blanchet@58058: (*  Title:      HOL/Library/Old_SMT/old_z3_proof_literals.ML
boehmes@36898:     Author:     Sascha Boehme, TU Muenchen
boehmes@36898: 
boehmes@36898: Proof tools related to conjunctions and disjunctions.
boehmes@36898: *)
boehmes@36898: 
blanchet@58058: signature OLD_Z3_PROOF_LITERALS =
boehmes@36898: sig
boehmes@41123:   (*literal table*)
boehmes@36898:   type littab = thm Termtab.table
boehmes@36898:   val make_littab: thm list -> littab
boehmes@36898:   val insert_lit: thm -> littab -> littab
boehmes@36898:   val delete_lit: thm -> littab -> littab
boehmes@36898:   val lookup_lit: littab -> term -> thm option
boehmes@36898:   val get_first_lit: (term -> bool) -> littab -> thm option
boehmes@36898: 
boehmes@41123:   (*rules*)
boehmes@36898:   val true_thm: thm
boehmes@36898:   val rewrite_true: thm
boehmes@36898: 
boehmes@41123:   (*properties*)
boehmes@36898:   val is_conj: term -> bool
boehmes@36898:   val is_disj: term -> bool
boehmes@36898:   val exists_lit: bool -> (term -> bool) -> term -> bool
boehmes@40579:   val negate: cterm -> cterm
boehmes@36898: 
boehmes@41123:   (*proof tools*)
boehmes@36898:   val explode: bool -> bool -> bool -> term list -> thm -> thm list
boehmes@36898:   val join: bool -> littab -> term -> thm
boehmes@36898:   val prove_conj_disj_eq: cterm -> thm
boehmes@36898: end
boehmes@36898: 
blanchet@58058: structure Old_Z3_Proof_Literals: OLD_Z3_PROOF_LITERALS =
boehmes@36898: struct
boehmes@36898: 
boehmes@36898: 
boehmes@36898: 
boehmes@41123: (* literal table *)
boehmes@36898: 
boehmes@36898: type littab = thm Termtab.table
boehmes@36898: 
boehmes@41328: fun make_littab thms =
blanchet@58058:   fold (Termtab.update o `Old_SMT_Utils.prop_of) thms Termtab.empty
boehmes@36898: 
blanchet@58058: fun insert_lit thm = Termtab.update (`Old_SMT_Utils.prop_of thm)
blanchet@58058: fun delete_lit thm = Termtab.delete (Old_SMT_Utils.prop_of thm)
boehmes@36898: fun lookup_lit lits = Termtab.lookup lits
boehmes@36898: fun get_first_lit f =
boehmes@36898:   Termtab.get_first (fn (t, thm) => if f t then SOME thm else NONE)
boehmes@36898: 
boehmes@36898: 
boehmes@36898: 
boehmes@41123: (* rules *)
boehmes@36898: 
boehmes@36898: val true_thm = @{lemma "~False" by simp}
boehmes@36898: val rewrite_true = @{lemma "True == ~ False" by simp}
boehmes@36898: 
boehmes@36898: 
boehmes@36898: 
boehmes@41123: (* properties and term operations *)
boehmes@36898: 
boehmes@40579: val is_neg = (fn @{const Not} $ _ => true | _ => false)
boehmes@40579: fun is_neg' f = (fn @{const Not} $ t => f t | _ => false)
boehmes@36898: val is_dneg = is_neg' is_neg
boehmes@40579: val is_conj = (fn @{const HOL.conj} $ _ $ _ => true | _ => false)
boehmes@40579: val is_disj = (fn @{const HOL.disj} $ _ $ _ => true | _ => false)
boehmes@36898: 
boehmes@36898: fun dest_disj_term' f = (fn
boehmes@40579:     @{const Not} $ (@{const HOL.disj} $ t $ u) => SOME (f t, f u)
boehmes@36898:   | _ => NONE)
boehmes@36898: 
boehmes@40579: val dest_conj_term = (fn @{const HOL.conj} $ t $ u => SOME (t, u) | _ => NONE)
boehmes@36898: val dest_disj_term =
boehmes@40579:   dest_disj_term' (fn @{const Not} $ t => t | t => @{const Not} $ t)
boehmes@36898: 
boehmes@36898: fun exists_lit is_conj P =
boehmes@36898:   let
boehmes@36898:     val dest = if is_conj then dest_conj_term else dest_disj_term
boehmes@36898:     fun exists t = P t orelse
boehmes@36898:       (case dest t of
boehmes@36898:         SOME (t1, t2) => exists t1 orelse exists t2
boehmes@36898:       | NONE => false)
boehmes@36898:   in exists end
boehmes@36898: 
wenzelm@46497: val negate = Thm.apply (Thm.cterm_of @{theory} @{const Not})
boehmes@40579: 
boehmes@36898: 
boehmes@36898: 
boehmes@41123: (* proof tools *)
boehmes@36898: 
boehmes@41123: (** explosion of conjunctions and disjunctions **)
boehmes@36898: 
boehmes@36898: local
blanchet@58058:   val precomp = Old_Z3_Proof_Tools.precompose2
boehmes@41328: 
boehmes@36898:   fun destc ct = Thm.dest_binop (Thm.dest_arg ct)
boehmes@41328:   val dest_conj1 = precomp destc @{thm conjunct1}
boehmes@41328:   val dest_conj2 = precomp destc @{thm conjunct2}
boehmes@36898:   fun dest_conj_rules t =
boehmes@36898:     dest_conj_term t |> Option.map (K (dest_conj1, dest_conj2))
boehmes@36898:     
boehmes@36898:   fun destd f ct = f (Thm.dest_binop (Thm.dest_arg (Thm.dest_arg ct)))
boehmes@36898:   val dn1 = apfst Thm.dest_arg and dn2 = apsnd Thm.dest_arg
boehmes@41328:   val dest_disj1 = precomp (destd I) @{lemma "~(P | Q) ==> ~P" by fast}
boehmes@41328:   val dest_disj2 = precomp (destd dn1) @{lemma "~(~P | Q) ==> P" by fast}
boehmes@41328:   val dest_disj3 = precomp (destd I) @{lemma "~(P | Q) ==> ~Q" by fast}
boehmes@41328:   val dest_disj4 = precomp (destd dn2) @{lemma "~(P | ~Q) ==> Q" by fast}
boehmes@36898: 
boehmes@36898:   fun dest_disj_rules t =
boehmes@36898:     (case dest_disj_term' is_neg t of
boehmes@36898:       SOME (true, true) => SOME (dest_disj2, dest_disj4)
boehmes@36898:     | SOME (true, false) => SOME (dest_disj2, dest_disj3)
boehmes@36898:     | SOME (false, true) => SOME (dest_disj1, dest_disj4)
boehmes@36898:     | SOME (false, false) => SOME (dest_disj1, dest_disj3)
boehmes@36898:     | NONE => NONE)
boehmes@36898: 
boehmes@36898:   fun destn ct = [Thm.dest_arg (Thm.dest_arg (Thm.dest_arg ct))]
blanchet@58058:   val dneg_rule = Old_Z3_Proof_Tools.precompose destn @{thm notnotD}
boehmes@36898: in
boehmes@36898: 
boehmes@41123: (*
boehmes@41123:   explode a term into literals and collect all rules to be able to deduce
boehmes@41123:   particular literals afterwards
boehmes@41123: *)
boehmes@36898: fun explode_term is_conj =
boehmes@36898:   let
boehmes@36898:     val dest = if is_conj then dest_conj_term else dest_disj_term
boehmes@36898:     val dest_rules = if is_conj then dest_conj_rules else dest_disj_rules
boehmes@36898: 
boehmes@36898:     fun add (t, rs) = Termtab.map_default (t, rs)
boehmes@36898:       (fn rs' => if length rs' < length rs then rs' else rs)
boehmes@36898: 
boehmes@36898:     fun explode1 rules t =
boehmes@36898:       (case dest t of
boehmes@36898:         SOME (t1, t2) =>
boehmes@36898:           let val (rule1, rule2) = the (dest_rules t)
boehmes@36898:           in
boehmes@36898:             explode1 (rule1 :: rules) t1 #>
boehmes@36898:             explode1 (rule2 :: rules) t2 #>
boehmes@36898:             add (t, rev rules)
boehmes@36898:           end
boehmes@36898:       | NONE => add (t, rev rules))
boehmes@36898: 
boehmes@40579:     fun explode0 (@{const Not} $ (@{const Not} $ t)) =
boehmes@36898:           Termtab.make [(t, [dneg_rule])]
boehmes@36898:       | explode0 t = explode1 [] t Termtab.empty
boehmes@36898: 
boehmes@36898:   in explode0 end
boehmes@36898: 
boehmes@41123: (*
boehmes@41123:   extract a literal by applying previously collected rules
boehmes@41123: *)
blanchet@58058: fun extract_lit thm rules = fold Old_Z3_Proof_Tools.compose rules thm
boehmes@36898: 
boehmes@36898: 
boehmes@41123: (*
boehmes@41123:   explode a theorem into its literals
boehmes@41123: *)
boehmes@36898: fun explode is_conj full keep_intermediate stop_lits =
boehmes@36898:   let
boehmes@36898:     val dest_rules = if is_conj then dest_conj_rules else dest_disj_rules
boehmes@36898:     val tab = fold (Termtab.update o rpair ()) stop_lits Termtab.empty
boehmes@36898: 
boehmes@36898:     fun explode1 thm =
blanchet@58058:       if Termtab.defined tab (Old_SMT_Utils.prop_of thm) then cons thm
boehmes@36898:       else
blanchet@58058:         (case dest_rules (Old_SMT_Utils.prop_of thm) of
boehmes@36898:           SOME (rule1, rule2) =>
boehmes@36898:             explode2 rule1 thm #>
boehmes@36898:             explode2 rule2 thm #>
boehmes@36898:             keep_intermediate ? cons thm
boehmes@36898:         | NONE => cons thm)
boehmes@36898: 
boehmes@36898:     and explode2 dest_rule thm =
boehmes@41328:       if full orelse
blanchet@58058:         exists_lit is_conj (Termtab.defined tab) (Old_SMT_Utils.prop_of thm)
blanchet@58058:       then explode1 (Old_Z3_Proof_Tools.compose dest_rule thm)
blanchet@58058:       else cons (Old_Z3_Proof_Tools.compose dest_rule thm)
boehmes@36898: 
boehmes@36898:     fun explode0 thm =
blanchet@58058:       if not is_conj andalso is_dneg (Old_SMT_Utils.prop_of thm)
blanchet@58058:       then [Old_Z3_Proof_Tools.compose dneg_rule thm]
boehmes@36898:       else explode1 thm []
boehmes@36898: 
boehmes@36898:   in explode0 end
boehmes@36898: 
boehmes@36898: end
boehmes@36898: 
boehmes@36898: 
boehmes@36898: 
boehmes@41123: (** joining of literals to conjunctions or disjunctions **)
boehmes@36898: 
boehmes@36898: local
boehmes@36898:   fun on_cprem i f thm = f (Thm.cprem_of thm i)
boehmes@36898:   fun on_cprop f thm = f (Thm.cprop_of thm)
boehmes@36898:   fun precomp2 f g thm = (on_cprem 1 f thm, on_cprem 2 g thm, f, g, thm)
boehmes@36898:   fun comp2 (cv1, cv2, f, g, rule) thm1 thm2 =
boehmes@36898:     Thm.instantiate ([], [(cv1, on_cprop f thm1), (cv2, on_cprop g thm2)]) rule
blanchet@58058:     |> Old_Z3_Proof_Tools.discharge thm1 |> Old_Z3_Proof_Tools.discharge thm2
boehmes@36898: 
boehmes@36898:   fun d1 ct = Thm.dest_arg ct and d2 ct = Thm.dest_arg (Thm.dest_arg ct)
boehmes@36898: 
boehmes@36898:   val conj_rule = precomp2 d1 d1 @{thm conjI}
boehmes@36898:   fun comp_conj ((_, thm1), (_, thm2)) = comp2 conj_rule thm1 thm2
boehmes@36898: 
boehmes@36898:   val disj1 = precomp2 d2 d2 @{lemma "~P ==> ~Q ==> ~(P | Q)" by fast}
boehmes@36898:   val disj2 = precomp2 d2 d1 @{lemma "~P ==> Q ==> ~(P | ~Q)" by fast}
boehmes@36898:   val disj3 = precomp2 d1 d2 @{lemma "P ==> ~Q ==> ~(~P | Q)" by fast}
boehmes@36898:   val disj4 = precomp2 d1 d1 @{lemma "P ==> Q ==> ~(~P | ~Q)" by fast}
boehmes@36898: 
boehmes@36898:   fun comp_disj ((false, thm1), (false, thm2)) = comp2 disj1 thm1 thm2
boehmes@36898:     | comp_disj ((false, thm1), (true, thm2)) = comp2 disj2 thm1 thm2
boehmes@36898:     | comp_disj ((true, thm1), (false, thm2)) = comp2 disj3 thm1 thm2
boehmes@36898:     | comp_disj ((true, thm1), (true, thm2)) = comp2 disj4 thm1 thm2
boehmes@36898: 
boehmes@40579:   fun dest_conj (@{const HOL.conj} $ t $ u) = ((false, t), (false, u))
boehmes@36898:     | dest_conj t = raise TERM ("dest_conj", [t])
boehmes@36898: 
boehmes@40579:   val neg = (fn @{const Not} $ t => (true, t) | t => (false, @{const Not} $ t))
boehmes@40579:   fun dest_disj (@{const Not} $ (@{const HOL.disj} $ t $ u)) = (neg t, neg u)
boehmes@36898:     | dest_disj t = raise TERM ("dest_disj", [t])
boehmes@36898: 
blanchet@58058:   val precomp = Old_Z3_Proof_Tools.precompose
boehmes@41328:   val dnegE = precomp (single o d2 o d1) @{thm notnotD}
boehmes@41328:   val dnegI = precomp (single o d1) @{lemma "P ==> ~~P" by fast}
boehmes@40579:   fun as_dneg f t = f (@{const Not} $ (@{const Not} $ t))
boehmes@36898: 
blanchet@58058:   val precomp2 = Old_Z3_Proof_Tools.precompose2
boehmes@36898:   fun dni f = apsnd f o Thm.dest_binop o f o d1
boehmes@41328:   val negIffE = precomp2 (dni d1) @{lemma "~(P = (~Q)) ==> Q = P" by fast}
boehmes@41328:   val negIffI = precomp2 (dni I) @{lemma "P = Q ==> ~(Q = (~P))" by fast}
boehmes@40579:   val iff_const = @{const HOL.eq (bool)}
boehmes@40579:   fun as_negIff f (@{const HOL.eq (bool)} $ t $ u) =
boehmes@40579:         f (@{const Not} $ (iff_const $ u $ (@{const Not} $ t)))
boehmes@36898:     | as_negIff _ _ = NONE
boehmes@36898: in
boehmes@36898: 
boehmes@36898: fun join is_conj littab t =
boehmes@36898:   let
boehmes@36898:     val comp = if is_conj then comp_conj else comp_disj
boehmes@36898:     val dest = if is_conj then dest_conj else dest_disj
boehmes@36898: 
boehmes@36898:     val lookup = lookup_lit littab
boehmes@36898: 
boehmes@36898:     fun lookup_rule t =
boehmes@36898:       (case t of
boehmes@41328:         @{const Not} $ (@{const Not} $ t) =>
blanchet@58058:           (Old_Z3_Proof_Tools.compose dnegI, lookup t)
boehmes@40579:       | @{const Not} $ (@{const HOL.eq (bool)} $ t $ (@{const Not} $ u)) =>
blanchet@58058:           (Old_Z3_Proof_Tools.compose negIffI, lookup (iff_const $ u $ t))
boehmes@40579:       | @{const Not} $ ((eq as Const (@{const_name HOL.eq}, _)) $ t $ u) =>
boehmes@36898:           let fun rewr lit = lit COMP @{thm not_sym}
boehmes@40579:           in (rewr, lookup (@{const Not} $ (eq $ u $ t))) end
boehmes@36898:       | _ =>
boehmes@36898:           (case as_dneg lookup t of
blanchet@58058:             NONE => (Old_Z3_Proof_Tools.compose negIffE, as_negIff lookup t)
blanchet@58058:           | x => (Old_Z3_Proof_Tools.compose dnegE, x)))
boehmes@36898: 
boehmes@36898:     fun join1 (s, t) =
boehmes@36898:       (case lookup t of
boehmes@36898:         SOME lit => (s, lit)
boehmes@36898:       | NONE => 
boehmes@36898:           (case lookup_rule t of
boehmes@36898:             (rewrite, SOME lit) => (s, rewrite lit)
boehmes@36898:           | (_, NONE) => (s, comp (pairself join1 (dest t)))))
boehmes@36898: 
boehmes@36898:   in snd (join1 (if is_conj then (false, t) else (true, t))) end
boehmes@36898: 
boehmes@36898: end
boehmes@36898: 
boehmes@36898: 
boehmes@36898: 
boehmes@41123: (** proving equality of conjunctions or disjunctions **)
boehmes@36898: 
boehmes@36898: fun iff_intro thm1 thm2 = thm2 COMP (thm1 COMP @{thm iffI})
boehmes@36898: 
boehmes@36898: local
boehmes@36898:   val cp1 = @{lemma "(~P) = (~Q) ==> P = Q" by simp}
nipkow@44890:   val cp2 = @{lemma "(~P) = Q ==> P = (~Q)" by fastforce}
boehmes@36898:   val cp3 = @{lemma "P = (~Q) ==> (~P) = Q" by simp}
boehmes@36898: in
boehmes@40579: fun contrapos1 prove (ct, cu) = prove (negate ct, negate cu) COMP cp1
boehmes@40579: fun contrapos2 prove (ct, cu) = prove (negate ct, Thm.dest_arg cu) COMP cp2
boehmes@40579: fun contrapos3 prove (ct, cu) = prove (Thm.dest_arg ct, negate cu) COMP cp3
boehmes@36898: end
boehmes@36898: 
boehmes@36898: 
boehmes@36898: local
boehmes@36898:   val contra_rule = @{lemma "P ==> ~P ==> False" by (rule notE)}
boehmes@36898:   fun contra_left conj thm =
boehmes@36898:     let
blanchet@58058:       val rules = explode_term conj (Old_SMT_Utils.prop_of thm)
boehmes@36898:       fun contra_lits (t, rs) =
boehmes@36898:         (case t of
boehmes@40579:           @{const Not} $ u => Termtab.lookup rules u |> Option.map (pair rs)
boehmes@36898:         | _ => NONE)
boehmes@36898:     in
boehmes@40579:       (case Termtab.lookup rules @{const False} of
boehmes@36898:         SOME rs => extract_lit thm rs
boehmes@36898:       | NONE =>
boehmes@36898:           the (Termtab.get_first contra_lits rules)
boehmes@36898:           |> pairself (extract_lit thm)
boehmes@36898:           |> (fn (nlit, plit) => nlit COMP (plit COMP contra_rule)))
boehmes@36898:     end
boehmes@36898: 
boehmes@36898:   val falseE_v = Thm.dest_arg (Thm.dest_arg (Thm.cprop_of @{thm FalseE}))
boehmes@36898:   fun contra_right ct = Thm.instantiate ([], [(falseE_v, ct)]) @{thm FalseE}
boehmes@36898: in
boehmes@36898: fun contradict conj ct =
blanchet@58058:   iff_intro (Old_Z3_Proof_Tools.under_assumption (contra_left conj) ct)
boehmes@41328:     (contra_right ct)
boehmes@36898: end
boehmes@36898: 
boehmes@36898: 
boehmes@36898: local
boehmes@36898:   fun prove_eq l r (cl, cr) =
boehmes@36898:     let
boehmes@36898:       fun explode' is_conj = explode is_conj true (l <> r) []
boehmes@36898:       fun make_tab is_conj thm = make_littab (true_thm :: explode' is_conj thm)
boehmes@36898:       fun prove is_conj ct tab = join is_conj tab (Thm.term_of ct)
boehmes@36898: 
blanchet@58058:       val thm1 = Old_Z3_Proof_Tools.under_assumption (prove r cr o make_tab l) cl
blanchet@58058:       val thm2 = Old_Z3_Proof_Tools.under_assumption (prove l cl o make_tab r) cr
boehmes@36898:     in iff_intro thm1 thm2 end
boehmes@36898: 
boehmes@36898:   datatype conj_disj = CONJ | DISJ | NCON | NDIS
boehmes@36898:   fun kind_of t =
boehmes@36898:     if is_conj t then SOME CONJ
boehmes@36898:     else if is_disj t then SOME DISJ
boehmes@36898:     else if is_neg' is_conj t then SOME NCON
boehmes@36898:     else if is_neg' is_disj t then SOME NDIS
boehmes@36898:     else NONE
boehmes@36898: in
boehmes@36898: 
boehmes@36898: fun prove_conj_disj_eq ct =
boehmes@36898:   let val cp as (cl, cr) = Thm.dest_binop (Thm.dest_arg ct)
boehmes@36898:   in
boehmes@36898:     (case (kind_of (Thm.term_of cl), Thm.term_of cr) of
boehmes@40579:       (SOME CONJ, @{const False}) => contradict true cl
boehmes@40579:     | (SOME DISJ, @{const Not} $ @{const False}) =>
boehmes@40579:         contrapos2 (contradict false o fst) cp
boehmes@36898:     | (kl, _) =>
boehmes@36898:         (case (kl, kind_of (Thm.term_of cr)) of
boehmes@36898:           (SOME CONJ, SOME CONJ) => prove_eq true true cp
boehmes@36898:         | (SOME CONJ, SOME NDIS) => prove_eq true false cp
boehmes@36898:         | (SOME CONJ, _) => prove_eq true true cp
boehmes@36898:         | (SOME DISJ, SOME DISJ) => contrapos1 (prove_eq false false) cp
boehmes@36898:         | (SOME DISJ, SOME NCON) => contrapos2 (prove_eq false true) cp
boehmes@36898:         | (SOME DISJ, _) => contrapos1 (prove_eq false false) cp
boehmes@36898:         | (SOME NCON, SOME NCON) => contrapos1 (prove_eq true true) cp
boehmes@36898:         | (SOME NCON, SOME DISJ) => contrapos3 (prove_eq true false) cp
boehmes@36898:         | (SOME NCON, NONE) => contrapos3 (prove_eq true false) cp
boehmes@36898:         | (SOME NDIS, SOME NDIS) => prove_eq false false cp
boehmes@36898:         | (SOME NDIS, SOME CONJ) => prove_eq false true cp
boehmes@36898:         | (SOME NDIS, NONE) => prove_eq false true cp
boehmes@36898:         | _ => raise CTERM ("prove_conj_disj_eq", [ct])))
boehmes@36898:   end
boehmes@36898: 
boehmes@36898: end
boehmes@36898: 
boehmes@36898: end