author  nipkow 
Wed, 30 Aug 2000 10:21:19 +0200  
changeset 9736  332fab43628f 
parent 9713  2c5b42311eb0 
child 9839  da5ca8b30244 
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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theory HOL = CPure 
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files ("HOL_lemmas.ML") ("cladata.ML") ("blastdata.ML") ("simpdata.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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(* 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 power < "term" 

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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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abs :: "('a::minus) => 'a" 
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* :: "['a::times, 'a] => 'a" (infixl 70) 
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(*See Nat.thy for "^"*) 
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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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"_Eps" :: "[pttrn, bool] => 'a" ("(3\\<epsilon>_./ _)" [0, 10] 10) 

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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 (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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syntax (HTML output) 
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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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local 
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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 metalogic, and this rule expresses 

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a related property. It is an etaexpanded 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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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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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 == @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 == (@x. False)" 
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(* theory and package setup *) 
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use "HOL_lemmas.ML" 
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use "cladata.ML" setup hypsubst_setup setup Classical.setup setup clasetup 
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lemma all_eq: "(!!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" .. 

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next 

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assume "ALL x. P x" 

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thus "!!x. P x" .. 

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qed 

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lemma imp_eq: "(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" 

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by (rule) (rule r) 

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next 

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assume "A > B" and A 

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thus B .. 

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qed 

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lemmas atomize = all_eq imp_eq 
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use "blastdata.ML" setup Blast.setup 
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use "simpdata.ML" 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 setup iff_attrib_setup 
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setup attrib_setup 
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end 