src/HOL/HOL.thy
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(*  Title:      HOL/HOL.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow
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    Copyright   1993  University of Cambridge
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Higher-Order Logic.
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*)
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theory HOL = CPure
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files ("HOL_lemmas.ML") ("cladata.ML") ("blastdata.ML") ("simpdata.ML")
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  ("meson_lemmas.ML") ("Tools/meson.ML"):
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(** Core syntax **)
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global
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classes "term" < logic
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defaultsort "term"
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typedecl bool
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arities
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  bool :: "term"
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  fun :: ("term", "term") "term"
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consts
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  (* Constants *)
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  Trueprop      :: "bool => prop"                   ("(_)" 5)
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  Not           :: "bool => bool"                   ("~ _" [40] 40)
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  True          :: bool
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  False         :: bool
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  If            :: "[bool, 'a, 'a] => 'a"           ("(if (_)/ then (_)/ else (_))" 10)
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  arbitrary     :: 'a
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  (* Binders *)
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  Eps           :: "('a => bool) => 'a"
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  All           :: "('a => bool) => bool"           (binder "ALL " 10)
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  Ex            :: "('a => bool) => bool"           (binder "EX " 10)
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  Ex1           :: "('a => bool) => bool"           (binder "EX! " 10)
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  Let           :: "['a, 'a => 'b] => 'b"
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  (* Infixes *)
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  "="           :: "['a, 'a] => bool"               (infixl 50)
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  &             :: "[bool, bool] => bool"           (infixr 35)
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  "|"           :: "[bool, bool] => bool"           (infixr 30)
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  -->           :: "[bool, bool] => bool"           (infixr 25)
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local
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(* Overloaded Constants *)
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axclass zero  < "term"
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axclass plus  < "term"
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axclass minus < "term"
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axclass times < "term"
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axclass inverse < "term"
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global
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consts
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  "0"           :: "'a::zero"                       ("0")
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  "+"           :: "['a::plus, 'a]  => 'a"          (infixl 65)
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  -             :: "['a::minus, 'a] => 'a"          (infixl 65)
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  uminus        :: "['a::minus] => 'a"              ("- _" [81] 80)
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  *             :: "['a::times, 'a] => 'a"          (infixl 70)
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local
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consts
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  abs           :: "'a::minus => 'a"
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  inverse       :: "'a::inverse => 'a"
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  divide        :: "['a::inverse, 'a] => 'a"        (infixl "'/" 70)
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syntax (xsymbols)
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  abs :: "'a::minus => 'a"    ("\<bar>_\<bar>")
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syntax (HTML output)
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  abs :: "'a::minus => 'a"    ("\<bar>_\<bar>")
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axclass plus_ac0 < plus, zero
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  commute: "x + y = y + x"
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  assoc:   "(x + y) + z = x + (y + z)"
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  zero:    "0 + x = x"
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(** Additional concrete syntax **)
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nonterminals
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  letbinds  letbind
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  case_syn  cases_syn
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syntax
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  ~=            :: "['a, 'a] => bool"                    (infixl 50)
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  "_Eps"        :: "[pttrn, bool] => 'a"                 ("(3SOME _./ _)" [0, 10] 10)
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  (* Let expressions *)
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  "_bind"       :: "[pttrn, 'a] => letbind"              ("(2_ =/ _)" 10)
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  ""            :: "letbind => letbinds"                 ("_")
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  "_binds"      :: "[letbind, letbinds] => letbinds"     ("_;/ _")
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  "_Let"        :: "[letbinds, 'a] => 'a"                ("(let (_)/ in (_))" 10)
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  (* Case expressions *)
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  "_case_syntax":: "['a, cases_syn] => 'b"               ("(case _ of/ _)" 10)
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  "_case1"      :: "['a, 'b] => case_syn"                ("(2_ =>/ _)" 10)
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  ""            :: "case_syn => cases_syn"               ("_")
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  "_case2"      :: "[case_syn, cases_syn] => cases_syn"  ("_/ | _")
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translations
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  "x ~= y"                == "~ (x = y)"
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  "SOME x. P"             == "Eps (%x. P)"
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  "_Let (_binds b bs) e"  == "_Let b (_Let bs e)"
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  "let x = a in e"        == "Let a (%x. e)"
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syntax ("" output)
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  "op ="        :: "['a, 'a] => bool"                    ("(_ =/ _)" [51, 51] 50)
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  "op ~="       :: "['a, 'a] => bool"                    ("(_ ~=/ _)" [51, 51] 50)
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syntax (symbols)
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  Not           :: "bool => bool"                        ("\<not> _" [40] 40)
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  "op &"        :: "[bool, bool] => bool"                (infixr "\<and>" 35)
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  "op |"        :: "[bool, bool] => bool"                (infixr "\<or>" 30)
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  "op -->"      :: "[bool, bool] => bool"                (infixr "\<midarrow>\<rightarrow>" 25)
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  "op ~="       :: "['a, 'a] => bool"                    (infixl "\<noteq>" 50)
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  "ALL "        :: "[idts, bool] => bool"                ("(3\<forall>_./ _)" [0, 10] 10)
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  "EX "         :: "[idts, bool] => bool"                ("(3\<exists>_./ _)" [0, 10] 10)
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  "EX! "        :: "[idts, bool] => bool"                ("(3\<exists>!_./ _)" [0, 10] 10)
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  "_case1"      :: "['a, 'b] => case_syn"                ("(2_ \<Rightarrow>/ _)" 10)
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(*"_case2"      :: "[case_syn, cases_syn] => cases_syn"  ("_/ \\<orelse> _")*)
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syntax (input)
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  "_Eps"        :: "[pttrn, bool] => 'a"                 ("(3\<epsilon>_./ _)" [0, 10] 10)
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syntax (symbols output)
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  "op ~="       :: "['a, 'a] => bool"                    ("(_ \<noteq>/ _)" [51, 51] 50)
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syntax (xsymbols)
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  "op -->"      :: "[bool, bool] => bool"                (infixr "\<longrightarrow>" 25)
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  Not           :: "bool => bool"                        ("\<not> _" [40] 40)
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syntax (HOL)
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  "_Eps"        :: "[pttrn, bool] => 'a"                 ("(3@ _./ _)" [0, 10] 10)
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  "ALL "        :: "[idts, bool] => bool"                ("(3! _./ _)" [0, 10] 10)
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  "EX "         :: "[idts, bool] => bool"                ("(3? _./ _)" [0, 10] 10)
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  "EX! "        :: "[idts, bool] => bool"                ("(3?! _./ _)" [0, 10] 10)
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(** Rules and definitions **)
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axioms
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  eq_reflection: "(x=y) ==> (x==y)"
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  (* Basic Rules *)
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  refl:         "t = (t::'a)"
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  subst:        "[| s = t; P(s) |] ==> P(t::'a)"
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  (*Extensionality is built into the meta-logic, and this rule expresses
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    a related property.  It is an eta-expanded version of the traditional
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    rule, and similar to the ABS rule of HOL.*)
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  ext:          "(!!x::'a. (f x ::'b) = g x) ==> (%x. f x) = (%x. g x)"
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  someI:        "P (x::'a) ==> P (SOME x. P x)"
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  impI:         "(P ==> Q) ==> P-->Q"
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  mp:           "[| P-->Q;  P |] ==> Q"
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defs
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  True_def:     "True      == ((%x::bool. x) = (%x. x))"
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  All_def:      "All(P)    == (P = (%x. True))"
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  Ex_def:       "Ex(P)     == P (SOME x. P x)"
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  False_def:    "False     == (!P. P)"
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  not_def:      "~ P       == P-->False"
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  and_def:      "P & Q     == !R. (P-->Q-->R) --> R"
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  or_def:       "P | Q     == !R. (P-->R) --> (Q-->R) --> R"
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  Ex1_def:      "Ex1(P)    == ? x. P(x) & (! y. P(y) --> y=x)"
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axioms
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  (* Axioms *)
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  iff:          "(P-->Q) --> (Q-->P) --> (P=Q)"
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  True_or_False:  "(P=True) | (P=False)"
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defs
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  (*misc definitions*)
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  Let_def:      "Let s f == f(s)"
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  if_def:       "If P x y == SOME z::'a. (P=True --> z=x) & (P=False --> z=y)"
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  (*arbitrary is completely unspecified, but is made to appear as a
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    definition syntactically*)
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  arbitrary_def:  "False ==> arbitrary == (SOME x. False)"
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(* theory and package setup *)
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use "HOL_lemmas.ML"
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lemma atomize_all: "(!!x. P x) == Trueprop (ALL x. P x)"
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proof (rule equal_intr_rule)
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  assume "!!x. P x"
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  show "ALL x. P x" by (rule allI)
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next
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  assume "ALL x. P x"
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  thus "!!x. P x" by (rule allE)
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qed
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lemma atomize_imp: "(A ==> B) == Trueprop (A --> B)"
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proof (rule equal_intr_rule)
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  assume r: "A ==> B"
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  show "A --> B" by (rule impI) (rule r)
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next
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  assume "A --> B" and A
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  thus B by (rule mp)
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qed
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lemma atomize_eq: "(x == y) == Trueprop (x = y)"
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proof (rule equal_intr_rule)
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  assume "x == y"
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  show "x = y" by (unfold prems) (rule refl)
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next
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  assume "x = y"
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  thus "x == y" by (rule eq_reflection)
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qed
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lemmas atomize = atomize_all atomize_imp
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lemmas atomize' = atomize atomize_eq
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use "cladata.ML"
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setup hypsubst_setup
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setup Classical.setup
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setup clasetup
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use "blastdata.ML"
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setup Blast.setup
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use "simpdata.ML"
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setup Simplifier.setup
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setup "Simplifier.method_setup Splitter.split_modifiers" setup simpsetup
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setup Splitter.setup setup Clasimp.setup
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use "meson_lemmas.ML"
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use "Tools/meson.ML"
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setup meson_setup
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end