src/Sequents/LK0.thy
changeset 7093 b2ee0e5d1a7f
child 7118 ee384c7b7416
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/Sequents/LK0.thy	Tue Jul 27 18:52:23 1999 +0200
@@ -0,0 +1,142 @@
+(*  Title:      LK/LK0
+    ID:         $Id$
+    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
+    Copyright   1993  University of Cambridge
+
+Classical First-Order Sequent Calculus
+
+There may be printing problems if a seqent is in expanded normal form
+	(eta-expanded, beta-contracted)
+*)
+
+LK0 = Sequents +
+
+global
+
+classes
+  term < logic
+
+default
+  term
+
+consts
+
+ Trueprop	:: "two_seqi"
+ "@Trueprop"	:: "two_seqe" ("((_)/ |- (_))" [6,6] 5)
+
+  (*Constant to allow definitions of SEQUENCES of formulas*)
+  "@Side"        :: "seq=>(seq'=>seq')"     ("<<(_)>>")
+
+  True,False   :: o
+  "="          :: ['a,'a] => o       (infixl 50)
+  Not          :: o => o             ("~ _" [40] 40)
+  "&"          :: [o,o] => o         (infixr 35)
+  "|"          :: [o,o] => o         (infixr 30)
+  "-->","<->"  :: [o,o] => o         (infixr 25)
+  The          :: ('a => o) => 'a    (binder "THE " 10)
+  All          :: ('a => o) => o     (binder "ALL " 10)
+  Ex           :: ('a => o) => o     (binder "EX " 10)
+
+syntax
+  "~="          :: ['a, 'a] => o                (infixl 50)
+
+translations
+  "x ~= y"      == "~ (x = y)"
+
+syntax (symbols)
+  Not           :: o => o               ("\\<not> _" [40] 40)
+  "op &"        :: [o, o] => o          (infixr "\\<and>" 35)
+  "op |"        :: [o, o] => o          (infixr "\\<or>" 30)
+  "op -->"      :: [o, o] => o          (infixr "\\<midarrow>\\<rightarrow>" 25)
+  "op <->"      :: [o, o] => o          (infixr "\\<leftarrow>\\<rightarrow>" 25)
+  "ALL "        :: [idts, o] => o       ("(3\\<forall>_./ _)" [0, 10] 10)
+  "EX "         :: [idts, o] => o       ("(3\\<exists>_./ _)" [0, 10] 10)
+  "EX! "        :: [idts, o] => o       ("(3\\<exists>!_./ _)" [0, 10] 10)
+  "op ~="       :: ['a, 'a] => o        (infixl "\\<noteq>" 50)
+
+syntax (xsymbols)
+  "op -->"      :: [o, o] => o          (infixr "\\<longrightarrow>" 25)
+  "op <->"      :: [o, o] => o          (infixr "\\<longleftrightarrow>" 25)
+
+syntax (HTML output)
+  Not           :: o => o               ("\\<not> _" [40] 40)
+
+
+local
+  
+rules
+
+  (*Structural rules: contraction, thinning, exchange [Soren Heilmann] *)
+
+  contRS "$H |- $E, $S, $S, $F ==> $H |- $E, $S, $F"
+  contLS "$H, $S, $S, $G |- $E ==> $H, $S, $G |- $E"
+
+  thinRS "$H |- $E, $F ==> $H |- $E, $S, $F"
+  thinLS "$H, $G |- $E ==> $H, $S, $G |- $E"
+
+  exchRS "$H |- $E, $R, $S, $F ==> $H |- $E, $S, $R, $F"
+  exchLS "$H, $R, $S, $G |- $E ==> $H, $S, $R, $G |- $E"
+
+  cut   "[| $H |- $E, P;  $H, P |- $E |] ==> $H |- $E"
+
+  (*Propositional rules*)
+
+  basic "$H, P, $G |- $E, P, $F"
+
+  conjR "[| $H|- $E, P, $F;  $H|- $E, Q, $F |] ==> $H|- $E, P&Q, $F"
+  conjL "$H, P, Q, $G |- $E ==> $H, P & Q, $G |- $E"
+
+  disjR "$H |- $E, P, Q, $F ==> $H |- $E, P|Q, $F"
+  disjL "[| $H, P, $G |- $E;  $H, Q, $G |- $E |] ==> $H, P|Q, $G |- $E"
+
+  impR  "$H, P |- $E, Q, $F ==> $H |- $E, P-->Q, $F"
+  impL  "[| $H,$G |- $E,P;  $H, Q, $G |- $E |] ==> $H, P-->Q, $G |- $E"
+
+  notR  "$H, P |- $E, $F ==> $H |- $E, ~P, $F"
+  notL  "$H, $G |- $E, P ==> $H, ~P, $G |- $E"
+
+  FalseL "$H, False, $G |- $E"
+
+  True_def "True == False-->False"
+  iff_def  "P<->Q == (P-->Q) & (Q-->P)"
+
+  (*Quantifiers*)
+
+  allR  "(!!x.$H |- $E, P(x), $F) ==> $H |- $E, ALL x. P(x), $F"
+  allL  "$H, P(x), $G, ALL x. P(x) |- $E ==> $H, ALL x. P(x), $G |- $E"
+
+  exR   "$H |- $E, P(x), $F, EX x. P(x) ==> $H |- $E, EX x. P(x), $F"
+  exL   "(!!x.$H, P(x), $G |- $E) ==> $H, EX x. P(x), $G |- $E"
+
+  (*Equality*)
+
+  refl  "$H |- $E, a=a, $F"
+  subst "$H(a), $G(a) |- $E(a) ==> $H(b), a=b, $G(b) |- $E(b)"
+
+  (* Reflection *)
+
+  eq_reflection  "|- x=y ==> (x==y)"
+  iff_reflection "|- P<->Q ==> (P==Q)"
+
+  (*Descriptions*)
+
+  The "[| $H |- $E, P(a), $F;  !!x.$H, P(x) |- $E, x=a, $F |] ==> 
+          $H |- $E, P(THE x. P(x)), $F"
+
+constdefs
+  If :: [o, 'a, 'a] => 'a   ("(if (_)/ then (_)/ else (_))" 10)
+   "If(P,x,y) == THE z::'a. (P --> z=x) & (~P --> z=y)"
+
+
+setup
+  Simplifier.setup
+
+setup
+  prover_setup
+
+end
+
+  ML
+
+val parse_translation = [("@Trueprop",Sequents.two_seq_tr "Trueprop")];
+val print_translation = [("Trueprop",Sequents.two_seq_tr' "@Trueprop")];