src/Sequents/LK0.thy

tuned;

(*  Title:      LK/LK0.thy
    ID:         $Id$
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1993  University of Cambridge

There may be printing problems if a seqent is in expanded normal form
        (eta-expanded, beta-contracted)
*)

header {* Classical First-Order Sequent Calculus *}

theory LK0
imports Sequents
begin

global

classes "term"
defaultsort "term"

consts

 Trueprop       :: "two_seqi"

  True         :: o
  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)
  "<->"        :: "[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
 "@Trueprop"    :: "two_seqe" ("((_)/ |- (_))" [6,6] 5)
  "_not_equal" :: "['a, 'a] => o"              (infixl "~=" 50)

parse_translation {* [("@Trueprop", two_seq_tr "Trueprop")] *}
print_translation {* [("Trueprop", two_seq_tr' "@Trueprop")] *}

translations
  "x ~= y"      == "~ (x = y)"

syntax (xsymbols)
  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 "\<longrightarrow>" 25)
  "op <->"      :: "[o, o] => o"          (infixr "\<longleftrightarrow>" 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)
  "_not_equal"  :: "['a, 'a] => o"        (infixl "\<noteq>" 50)

syntax (HTML output)
  Not           :: "o => o"               ("\<not> _" [40] 40)
  "op &"        :: "[o, o] => o"          (infixr "\<and>" 35)
  "op |"        :: "[o, o] => o"          (infixr "\<or>" 30)
  "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)
  "_not_equal"  :: "['a, 'a] => o"        (infixl "\<noteq>" 50)

local

axioms

  (*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
  prover_setup

ML {* use_legacy_bindings (the_context ()) *}

end