# Theory LK0

Up to index of Isabelle/Sequents

theory LK0
imports Sequents
uses [LK0.ML]
begin

    (*  Title:      LK/LK0.thy
        ID:         $Id: LK0.thy,v 1.10 2005/09/18 13:20:08 wenzelm Exp $
        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"               ("¬ _" [40] 40)
      "op &"        :: "[o, o] => o"          (infixr "∧" 35)
      "op |"        :: "[o, o] => o"          (infixr "∨" 30)
      "op -->"      :: "[o, o] => o"          (infixr "-->" 25)
      "op <->"      :: "[o, o] => o"          (infixr "<->" 25)
      "ALL "        :: "[idts, o] => o"       ("(3∀_./ _)" [0, 10] 10)
      "EX "         :: "[idts, o] => o"       ("(3∃_./ _)" [0, 10] 10)
      "EX! "        :: "[idts, o] => o"       ("(3∃!_./ _)" [0, 10] 10)
      "_not_equal"  :: "['a, 'a] => o"        (infixl "≠" 50)

    syntax (HTML output)
      Not           :: "o => o"               ("¬ _" [40] 40)
      "op &"        :: "[o, o] => o"          (infixr "∧" 35)
      "op |"        :: "[o, o] => o"          (infixr "∨" 30)
      "ALL "        :: "[idts, o] => o"       ("(3∀_./ _)" [0, 10] 10)
      "EX "         :: "[idts, o] => o"       ("(3∃_./ _)" [0, 10] 10)
      "EX! "        :: "[idts, o] => o"       ("(3∃!_./ _)" [0, 10] 10)
      "_not_equal"  :: "['a, 'a] => o"        (infixl "≠" 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

theorem contR:

      $H |- $E, P, P, $F ==> $H |- $E, P, $F

theorem contL:

      $H, P, P, $G |- $E ==> $H, P, $G |- $E

theorem thinR:

      $H |- $E, $F ==> $H |- $E, P, $F

theorem thinL:

      $H, $G |- $E ==> $H, P, $G |- $E

theorem exchR:

      $H |- $E, Q, P, $F ==> $H |- $E, P, Q, $F

theorem exchL:

      $H, Q, P, $G |- $E ==> $H, P, Q, $G |- $E

theorem iffR:

      [| $H, P |- $E, Q, $F; $H, Q |- $E, P, $F |] ==> $H |- $E, P <-> Q, $F

theorem iffL:

      [| $H, $G |- $E, P, Q; $H, Q, P, $G |- $E |] ==> $H, P <-> Q, $G |- $E

theorem iff_refl:

      $H |- $E, P <-> P, $F

theorem TrueR:

      $H |- $E, True, $F

theorem the_equality:

      [| $H |- $E, P(a), $F; !!x. $H, P(x) |- $E, x = a, $F |]
      ==> $H |- $E, (THE x. P(x)) = a, $F

theorem allL_thin:

      $H, P(x), $G |- $E ==> $H, ∀x. P(x), $G |- $E

theorem exR_thin:

      $H |- $E, P(x), $F ==> $H |- $E, ∃x. P(x), $F

theorem mp_R:

      [| $H |- $E, $F, P --> Q; $H |- $E, $F, P |] ==> $H |- $E, Q, $F

theorem mp_L:

      [| $H, $G |- $E, P --> Q; $H, $G, Q |- $E |] ==> $H, P, $G |- $E

theorem R_of_imp:

      [|  |- P --> Q; $H |- $E, $F, P |] ==> $H |- $E, Q, $F

theorem L_of_imp:

      [|  |- P --> Q; $H, $G, Q |- $E |] ==> $H, P, $G |- $E

theorem backwards_impR:

      $H, $G |- $E, $F, P --> Q ==> $H, P, $G |- $E, Q, $F

theorem conjunct1:

       |- P ∧ Q ==>  |- P

theorem conjunct2:

       |- P ∧ Q ==>  |- Q

theorem spec:

       |- ∀x. P(x) ==>  |- P(x)

theorem sym:

       |- a = b --> b = a

theorem trans:

       |- a = b --> b = c --> a = c

theorem symL:

      $H, $G, b = a |- $E ==> $H, a = b, $G |- $E

theorem symR:

      $H |- $E, $F, a = b ==> $H |- $E, b = a, $F

theorem transR:

      [| $H |- $E, $F, a = b; $H |- $E, $F, b = c |] ==> $H |- $E, a = c, $F

theorem def_imp_iff:

      A == B ==>  |- A <-> B

theorem meta_eq_to_obj_eq:

      A == B ==>  |- A = B