src/HOL/Tools/cnf_funcs.ML
author webertj
Wed Aug 30 16:27:53 2006 +0200 (2006-08-30)
changeset 20440 e6fe74eebda3
parent 19236 150e8b0fb991
child 20547 796ae7fa1049
permissions -rw-r--r--
faster clause representation (again): full CNF formula as a hypothesis, instead of separate clauses
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(*  Title:      HOL/Tools/cnf_funcs.ML
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    ID:         $Id$
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    Author:     Alwen Tiu, QSL Team, LORIA (http://qsl.loria.fr)
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    Author:     Tjark Weber
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    Copyright   2005-2006
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  Description:
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  This file contains functions and tactics to transform a formula into
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  Conjunctive Normal Form (CNF).
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  A formula in CNF is of the following form:
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      (x11 | x12 | ... | x1n) & ... & (xm1 | xm2 | ... | xmk)
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      False
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      True
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  where each xij is a literal (a positive or negative atomic Boolean term),
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  i.e. the formula is a conjunction of disjunctions of literals, or
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  "False", or "True".
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  A (non-empty) disjunction of literals is referred to as "clause".
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  For the purpose of SAT proof reconstruction, we also make use of another
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  representation of clauses, which we call the "raw clauses".
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  Raw clauses are of the form
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      [..., x1', x2', ..., xn'] |- False ,
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  where each xi is a literal, and each xi' is the negation normal form of ~xi.
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  Literals are successively removed from the hyps of raw clauses by resolution
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  during SAT proof reconstruction.
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*)
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signature CNF =
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sig
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	val is_atom           : Term.term -> bool
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	val is_literal        : Term.term -> bool
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	val is_clause         : Term.term -> bool
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	val clause_is_trivial : Term.term -> bool
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	val clause2raw_thm : Thm.thm -> Thm.thm
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	val weakening_tac : int -> Tactical.tactic  (* removes the first hypothesis of a subgoal *)
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	val make_cnf_thm  : Theory.theory -> Term.term -> Thm.thm
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	val make_cnfx_thm : Theory.theory -> Term.term ->  Thm.thm
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	val cnf_rewrite_tac  : int -> Tactical.tactic  (* converts all prems of a subgoal to CNF *)
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	val cnfx_rewrite_tac : int -> Tactical.tactic  (* converts all prems of a subgoal to (almost) definitional CNF *)
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end;
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structure cnf : CNF =
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struct
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fun thm_by_auto (G : string) : thm =
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	prove_goal (the_context ()) G (fn prems => [cut_facts_tac prems 1, Auto_tac]);
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(* Thm.thm *)
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val clause2raw_notE      = thm_by_auto "[| P; ~P |] ==> False";
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val clause2raw_not_disj  = thm_by_auto "[| ~P; ~Q |] ==> ~(P | Q)";
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val clause2raw_not_not   = thm_by_auto "P ==> ~~P";
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val iff_refl             = thm_by_auto "(P::bool) = P";
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val iff_trans            = thm_by_auto "[| (P::bool) = Q; Q = R |] ==> P = R";
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val conj_cong            = thm_by_auto "[| P = P'; Q = Q' |] ==> (P & Q) = (P' & Q')";
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val disj_cong            = thm_by_auto "[| P = P'; Q = Q' |] ==> (P | Q) = (P' | Q')";
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val make_nnf_imp         = thm_by_auto "[| (~P) = P'; Q = Q' |] ==> (P --> Q) = (P' | Q')";
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val make_nnf_iff         = thm_by_auto "[| P = P'; (~P) = NP; Q = Q'; (~Q) = NQ |] ==> (P = Q) = ((P' | NQ) & (NP | Q'))";
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val make_nnf_not_false   = thm_by_auto "(~False) = True";
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val make_nnf_not_true    = thm_by_auto "(~True) = False";
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val make_nnf_not_conj    = thm_by_auto "[| (~P) = P'; (~Q) = Q' |] ==> (~(P & Q)) = (P' | Q')";
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val make_nnf_not_disj    = thm_by_auto "[| (~P) = P'; (~Q) = Q' |] ==> (~(P | Q)) = (P' & Q')";
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val make_nnf_not_imp     = thm_by_auto "[| P = P'; (~Q) = Q' |] ==> (~(P --> Q)) = (P' & Q')";
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val make_nnf_not_iff     = thm_by_auto "[| P = P'; (~P) = NP; Q = Q'; (~Q) = NQ |] ==> (~(P = Q)) = ((P' | Q') & (NP | NQ))";
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val make_nnf_not_not     = thm_by_auto "P = P' ==> (~~P) = P'";
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val simp_TF_conj_True_l  = thm_by_auto "[| P = True; Q = Q' |] ==> (P & Q) = Q'";
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val simp_TF_conj_True_r  = thm_by_auto "[| P = P'; Q = True |] ==> (P & Q) = P'";
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val simp_TF_conj_False_l = thm_by_auto "P = False ==> (P & Q) = False";
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val simp_TF_conj_False_r = thm_by_auto "Q = False ==> (P & Q) = False";
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val simp_TF_disj_True_l  = thm_by_auto "P = True ==> (P | Q) = True";
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val simp_TF_disj_True_r  = thm_by_auto "Q = True ==> (P | Q) = True";
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val simp_TF_disj_False_l = thm_by_auto "[| P = False; Q = Q' |] ==> (P | Q) = Q'";
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val simp_TF_disj_False_r = thm_by_auto "[| P = P'; Q = False |] ==> (P | Q) = P'";
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val make_cnf_disj_conj_l = thm_by_auto "[| (P | R) = PR; (Q | R) = QR |] ==> ((P & Q) | R) = (PR & QR)";
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val make_cnf_disj_conj_r = thm_by_auto "[| (P | Q) = PQ; (P | R) = PR |] ==> (P | (Q & R)) = (PQ & PR)";
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val make_cnfx_disj_ex_l = thm_by_auto "((EX (x::bool). P x) | Q) = (EX x. P x | Q)";
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val make_cnfx_disj_ex_r = thm_by_auto "(P | (EX (x::bool). Q x)) = (EX x. P | Q x)";
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val make_cnfx_newlit    = thm_by_auto "(P | Q) = (EX x. (P | x) & (Q | ~x))";
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val make_cnfx_ex_cong   = thm_by_auto "(ALL (x::bool). P x = Q x) ==> (EX x. P x) = (EX x. Q x)";
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val weakening_thm        = thm_by_auto "[| P; Q |] ==> Q";
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val cnftac_eq_imp        = thm_by_auto "[| P = Q; P |] ==> Q";
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(* Term.term -> bool *)
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fun is_atom (Const ("False", _))                                           = false
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  | is_atom (Const ("True", _))                                            = false
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  | is_atom (Const ("op &", _) $ _ $ _)                                    = false
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  | is_atom (Const ("op |", _) $ _ $ _)                                    = false
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  | is_atom (Const ("op -->", _) $ _ $ _)                                  = false
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  | is_atom (Const ("op =", Type ("fun", Type ("bool", []) :: _)) $ _ $ _) = false
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  | is_atom (Const ("Not", _) $ _)                                         = false
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  | is_atom _                                                              = true;
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(* Term.term -> bool *)
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fun is_literal (Const ("Not", _) $ x) = is_atom x
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  | is_literal x                      = is_atom x;
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(* Term.term -> bool *)
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fun is_clause (Const ("op |", _) $ x $ y) = is_clause x andalso is_clause y
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  | is_clause x                           = is_literal x;
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(* ------------------------------------------------------------------------- *)
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(* clause_is_trivial: a clause is trivially true if it contains both an atom *)
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(*      and the atom's negation                                              *)
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(* ------------------------------------------------------------------------- *)
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(* Term.term -> bool *)
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fun clause_is_trivial c =
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	let
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		(* Term.term -> Term.term list -> Term.term list *)
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		fun collect_literals (Const ("op |", _) $ x $ y) ls = collect_literals x (collect_literals y ls)
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		  | collect_literals x                           ls = x :: ls
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		(* Term.term -> Term.term *)
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		fun dual (Const ("Not", _) $ x) = x
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		  | dual x                      = HOLogic.Not $ x
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		(* Term.term list -> bool *)
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		fun has_duals []      = false
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		  | has_duals (x::xs) = (dual x) mem xs orelse has_duals xs
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	in
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		has_duals (collect_literals c [])
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	end;
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(* ------------------------------------------------------------------------- *)
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(* clause2raw_thm: translates a clause into a raw clause, i.e.               *)
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(*        [...] |- x1 | ... | xn                                             *)
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(*      (where each xi is a literal) is translated to                        *)
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(*        [..., x1', ..., xn'] |- False ,                                    *)
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(*      where each xi' is the negation normal form of ~xi                    *)
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(* ------------------------------------------------------------------------- *)
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(* Thm.thm -> Thm.thm *)
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fun clause2raw_thm clause =
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let
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	(* eliminates negated disjunctions from the i-th premise, possibly *)
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	(* adding new premises, then continues with the (i+1)-th premise   *)
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	(* int -> Thm.thm -> Thm.thm *)
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	fun not_disj_to_prem i thm =
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		if i > nprems_of thm then
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			thm
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		else
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			not_disj_to_prem (i+1) (Seq.hd (REPEAT_DETERM (rtac clause2raw_not_disj i) thm))
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	(* moves all premises to hyps, i.e. "[...] |- A1 ==> ... ==> An ==> B" *)
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	(* becomes "[..., A1, ..., An] |- B"                                   *)
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	(* Thm.thm -> Thm.thm *)
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	fun prems_to_hyps thm =
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		fold (fn cprem => fn thm' =>
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			Thm.implies_elim thm' (Thm.assume cprem)) (cprems_of thm) thm
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in
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	(* [...] |- ~(x1 | ... | xn) ==> False *)
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	(clause2raw_notE OF [clause])
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	(* [...] |- ~x1 ==> ... ==> ~xn ==> False *)
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	|> not_disj_to_prem 1
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	(* [...] |- x1' ==> ... ==> xn' ==> False *)
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	|> Seq.hd o TRYALL (rtac clause2raw_not_not)
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	(* [..., x1', ..., xn'] |- False *)
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	|> prems_to_hyps
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end;
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(* ------------------------------------------------------------------------- *)
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(* inst_thm: instantiates a theorem with a list of terms                     *)
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(* ------------------------------------------------------------------------- *)
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(* Theory.theory -> Term.term list -> Thm.thm -> Thm.thm *)
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fun inst_thm thy ts thm =
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	instantiate' [] (map (SOME o cterm_of thy) ts) thm;
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(* ------------------------------------------------------------------------- *)
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(*                         Naive CNF transformation                          *)
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(* ------------------------------------------------------------------------- *)
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(* ------------------------------------------------------------------------- *)
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(* make_nnf_thm: produces a theorem of the form t = t', where t' is the      *)
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(*      negation normal form (i.e. negation only occurs in front of atoms)   *)
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(*      of t; implications ("-->") and equivalences ("=" on bool) are        *)
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(*      eliminated (possibly causing an exponential blowup)                  *)
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(* ------------------------------------------------------------------------- *)
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(* Theory.theory -> Term.term -> Thm.thm *)
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fun make_nnf_thm thy (Const ("op &", _) $ x $ y) =
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	let
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		val thm1 = make_nnf_thm thy x
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		val thm2 = make_nnf_thm thy y
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	in
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		conj_cong OF [thm1, thm2]
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	end
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  | make_nnf_thm thy (Const ("op |", _) $ x $ y) =
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	let
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		val thm1 = make_nnf_thm thy x
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		val thm2 = make_nnf_thm thy y
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	in
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		disj_cong OF [thm1, thm2]
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	end
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  | make_nnf_thm thy (Const ("op -->", _) $ x $ y) =
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	let
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		val thm1 = make_nnf_thm thy (HOLogic.Not $ x)
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		val thm2 = make_nnf_thm thy y
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	in
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		make_nnf_imp OF [thm1, thm2]
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	end
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  | make_nnf_thm thy (Const ("op =", Type ("fun", Type ("bool", []) :: _)) $ x $ y) =
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	let
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		val thm1 = make_nnf_thm thy x
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		val thm2 = make_nnf_thm thy (HOLogic.Not $ x)
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		val thm3 = make_nnf_thm thy y
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		val thm4 = make_nnf_thm thy (HOLogic.Not $ y)
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	in
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		make_nnf_iff OF [thm1, thm2, thm3, thm4]
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	end
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  | make_nnf_thm thy (Const ("Not", _) $ Const ("False", _)) =
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	make_nnf_not_false
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  | make_nnf_thm thy (Const ("Not", _) $ Const ("True", _)) =
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	make_nnf_not_true
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  | make_nnf_thm thy (Const ("Not", _) $ (Const ("op &", _) $ x $ y)) =
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	let
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		val thm1 = make_nnf_thm thy (HOLogic.Not $ x)
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		val thm2 = make_nnf_thm thy (HOLogic.Not $ y)
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	in
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		make_nnf_not_conj OF [thm1, thm2]
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	end
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  | make_nnf_thm thy (Const ("Not", _) $ (Const ("op |", _) $ x $ y)) =
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	let
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		val thm1 = make_nnf_thm thy (HOLogic.Not $ x)
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		val thm2 = make_nnf_thm thy (HOLogic.Not $ y)
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	in
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		make_nnf_not_disj OF [thm1, thm2]
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	end
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  | make_nnf_thm thy (Const ("Not", _) $ (Const ("op -->", _) $ x $ y)) =
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	let
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		val thm1 = make_nnf_thm thy x
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		val thm2 = make_nnf_thm thy (HOLogic.Not $ y)
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	in
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		make_nnf_not_imp OF [thm1, thm2]
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	end
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  | make_nnf_thm thy (Const ("Not", _) $ (Const ("op =", Type ("fun", Type ("bool", []) :: _)) $ x $ y)) =
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	let
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		val thm1 = make_nnf_thm thy x
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		val thm2 = make_nnf_thm thy (HOLogic.Not $ x)
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		val thm3 = make_nnf_thm thy y
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		val thm4 = make_nnf_thm thy (HOLogic.Not $ y)
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	in
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		make_nnf_not_iff OF [thm1, thm2, thm3, thm4]
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	end
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  | make_nnf_thm thy (Const ("Not", _) $ (Const ("Not", _) $ x)) =
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	let
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		val thm1 = make_nnf_thm thy x
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	in
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		make_nnf_not_not OF [thm1]
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	end
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  | make_nnf_thm thy t =
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	inst_thm thy [t] iff_refl;
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(* ------------------------------------------------------------------------- *)
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(* simp_True_False_thm: produces a theorem t = t', where t' is equivalent to *)
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(*      t, but simplified wrt. the following theorems:                       *)
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(*        (True & x) = x                                                     *)
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(*        (x & True) = x                                                     *)
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(*        (False & x) = False                                                *)
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(*        (x & False) = False                                                *)
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(*        (True | x) = True                                                  *)
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(*        (x | True) = True                                                  *)
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(*        (False | x) = x                                                    *)
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(*        (x | False) = x                                                    *)
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(*      No simplification is performed below connectives other than & and |. *)
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(*      Optimization: The right-hand side of a conjunction (disjunction) is  *)
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(*      simplified only if the left-hand side does not simplify to False     *)
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(*      (True, respectively).                                                *)
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   285
(* ------------------------------------------------------------------------- *)
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   287
(* Theory.theory -> Term.term -> Thm.thm *)
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   288
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   289
fun simp_True_False_thm thy (Const ("op &", _) $ x $ y) =
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   290
	let
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   291
		val thm1 = simp_True_False_thm thy x
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   292
		val x'   = (snd o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) thm1
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   293
	in
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   294
		if x' = HOLogic.false_const then
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   295
			simp_TF_conj_False_l OF [thm1]  (* (x & y) = False *)
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   296
		else
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   297
			let
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   298
				val thm2 = simp_True_False_thm thy y
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   299
				val y'   = (snd o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) thm2
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   300
			in
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   301
				if x' = HOLogic.true_const then
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   302
					simp_TF_conj_True_l OF [thm1, thm2]  (* (x & y) = y' *)
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   303
				else if y' = HOLogic.false_const then
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   304
					simp_TF_conj_False_r OF [thm2]  (* (x & y) = False *)
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   305
				else if y' = HOLogic.true_const then
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   306
					simp_TF_conj_True_r OF [thm1, thm2]  (* (x & y) = x' *)
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   307
				else
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   308
					conj_cong OF [thm1, thm2]  (* (x & y) = (x' & y') *)
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   309
			end
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   310
	end
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   311
  | simp_True_False_thm thy (Const ("op |", _) $ x $ y) =
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   312
	let
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		val thm1 = simp_True_False_thm thy x
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   314
		val x'   = (snd o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) thm1
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   315
	in
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   316
		if x' = HOLogic.true_const then
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   317
			simp_TF_disj_True_l OF [thm1]  (* (x | y) = True *)
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   318
		else
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   319
			let
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   320
				val thm2 = simp_True_False_thm thy y
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   321
				val y'   = (snd o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) thm2
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   322
			in
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   323
				if x' = HOLogic.false_const then
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   324
					simp_TF_disj_False_l OF [thm1, thm2]  (* (x | y) = y' *)
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   325
				else if y' = HOLogic.true_const then
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   326
					simp_TF_disj_True_r OF [thm2]  (* (x | y) = True *)
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   327
				else if y' = HOLogic.false_const then
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   328
					simp_TF_disj_False_r OF [thm1, thm2]  (* (x | y) = x' *)
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   329
				else
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   330
					disj_cong OF [thm1, thm2]  (* (x | y) = (x' | y') *)
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   331
			end
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   332
	end
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   333
  | simp_True_False_thm thy t =
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   334
	inst_thm thy [t] iff_refl;  (* t = t *)
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   335
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   336
(* ------------------------------------------------------------------------- *)
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   337
(* make_cnf_thm: given any HOL term 't', produces a theorem t = t', where t' *)
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   338
(*      is in conjunction normal form.  May cause an exponential blowup      *)
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   339
(*      in the length of the term.                                           *)
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   340
(* ------------------------------------------------------------------------- *)
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   341
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   342
(* Theory.theory -> Term.term -> Thm.thm *)
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   343
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   344
fun make_cnf_thm thy t =
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   345
let
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   346
	(* Term.term -> Thm.thm *)
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   347
	fun make_cnf_thm_from_nnf (Const ("op &", _) $ x $ y) =
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   348
		let
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   349
			val thm1 = make_cnf_thm_from_nnf x
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   350
			val thm2 = make_cnf_thm_from_nnf y
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   351
		in
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   352
			conj_cong OF [thm1, thm2]
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   353
		end
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   354
	  | make_cnf_thm_from_nnf (Const ("op |", _) $ x $ y) =
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   355
		let
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   356
			(* produces a theorem "(x' | y') = t'", where x', y', and t' are in CNF *)
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   357
			fun make_cnf_disj_thm (Const ("op &", _) $ x1 $ x2) y' =
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   358
				let
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   359
					val thm1 = make_cnf_disj_thm x1 y'
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   360
					val thm2 = make_cnf_disj_thm x2 y'
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   361
				in
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   362
					make_cnf_disj_conj_l OF [thm1, thm2]  (* ((x1 & x2) | y') = ((x1 | y')' & (x2 | y')') *)
webertj@17809
   363
				end
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   364
			  | make_cnf_disj_thm x' (Const ("op &", _) $ y1 $ y2) =
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   365
				let
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   366
					val thm1 = make_cnf_disj_thm x' y1
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   367
					val thm2 = make_cnf_disj_thm x' y2
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   368
				in
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   369
					make_cnf_disj_conj_r OF [thm1, thm2]  (* (x' | (y1 & y2)) = ((x' | y1)' & (x' | y2)') *)
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   370
				end
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   371
			  | make_cnf_disj_thm x' y' =
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   372
				inst_thm thy [HOLogic.mk_disj (x', y')] iff_refl  (* (x' | y') = (x' | y') *)
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   373
			val thm1     = make_cnf_thm_from_nnf x
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   374
			val thm2     = make_cnf_thm_from_nnf y
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   375
			val x'       = (snd o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) thm1
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   376
			val y'       = (snd o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) thm2
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   377
			val disj_thm = disj_cong OF [thm1, thm2]  (* (x | y) = (x' | y') *)
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   378
		in
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   379
			iff_trans OF [disj_thm, make_cnf_disj_thm x' y']
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   380
		end
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   381
	  | make_cnf_thm_from_nnf t =
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   382
		inst_thm thy [t] iff_refl
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   383
	(* convert 't' to NNF first *)
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   384
	val nnf_thm  = make_nnf_thm thy t
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   385
	val nnf      = (snd o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) nnf_thm
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   386
	(* then simplify wrt. True/False (this should preserve NNF) *)
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   387
	val simp_thm = simp_True_False_thm thy nnf
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   388
	val simp     = (snd o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) simp_thm
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   389
	(* finally, convert to CNF (this should preserve the simplification) *)
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   390
	val cnf_thm  = make_cnf_thm_from_nnf simp
webertj@17618
   391
in
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   392
	iff_trans OF [iff_trans OF [nnf_thm, simp_thm], cnf_thm]
webertj@17809
   393
end;
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   394
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   395
(* ------------------------------------------------------------------------- *)
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   396
(*            CNF transformation by introducing new literals                 *)
webertj@17809
   397
(* ------------------------------------------------------------------------- *)
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   398
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   399
(* ------------------------------------------------------------------------- *)
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   400
(* make_cnfx_thm: given any HOL term 't', produces a theorem t = t', where   *)
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   401
(*      t' is almost in conjunction normal form, except that conjunctions    *)
webertj@17809
   402
(*      and existential quantifiers may be nested.  (Use e.g. 'REPEAT_DETERM *)
webertj@17809
   403
(*      (etac exE i ORELSE etac conjE i)' afterwards to normalize.)  May     *)
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   404
(*      introduce new (existentially bound) literals.  Note: the current     *)
webertj@17809
   405
(*      implementation calls 'make_nnf_thm', causing an exponential blowup   *)
webertj@17809
   406
(*      in the case of nested equivalences.                                  *)
webertj@17809
   407
(* ------------------------------------------------------------------------- *)
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   408
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   409
(* Theory.theory -> Term.term -> Thm.thm *)
webertj@17618
   410
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   411
fun make_cnfx_thm thy t =
webertj@17809
   412
let
webertj@17809
   413
	val var_id = ref 0  (* properly initialized below *)
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   414
	(* unit -> Term.term *)
webertj@17809
   415
	fun new_free () =
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   416
		Free ("cnfx_" ^ string_of_int (inc var_id), HOLogic.boolT)
webertj@17809
   417
	(* Term.term -> Thm.thm *)
webertj@17809
   418
	fun make_cnfx_thm_from_nnf (Const ("op &", _) $ x $ y) =
webertj@17809
   419
		let
webertj@17809
   420
			val thm1 = make_cnfx_thm_from_nnf x
webertj@17809
   421
			val thm2 = make_cnfx_thm_from_nnf y
webertj@17809
   422
		in
webertj@17809
   423
			conj_cong OF [thm1, thm2]
webertj@17809
   424
		end
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   425
	  | make_cnfx_thm_from_nnf (Const ("op |", _) $ x $ y) =
webertj@17809
   426
		if is_clause x andalso is_clause y then
webertj@17809
   427
			inst_thm thy [HOLogic.mk_disj (x, y)] iff_refl
webertj@17809
   428
		else if is_literal y orelse is_literal x then let
webertj@17809
   429
			(* produces a theorem "(x' | y') = t'", where x', y', and t' are *)
webertj@17809
   430
			(* almost in CNF, and x' or y' is a literal                      *)
webertj@17809
   431
			fun make_cnfx_disj_thm (Const ("op &", _) $ x1 $ x2) y' =
webertj@17809
   432
				let
webertj@17809
   433
					val thm1 = make_cnfx_disj_thm x1 y'
webertj@17809
   434
					val thm2 = make_cnfx_disj_thm x2 y'
webertj@17809
   435
				in
webertj@17809
   436
					make_cnf_disj_conj_l OF [thm1, thm2]  (* ((x1 & x2) | y') = ((x1 | y')' & (x2 | y')') *)
webertj@17809
   437
				end
webertj@17809
   438
			  | make_cnfx_disj_thm x' (Const ("op &", _) $ y1 $ y2) =
webertj@17809
   439
				let
webertj@17809
   440
					val thm1 = make_cnfx_disj_thm x' y1
webertj@17809
   441
					val thm2 = make_cnfx_disj_thm x' y2
webertj@17809
   442
				in
webertj@17809
   443
					make_cnf_disj_conj_r OF [thm1, thm2]  (* (x' | (y1 & y2)) = ((x' | y1)' & (x' | y2)') *)
webertj@17809
   444
				end
webertj@17809
   445
			  | make_cnfx_disj_thm (Const ("Ex", _) $ x') y' =
webertj@17809
   446
				let
webertj@17809
   447
					val thm1 = inst_thm thy [x', y'] make_cnfx_disj_ex_l   (* ((Ex x') | y') = (Ex (x' | y')) *)
webertj@17809
   448
					val var  = new_free ()
webertj@17809
   449
					val thm2 = make_cnfx_disj_thm (betapply (x', var)) y'  (* (x' | y') = body' *)
webertj@17809
   450
					val thm3 = forall_intr (cterm_of thy var) thm2         (* !!v. (x' | y') = body' *)
webertj@17809
   451
					val thm4 = strip_shyps (thm3 COMP allI)                (* ALL v. (x' | y') = body' *)
webertj@17809
   452
					val thm5 = strip_shyps (thm4 RS make_cnfx_ex_cong)     (* (EX v. (x' | y')) = (EX v. body') *)
webertj@17809
   453
				in
webertj@17809
   454
					iff_trans OF [thm1, thm5]  (* ((Ex x') | y') = (Ex v. body') *)
webertj@17809
   455
				end
webertj@17809
   456
			  | make_cnfx_disj_thm x' (Const ("Ex", _) $ y') =
webertj@17809
   457
				let
webertj@17809
   458
					val thm1 = inst_thm thy [x', y'] make_cnfx_disj_ex_r   (* (x' | (Ex y')) = (Ex (x' | y')) *)
webertj@17809
   459
					val var  = new_free ()
webertj@17809
   460
					val thm2 = make_cnfx_disj_thm x' (betapply (y', var))  (* (x' | y') = body' *)
webertj@17809
   461
					val thm3 = forall_intr (cterm_of thy var) thm2         (* !!v. (x' | y') = body' *)
webertj@17809
   462
					val thm4 = strip_shyps (thm3 COMP allI)                (* ALL v. (x' | y') = body' *)
webertj@17809
   463
					val thm5 = strip_shyps (thm4 RS make_cnfx_ex_cong)     (* (EX v. (x' | y')) = (EX v. body') *)
webertj@17809
   464
				in
webertj@17809
   465
					iff_trans OF [thm1, thm5]  (* (x' | (Ex y')) = (EX v. body') *)
webertj@17809
   466
				end
webertj@17809
   467
			  | make_cnfx_disj_thm x' y' =
webertj@17809
   468
				inst_thm thy [HOLogic.mk_disj (x', y')] iff_refl  (* (x' | y') = (x' | y') *)
webertj@17809
   469
			val thm1     = make_cnfx_thm_from_nnf x
webertj@17809
   470
			val thm2     = make_cnfx_thm_from_nnf y
webertj@17809
   471
			val x'       = (snd o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) thm1
webertj@17809
   472
			val y'       = (snd o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) thm2
webertj@17809
   473
			val disj_thm = disj_cong OF [thm1, thm2]  (* (x | y) = (x' | y') *)
webertj@17809
   474
		in
webertj@17809
   475
			iff_trans OF [disj_thm, make_cnfx_disj_thm x' y']
webertj@17809
   476
		end else let  (* neither 'x' nor 'y' is a literal: introduce a fresh variable *)
webertj@17809
   477
			val thm1 = inst_thm thy [x, y] make_cnfx_newlit     (* (x | y) = EX v. (x | v) & (y | ~v) *)
webertj@17809
   478
			val var  = new_free ()
webertj@17809
   479
			val body = HOLogic.mk_conj (HOLogic.mk_disj (x, var), HOLogic.mk_disj (y, HOLogic.Not $ var))
webertj@17809
   480
			val thm2 = make_cnfx_thm_from_nnf body              (* (x | v) & (y | ~v) = body' *)
webertj@17809
   481
			val thm3 = forall_intr (cterm_of thy var) thm2      (* !!v. (x | v) & (y | ~v) = body' *)
webertj@17809
   482
			val thm4 = strip_shyps (thm3 COMP allI)             (* ALL v. (x | v) & (y | ~v) = body' *)
webertj@17809
   483
			val thm5 = strip_shyps (thm4 RS make_cnfx_ex_cong)  (* (EX v. (x | v) & (y | ~v)) = (EX v. body') *)
webertj@17809
   484
		in
webertj@17809
   485
			iff_trans OF [thm1, thm5]
webertj@17809
   486
		end
webertj@17809
   487
	  | make_cnfx_thm_from_nnf t =
webertj@17809
   488
		inst_thm thy [t] iff_refl
webertj@17809
   489
	(* convert 't' to NNF first *)
webertj@17809
   490
	val nnf_thm  = make_nnf_thm thy t
webertj@17809
   491
	val nnf      = (snd o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) nnf_thm
webertj@17809
   492
	(* then simplify wrt. True/False (this should preserve NNF) *)
webertj@17809
   493
	val simp_thm = simp_True_False_thm thy nnf
webertj@17809
   494
	val simp     = (snd o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) simp_thm
webertj@17809
   495
	(* initialize var_id, in case the term already contains variables of the form "cnfx_<int>" *)
webertj@17809
   496
	val _        = (var_id := fold (fn free => fn max =>
webertj@17809
   497
		let
webertj@17809
   498
			val (name, _) = dest_Free free
webertj@17809
   499
			val idx       = if String.isPrefix "cnfx_" name then
webertj@17809
   500
					(Int.fromString o String.extract) (name, String.size "cnfx_", NONE)
webertj@17809
   501
				else
webertj@17809
   502
					NONE
webertj@17809
   503
		in
webertj@17809
   504
			Int.max (max, getOpt (idx, 0))
webertj@17809
   505
		end) (term_frees simp) 0)
webertj@17809
   506
	(* finally, convert to definitional CNF (this should preserve the simplification) *)
webertj@17809
   507
	val cnfx_thm = make_cnfx_thm_from_nnf simp
webertj@17809
   508
in
webertj@17809
   509
	iff_trans OF [iff_trans OF [nnf_thm, simp_thm], cnfx_thm]
webertj@17809
   510
end;
webertj@17618
   511
webertj@17809
   512
(* ------------------------------------------------------------------------- *)
webertj@17809
   513
(*                                  Tactics                                  *)
webertj@17809
   514
(* ------------------------------------------------------------------------- *)
webertj@17618
   515
webertj@17809
   516
(* ------------------------------------------------------------------------- *)
webertj@17809
   517
(* weakening_tac: removes the first hypothesis of the 'i'-th subgoal         *)
webertj@17809
   518
(* ------------------------------------------------------------------------- *)
webertj@17618
   519
webertj@17809
   520
(* int -> Tactical.tactic *)
webertj@17618
   521
webertj@17809
   522
fun weakening_tac i =
webertj@17809
   523
	dtac weakening_thm i THEN atac (i+1);
webertj@17618
   524
webertj@17809
   525
(* ------------------------------------------------------------------------- *)
webertj@17809
   526
(* cnf_rewrite_tac: converts all premises of the 'i'-th subgoal to CNF       *)
webertj@17809
   527
(*      (possibly causing an exponential blowup in the length of each        *)
webertj@17809
   528
(*      premise)                                                             *)
webertj@17809
   529
(* ------------------------------------------------------------------------- *)
webertj@17618
   530
webertj@17809
   531
(* int -> Tactical.tactic *)
webertj@17618
   532
webertj@17809
   533
fun cnf_rewrite_tac i =
webertj@17809
   534
	(* cut the CNF formulas as new premises *)
webertj@17809
   535
	METAHYPS (fn prems =>
webertj@17809
   536
		let
webertj@17809
   537
			val cnf_thms = map (fn pr => make_cnf_thm (theory_of_thm pr) ((HOLogic.dest_Trueprop o prop_of) pr)) prems
webertj@17809
   538
			val cut_thms = map (fn (th, pr) => cnftac_eq_imp OF [th, pr]) (cnf_thms ~~ prems)
webertj@17809
   539
		in
webertj@17809
   540
			cut_facts_tac cut_thms 1
webertj@17809
   541
		end) i
webertj@17809
   542
	(* remove the original premises *)
webertj@17809
   543
	THEN SELECT_GOAL (fn thm =>
webertj@17809
   544
		let
webertj@17809
   545
			val n = Logic.count_prems ((Term.strip_all_body o fst o Logic.dest_implies o prop_of) thm, 0)
webertj@17809
   546
		in
webertj@17809
   547
			PRIMITIVE (funpow (n div 2) (Seq.hd o weakening_tac 1)) thm
webertj@17809
   548
		end) i;
webertj@17618
   549
webertj@17809
   550
(* ------------------------------------------------------------------------- *)
webertj@17809
   551
(* cnfx_rewrite_tac: converts all premises of the 'i'-th subgoal to CNF      *)
webertj@17809
   552
(*      (possibly introducing new literals)                                  *)
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(* ------------------------------------------------------------------------- *)
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(* int -> Tactical.tactic *)
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fun cnfx_rewrite_tac i =
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	(* cut the CNF formulas as new premises *)
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	METAHYPS (fn prems =>
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		let
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			val cnfx_thms = map (fn pr => make_cnfx_thm (theory_of_thm pr) ((HOLogic.dest_Trueprop o prop_of) pr)) prems
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			val cut_thms  = map (fn (th, pr) => cnftac_eq_imp OF [th, pr]) (cnfx_thms ~~ prems)
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		in
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			cut_facts_tac cut_thms 1
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		end) i
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	(* remove the original premises *)
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	THEN SELECT_GOAL (fn thm =>
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		let
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			val n = Logic.count_prems ((Term.strip_all_body o fst o Logic.dest_implies o prop_of) thm, 0)
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		in
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			PRIMITIVE (funpow (n div 2) (Seq.hd o weakening_tac 1)) thm
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		end) i;
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end;  (* of structure *)