a
+ definition syntactically *}
+subsubsection {* Generic algebraic operations *}
-(* theory and package setup *)
+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*)
+declare trans [trans]
+declare impE [CPure.elim] iffD1 [CPure.elim] iffD2 [CPure.elim]
+
-lemma atomize_all: "(!!x. P x) == Trueprop (ALL x. P x)"
+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)
@@ -224,7 +214,7 @@
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)
@@ -233,7 +223,7 @@
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)
@@ -242,22 +232,344 @@
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