author  wenzelm 
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permissions  rwrr 
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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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HigherOrder Logic. 
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*) 
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HOL = CPure + 

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(** Core syntax **) 

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classes 
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term < logic 

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default 

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term 

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types 

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bool 

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arities 

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fun :: (term, term) term 

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bool :: 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, False :: bool 
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If :: [bool, 'a, 'a] => 'a ("(if (_)/ then (_)/ else (_))" 10) 
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(* Binders *) 

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Eps :: ('a => bool) => 'a 
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All :: ('a => bool) => bool (binder "! " 10) 
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Ex :: ('a => bool) => bool (binder "? " 10) 
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Ex1 :: ('a => bool) => bool (binder "?! " 10) 
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Let :: ['a, 'a => 'b] => 'b 
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(* Infixes *) 

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o :: ['b => 'c, 'a => 'b, 'a] => 'c (infixl 55) 
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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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(* Overloaded Constants *) 

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axclass 

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plus < term 

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axclass 
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minus < term 

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axclass 

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times < term 

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axclass 
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power < term 
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consts 
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"+" :: ['a::plus, 'a] => 'a (infixl 65) 
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"" :: ['a::minus, 'a] => 'a (infixl 65) 
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"*" :: ['a::times, 'a] => 'a (infixl 70) 
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(*See Nat.thy for "^"*) 
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(** Additional concrete syntax **) 
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types 
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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 ("(3@ _./ _)" [0, 10] 10) 
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(* Alternative Quantifiers *) 
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"*All" :: [idts, bool] => bool ("(3ALL _./ _)" [0, 10] 10) 
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"*Ex" :: [idts, bool] => bool ("(3EX _./ _)" [0, 10] 10) 

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"*Ex1" :: [idts, bool] => bool ("(3EX! _./ _)" [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" :: ['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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"@ x.b" == "Eps (%x. b)" 
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"ALL xs. P" => "! xs. P" 
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"EX xs. P" => "? xs. P" 

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"EX! xs. P" => "?! xs. 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 o" :: ['b => 'c, 'a => 'b, 'a] => 'c (infixl "\\<circ>" 55) 

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"op ~=" :: ['a, 'a] => bool (infixl "\\<noteq>" 50) 

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"@Eps" :: [pttrn, bool] => 'a ("(3\\<epsilon>_./ _)" [0, 10] 10) 
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"! " :: [idts, bool] => bool ("(3\\<forall>_./ _)" [0, 10] 10) 

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"? " :: [idts, bool] => bool ("(3\\<exists>_./ _)" [0, 10] 10) 

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"?! " :: [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 (symbols output) 
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"op ~=" :: ['a, 'a] => bool ("(_ \\<noteq>/ _)" [51, 51] 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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"*Ex1" :: [idts, bool] => bool ("(3\\<exists>!_./ _)" [0, 10] 10) 

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(** Rules and definitions **) 

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rules 
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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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ext "(!!x::'a. (f(x)::'b) = g(x)) ==> (%x.f(x)) = (%x.g(x))" 

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selectI "P(x::'a) ==> P(@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(@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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rules 

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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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o_def "(f::'b=>'c) o g == (%(x::'a). f(g(x)))" 

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if_def "If P x y == @z::'a. (P=True > z=x) & (P=False > z=y)" 
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consts 
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arbitrary :: 'a 

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end 
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ML 
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(** Choice between the HOL and Isabelle style of quantifiers **) 

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val HOL_quantifiers = ref true; 

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fun alt_ast_tr' (name, alt_name) = 

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let 

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fun ast_tr' (*name*) args = 

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if ! HOL_quantifiers then raise Match 

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else Syntax.mk_appl (Syntax.Constant alt_name) args; 

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in 

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(name, ast_tr') 

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end; 

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val print_ast_translation = 

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map alt_ast_tr' [("! ", "*All"), ("? ", "*Ex"), ("?! ", "*Ex1")]; 