(*  Title:      FOLP/FOLP.thy

    Author:     Martin D Coen, Cambridge University Computer Laboratory

    Copyright   1992  University of Cambridge

*)

header {* Classical First-Order Logic with Proofs *}

theory FOLP

imports IFOLP

uses

  ("classical.ML") ("simp.ML") ("simpdata.ML")

begin

axiomatization cla :: "[p=>p]=>p"

  where classical: "(!!x. x:~P ==> f(x):P) ==> cla(f):P"

(*** Classical introduction rules for | and EX ***)

schematic_lemma disjCI:

  assumes "!!x. x:~Q ==> f(x):P"

  shows "?p : P|Q"

  apply (rule classical)

  apply (assumption | rule assms disjI1 notI)+

  apply (assumption | rule disjI2 notE)+

  done

(*introduction rule involving only EX*)

schematic_lemma ex_classical:

  assumes "!!u. u:~(EX x. P(x)) ==> f(u):P(a)"

  shows "?p : EX x. P(x)"

  apply (rule classical)

  apply (rule exI, rule assms, assumption)

  done

(*version of above, simplifying ~EX to ALL~ *)

schematic_lemma exCI:

  assumes "!!u. u:ALL x. ~P(x) ==> f(u):P(a)"

  shows "?p : EX x. P(x)"

  apply (rule ex_classical)

  apply (rule notI [THEN allI, THEN assms])

  apply (erule notE)

  apply (erule exI)

  done

schematic_lemma excluded_middle: "?p : ~P | P"

  apply (rule disjCI)

  apply assumption

  done

(*** Special elimination rules *)

(*Classical implies (-->) elimination. *)

schematic_lemma impCE:

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

    and r1: "!!x. x:~P ==> f(x):R"

    and r2: "!!y. y:Q ==> g(y):R"

  shows "?p : R"

  apply (rule excluded_middle [THEN disjE])

   apply (tactic {* DEPTH_SOLVE (atac 1 ORELSE

       resolve_tac [@{thm r1}, @{thm r2}, @{thm major} RS @{thm mp}] 1) *})

  done

(*Double negation law*)

schematic_lemma notnotD: "p:~~P ==> ?p : P"

  apply (rule classical)

  apply (erule notE)

  apply assumption

  done

(*** Tactics for implication and contradiction ***)

(*Classical <-> elimination.  Proof substitutes P=Q in

    ~P ==> ~Q    and    P ==> Q  *)

schematic_lemma iffCE:

  assumes major: "p:P<->Q"

    and r1: "!!x y.[| x:P; y:Q |] ==> f(x,y):R"

    and r2: "!!x y.[| x:~P; y:~Q |] ==> g(x,y):R"

  shows "?p : R"

  apply (insert major)

  apply (unfold iff_def)

  apply (rule conjE)

  apply (tactic {* DEPTH_SOLVE_1 (etac @{thm impCE} 1 ORELSE

      eresolve_tac [@{thm notE}, @{thm impE}] 1 THEN atac 1 ORELSE atac 1 ORELSE

      resolve_tac [@{thm r1}, @{thm r2}] 1) *})+

  done

(*Should be used as swap since ~P becomes redundant*)

schematic_lemma swap:

  assumes major: "p:~P"

    and r: "!!x. x:~Q ==> f(x):P"

  shows "?p : Q"

  apply (rule classical)

  apply (rule major [THEN notE])

  apply (rule r)

  apply assumption

  done

use "classical.ML"      (* Patched 'cos matching won't instantiate proof *)

use "simp.ML"           (* Patched 'cos matching won't instantiate proof *)

ML {*

structure Cla = Classical

(

  val sizef = size_of_thm

  val mp = @{thm mp}

  val not_elim = @{thm notE}

  val swap = @{thm swap}

  val hyp_subst_tacs = [hyp_subst_tac]

);

open Cla;

(*Propositional rules

  -- iffCE might seem better, but in the examples in ex/cla

     run about 7% slower than with iffE*)

val prop_cs =

  empty_cs addSIs [@{thm refl}, @{thm TrueI}, @{thm conjI}, @{thm disjCI},

      @{thm impI}, @{thm notI}, @{thm iffI}]

    addSEs [@{thm conjE}, @{thm disjE}, @{thm impCE}, @{thm FalseE}, @{thm iffE}];

(*Quantifier rules*)

val FOLP_cs =

  prop_cs addSIs [@{thm allI}] addIs [@{thm exI}, @{thm ex1I}]

    addSEs [@{thm exE}, @{thm ex1E}] addEs [@{thm allE}];

val FOLP_dup_cs =

  prop_cs addSIs [@{thm allI}] addIs [@{thm exCI}, @{thm ex1I}]

    addSEs [@{thm exE}, @{thm ex1E}] addEs [@{thm all_dupE}];

*}

schematic_lemma cla_rews:

  "?p1 : P | ~P"

  "?p2 : ~P | P"

  "?p3 : ~ ~ P <-> P"

  "?p4 : (~P --> P) <-> P"

  apply (tactic {* ALLGOALS (Cla.fast_tac FOLP_cs) *})

  done

use "simpdata.ML"

end