(*  Title:      Sequents/LK0.thy

     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

 classes "term"

 default_sort "term"

 consts

   Trueprop       :: "two_seqi"

   True         :: o

   False        :: o

   equal        :: "['a,'a] => o"     (infixl "=" 50)

   Not          :: "o => o"           ("~ _" [40] 40)

   conj         :: "[o,o] => o"       (infixr "&" 35)

   disj         :: "[o,o] => o"       (infixr "|" 30)

   imp          :: "[o,o] => o"       (infixr "-->" 25)

   iff          :: "[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)

 parse_translation {* [(@{syntax_const "_Trueprop"}, two_seq_tr @{const_syntax Trueprop})] *}

 print_translation {* [(@{const_syntax Trueprop}, two_seq_tr' @{syntax_const "_Trueprop"})] *}

 abbreviation

   not_equal  (infixl "~=" 50) where

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

 notation (xsymbols)

   Not  ("\<not> _" [40] 40) and

   conj  (infixr "\<and>" 35) and

   disj  (infixr "\<or>" 30) and

   imp  (infixr "\<longrightarrow>" 25) and

   iff  (infixr "\<longleftrightarrow>" 25) and

   All  (binder "\<forall>" 10) and

   Ex  (binder "\<exists>" 10) and

   not_equal  (infixl "\<noteq>" 50)

 notation (HTML output)

   Not  ("\<not> _" [40] 40) and

   conj  (infixr "\<and>" 35) and

   disj  (infixr "\<or>" 30) and

   All  (binder "\<forall>" 10) and

   Ex  (binder "\<exists>" 10) and

   not_equal  (infixl "\<noteq>" 50)

 axiomatization where

   (*Structural rules: contraction, thinning, exchange [Soren Heilmann] *)

   contRS: "$H |- $E, $S, $S, $F ==> $H |- $E, $S, $F" and

   contLS: "$H, $S, $S, $G |- $E ==> $H, $S, $G |- $E" and

   thinRS: "$H |- $E, $F ==> $H |- $E, $S, $F" and

   thinLS: "$H, $G |- $E ==> $H, $S, $G |- $E" and

   exchRS: "$H |- $E, $R, $S, $F ==> $H |- $E, $S, $R, $F" and

   exchLS: "$H, $R, $S, $G |- $E ==> $H, $S, $R, $G |- $E" and

   cut:   "[| $H |- $E, P;  $H, P |- $E |] ==> $H |- $E" and

   (*Propositional rules*)

   basic: "$H, P, $G |- $E, P, $F" and

   conjR: "[| $H|- $E, P, $F;  $H|- $E, Q, $F |] ==> $H|- $E, P&Q, $F" and

   conjL: "$H, P, Q, $G |- $E ==> $H, P & Q, $G |- $E" and

   disjR: "$H |- $E, P, Q, $F ==> $H |- $E, P|Q, $F" and

   disjL: "[| $H, P, $G |- $E;  $H, Q, $G |- $E |] ==> $H, P|Q, $G |- $E" and

   impR:  "$H, P |- $E, Q, $F ==> $H |- $E, P-->Q, $F" and

   impL:  "[| $H,$G |- $E,P;  $H, Q, $G |- $E |] ==> $H, P-->Q, $G |- $E" and

   notR:  "$H, P |- $E, $F ==> $H |- $E, ~P, $F" and

   notL:  "$H, $G |- $E, P ==> $H, ~P, $G |- $E" and

   FalseL: "$H, False, $G |- $E" and

   True_def: "True == False-->False" and

   iff_def:  "P<->Q == (P-->Q) & (Q-->P)"

 axiomatization where

   (*Quantifiers*)

   allR:  "(!!x.$H |- $E, P(x), $F) ==> $H |- $E, ALL x. P(x), $F" and

   allL:  "$H, P(x), $G, ALL x. P(x) |- $E ==> $H, ALL x. P(x), $G |- $E" and

   exR:   "$H |- $E, P(x), $F, EX x. P(x) ==> $H |- $E, EX x. P(x), $F" and

   exL:   "(!!x.$H, P(x), $G |- $E) ==> $H, EX x. P(x), $G |- $E" and

   (*Equality*)

   refl:  "$H |- $E, a=a, $F" and

   subst: "\<And>G H E. $H(a), $G(a) |- $E(a) ==> $H(b), a=b, $G(b) |- $E(b)"

   (* Reflection *)

 axiomatization where

   eq_reflection:  "|- x=y ==> (x==y)" and

   iff_reflection: "|- P<->Q ==> (P==Q)"

   (*Descriptions*)

 axiomatization where

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

           $H |- $E, P(THE x. P(x)), $F"

 definition If :: "[o, 'a, 'a] => 'a" ("(if (_)/ then (_)/ else (_))" 10)

   where "If(P,x,y) == THE z::'a. (P --> z=x) & (~P --> z=y)"

 (** Structural Rules on formulas **)

 (*contraction*)

 lemma contR: "$H |- $E, P, P, $F ==> $H |- $E, P, $F"

   by (rule contRS)

 lemma contL: "$H, P, P, $G |- $E ==> $H, P, $G |- $E"

   by (rule contLS)

 (*thinning*)

 lemma thinR: "$H |- $E, $F ==> $H |- $E, P, $F"

   by (rule thinRS)

 lemma thinL: "$H, $G |- $E ==> $H, P, $G |- $E"

   by (rule thinLS)

 (*exchange*)

 lemma exchR: "$H |- $E, Q, P, $F ==> $H |- $E, P, Q, $F"

   by (rule exchRS)

 lemma exchL: "$H, Q, P, $G |- $E ==> $H, P, Q, $G |- $E"

   by (rule exchLS)

 ML {*

 (*Cut and thin, replacing the right-side formula*)

 fun cutR_tac ctxt s i =

   res_inst_tac ctxt [(("P", 0), s) ] @{thm cut} i  THEN  rtac @{thm thinR} i

 (*Cut and thin, replacing the left-side formula*)

 fun cutL_tac ctxt s i =

   res_inst_tac ctxt [(("P", 0), s)] @{thm cut} i  THEN  rtac @{thm thinL} (i+1)

 *}

 (** If-and-only-if rules **)

 lemma iffR: 

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

   apply (unfold iff_def)

   apply (assumption | rule conjR impR)+

   done

 lemma iffL: 

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

   apply (unfold iff_def)

   apply (assumption | rule conjL impL basic)+

   done

 lemma iff_refl: "$H |- $E, (P <-> P), $F"

   apply (rule iffR basic)+

   done

 lemma TrueR: "$H |- $E, True, $F"

   apply (unfold True_def)

   apply (rule impR)

   apply (rule basic)

   done

 (*Descriptions*)

 lemma the_equality:

   assumes p1: "$H |- $E, P(a), $F"

     and p2: "!!x. $H, P(x) |- $E, x=a, $F"

   shows "$H |- $E, (THE x. P(x)) = a, $F"

   apply (rule cut)

    apply (rule_tac [2] p2)

   apply (rule The, rule thinR, rule exchRS, rule p1)

   apply (rule thinR, rule exchRS, rule p2)

   done

 (** Weakened quantifier rules.  Incomplete, they let the search terminate.**)

 lemma allL_thin: "$H, P(x), $G |- $E ==> $H, ALL x. P(x), $G |- $E"

   apply (rule allL)

   apply (erule thinL)

   done

 lemma exR_thin: "$H |- $E, P(x), $F ==> $H |- $E, EX x. P(x), $F"

   apply (rule exR)

   apply (erule thinR)

   done

 (*The rules of LK*)

 ML {*

 val prop_pack = empty_pack add_safes

                 [@{thm basic}, @{thm refl}, @{thm TrueR}, @{thm FalseL},

                  @{thm conjL}, @{thm conjR}, @{thm disjL}, @{thm disjR}, @{thm impL}, @{thm impR},

                  @{thm notL}, @{thm notR}, @{thm iffL}, @{thm iffR}];

 val LK_pack = prop_pack add_safes   [@{thm allR}, @{thm exL}]

                         add_unsafes [@{thm allL_thin}, @{thm exR_thin}, @{thm the_equality}];

 val LK_dup_pack = prop_pack add_safes   [@{thm allR}, @{thm exL}]

                             add_unsafes [@{thm allL}, @{thm exR}, @{thm the_equality}];

 fun lemma_tac th i =

     rtac (@{thm thinR} RS @{thm cut}) i THEN REPEAT (rtac @{thm thinL} i) THEN rtac th i;

 *}

 method_setup fast_prop = {* Scan.succeed (K (SIMPLE_METHOD' (fast_tac prop_pack))) *}

 method_setup fast = {* Scan.succeed (K (SIMPLE_METHOD' (fast_tac LK_pack))) *}

 method_setup fast_dup = {* Scan.succeed (K (SIMPLE_METHOD' (fast_tac LK_dup_pack))) *}

 method_setup best = {* Scan.succeed (K (SIMPLE_METHOD' (best_tac LK_pack))) *}

 method_setup best_dup = {* Scan.succeed (K (SIMPLE_METHOD' (best_tac LK_dup_pack))) *}

 lemma mp_R:

   assumes major: "$H |- $E, $F, P --> Q"

     and minor: "$H |- $E, $F, P"

   shows "$H |- $E, Q, $F"

   apply (rule thinRS [THEN cut], rule major)

   apply (tactic "step_tac LK_pack 1")

   apply (rule thinR, rule minor)

   done

 lemma mp_L:

   assumes major: "$H, $G |- $E, P --> Q"

     and minor: "$H, $G, Q |- $E"

   shows "$H, P, $G |- $E"

   apply (rule thinL [THEN cut], rule major)

   apply (tactic "step_tac LK_pack 1")

   apply (rule thinL, rule minor)

   done

 (** Two rules to generate left- and right- rules from implications **)

 lemma R_of_imp:

   assumes major: "|- P --> Q"

     and minor: "$H |- $E, $F, P"

   shows "$H |- $E, Q, $F"

   apply (rule mp_R)

    apply (rule_tac [2] minor)

   apply (rule thinRS, rule major [THEN thinLS])

   done

 lemma L_of_imp:

   assumes major: "|- P --> Q"

     and minor: "$H, $G, Q |- $E"

   shows "$H, P, $G |- $E"

   apply (rule mp_L)

    apply (rule_tac [2] minor)

   apply (rule thinRS, rule major [THEN thinLS])

   done

 (*Can be used to create implications in a subgoal*)

 lemma backwards_impR:

   assumes prem: "$H, $G |- $E, $F, P --> Q"

   shows "$H, P, $G |- $E, Q, $F"

   apply (rule mp_L)

    apply (rule_tac [2] basic)

   apply (rule thinR, rule prem)

   done

 lemma conjunct1: "|-P&Q ==> |-P"

   apply (erule thinR [THEN cut])

   apply fast

   done

 lemma conjunct2: "|-P&Q ==> |-Q"

   apply (erule thinR [THEN cut])

   apply fast

   done

 lemma spec: "|- (ALL x. P(x)) ==> |- P(x)"

   apply (erule thinR [THEN cut])

   apply fast

   done

 (** Equality **)

 lemma sym: "|- a=b --> b=a"

   by (tactic {* safe_tac (LK_pack add_safes [@{thm subst}]) 1 *})

 lemma trans: "|- a=b --> b=c --> a=c"

   by (tactic {* safe_tac (LK_pack add_safes [@{thm subst}]) 1 *})

 (* Symmetry of equality in hypotheses *)

 lemmas symL = sym [THEN L_of_imp]

 (* Symmetry of equality in hypotheses *)

 lemmas symR = sym [THEN R_of_imp]

 lemma transR: "[| $H|- $E, $F, a=b;  $H|- $E, $F, b=c |] ==> $H|- $E, a=c, $F"

   by (rule trans [THEN R_of_imp, THEN mp_R])

 (* Two theorms for rewriting only one instance of a definition:

    the first for definitions of formulae and the second for terms *)

 lemma def_imp_iff: "(A == B) ==> |- A <-> B"

   apply unfold

   apply (rule iff_refl)

   done

 lemma meta_eq_to_obj_eq: "(A == B) ==> |- A = B"

   apply unfold

   apply (rule refl)

   done

 (** if-then-else rules **)

 lemma if_True: "|- (if True then x else y) = x"

   unfolding If_def by fast

 lemma if_False: "|- (if False then x else y) = y"

   unfolding If_def by fast

 lemma if_P: "|- P ==> |- (if P then x else y) = x"

   apply (unfold If_def)

   apply (erule thinR [THEN cut])

   apply fast

   done

 lemma if_not_P: "|- ~P ==> |- (if P then x else y) = y";

   apply (unfold If_def)

   apply (erule thinR [THEN cut])

   apply fast

   done

 end