src/HOL/HOL.thy
 author wenzelm Fri, 10 Nov 2000 19:02:37 +0100 changeset 10432 3dfbc913d184 parent 10383 a092ae7bb2a6 child 10489 a4684cf28edf permissions -rw-r--r--
added axclass inverse and consts inverse, divide (infix "/"); moved axclass power to Nat.thy;
```
(*  Title:      HOL/HOL.thy
ID:         \$Id\$
Author:     Tobias Nipkow
Copyright   1993  University of Cambridge

Higher-Order Logic.
*)

theory HOL = CPure
files ("HOL_lemmas.ML") ("cladata.ML") ("blastdata.ML") ("simpdata.ML")
("meson_lemmas.ML") ("Tools/meson.ML"):

(** Core syntax **)

global

classes "term" < logic
defaultsort "term"

typedecl bool

arities
bool :: "term"
fun :: ("term", "term") "term"

consts

(* Constants *)

Trueprop      :: "bool => prop"                   ("(_)" 5)
Not           :: "bool => bool"                   ("~ _"  40)
True          :: bool
False         :: bool
If            :: "[bool, 'a, 'a] => 'a"           ("(if (_)/ then (_)/ else (_))" 10)
arbitrary     :: 'a

(* Binders *)

Eps           :: "('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 plus  < "term"
axclass minus < "term"
axclass times < "term"
axclass inverse < "term"

global

consts
"0"           :: "'a::zero"                       ("0")
"+"           :: "['a::plus, 'a]  => 'a"          (infixl 65)
-             :: "['a::minus, 'a] => 'a"          (infixl 65)
uminus        :: "['a::minus] => 'a"              ("- _"  80)
*             :: "['a::times, 'a] => 'a"          (infixl 70)

local

consts
abs           :: "'a::minus => 'a"
inverse       :: "'a::inverse => 'a"
divide        :: "['a::inverse, 'a] => 'a"        (infixl "'/" 70)

axclass plus_ac0 < plus, zero
commute: "x + y = y + x"
assoc:   "(x + y) + z = x + (y + z)"
zero:    "0 + x = x"

(** Additional concrete syntax **)

nonterminals
letbinds  letbind
case_syn  cases_syn

syntax
~=            :: "['a, 'a] => bool"                    (infixl 50)
"_Eps"        :: "[pttrn, bool] => 'a"                 ("(3SOME _./ _)" [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"  ("_/ | _")

translations
"x ~= y"                == "~ (x = y)"
"SOME x. P"             == "Eps (%x. P)"
"_Let (_binds b bs) e"  == "_Let b (_Let bs e)"
"let x = a in e"        == "Let a (%x. e)"

syntax ("" output)
"op ="        :: "['a, 'a] => bool"                    ("(_ =/ _)" [51, 51] 50)
"op ~="       :: "['a, 'a] => bool"                    ("(_ ~=/ _)" [51, 51] 50)

syntax (symbols)
Not           :: "bool => bool"                        ("\<not> _"  40)
"op &"        :: "[bool, bool] => bool"                (infixr "\<and>" 35)
"op |"        :: "[bool, bool] => bool"                (infixr "\<or>" 30)
"op -->"      :: "[bool, bool] => bool"                (infixr "\<midarrow>\<rightarrow>" 25)
"op ~="       :: "['a, 'a] => bool"                    (infixl "\<noteq>" 50)
"ALL "        :: "[idts, bool] => bool"                ("(3\<forall>_./ _)" [0, 10] 10)
"EX "         :: "[idts, bool] => bool"                ("(3\<exists>_./ _)" [0, 10] 10)
"EX! "        :: "[idts, bool] => bool"                ("(3\<exists>!_./ _)" [0, 10] 10)
"_case1"      :: "['a, 'b] => case_syn"                ("(2_ \<Rightarrow>/ _)" 10)
(*"_case2"      :: "[case_syn, cases_syn] => cases_syn"  ("_/ \\<orelse> _")*)

syntax (input)
"_Eps"        :: "[pttrn, bool] => 'a"                 ("(3\<epsilon>_./ _)" [0, 10] 10)

syntax (symbols output)
"op ~="       :: "['a, 'a] => bool"                    ("(_ \<noteq>/ _)" [51, 51] 50)

syntax (xsymbols)
"op -->"      :: "[bool, bool] => bool"                (infixr "\<longrightarrow>" 25)

syntax (HTML output)
Not           :: "bool => bool"                        ("\<not> _"  40)

syntax (HOL)
"_Eps"        :: "[pttrn, bool] => 'a"                 ("(3@ _./ _)" [0, 10] 10)
"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 **)

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)"

someI:        "P (x::'a) ==> P (SOME x. P x)"

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)     == P (SOME x. P x)"
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 == SOME 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 == (SOME x. False)"

(* theory and package setup *)

use "HOL_lemmas.ML"

lemma atomize_all: "(!!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)"
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)"
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

use "cladata.ML"
setup hypsubst_setup
setup Classical.setup
setup clasetup

use "blastdata.ML"
setup Blast.setup

use "simpdata.ML"
setup Simplifier.setup
setup "Simplifier.method_setup Splitter.split_modifiers" setup simpsetup
setup Splitter.setup setup Clasimp.setup

use "meson_lemmas.ML"
use "Tools/meson.ML"
setup meson_setup

end
```