(*  Title:      HOL/Tools/set_comprehension_pointfree.ML

     Author:     Felix Kuperjans, Lukas Bulwahn, TU Muenchen

     Author:     Rafal Kolanski, NICTA

 Simproc for rewriting set comprehensions to pointfree expressions.

 *)

 signature SET_COMPREHENSION_POINTFREE =

 sig

   val base_simproc : simpset -> cterm -> thm option

   val code_simproc : simpset -> cterm -> thm option

   val simproc : simpset -> cterm -> thm option

 end

 structure Set_Comprehension_Pointfree : SET_COMPREHENSION_POINTFREE =

 struct

 (* syntactic operations *)

 fun mk_inf (t1, t2) =

   let

     val T = fastype_of t1

   in

     Const (@{const_name Lattices.inf_class.inf}, T --> T --> T) $ t1 $ t2

   end

 fun mk_sup (t1, t2) =

   let

     val T = fastype_of t1

   in

     Const (@{const_name Lattices.sup_class.sup}, T --> T --> T) $ t1 $ t2

   end

 fun mk_Compl t =

   let

     val T = fastype_of t

   in

     Const (@{const_name "Groups.uminus_class.uminus"}, T --> T) $ t

   end

 fun mk_image t1 t2 =

   let

     val T as Type (@{type_name fun}, [_ , R]) = fastype_of t1

   in

     Const (@{const_name image},

       T --> fastype_of t2 --> HOLogic.mk_setT R) $ t1 $ t2

   end;

 fun mk_sigma (t1, t2) =

   let

     val T1 = fastype_of t1

     val T2 = fastype_of t2

     val setT = HOLogic.dest_setT T1

     val resT = HOLogic.mk_setT (HOLogic.mk_prodT (setT, HOLogic.dest_setT T2))

   in

     Const (@{const_name Sigma},

       T1 --> (setT --> T2) --> resT) $ t1 $ absdummy setT t2

   end;

 fun dest_Collect (Const (@{const_name Collect}, _) $ Abs (_, _, t)) = t

   | dest_Collect t = raise TERM ("dest_Collect", [t])

 (* Copied from predicate_compile_aux.ML *)

 fun strip_ex (Const (@{const_name Ex}, _) $ Abs (x, T, t)) =

   let

     val (xTs, t') = strip_ex t

   in

     ((x, T) :: xTs, t')

   end

   | strip_ex t = ([], t)

 fun mk_prod1 Ts (t1, t2) =

   let

     val (T1, T2) = pairself (curry fastype_of1 Ts) (t1, t2)

   in

     HOLogic.pair_const T1 T2 $ t1 $ t2

   end;

 (* patterns *)

 datatype pattern = TBound of int | TPair of pattern * pattern;

 fun mk_pattern (Bound n) = TBound n

   | mk_pattern (Const (@{const_name "Product_Type.Pair"}, _) $ l $ r) =

       TPair (mk_pattern l, mk_pattern r)

   | mk_pattern t = raise TERM ("mk_pattern: only bound variable tuples currently supported", [t]);

 fun type_of_pattern Ts (TBound n) = nth Ts n

   | type_of_pattern Ts (TPair (l, r)) = HOLogic.mk_prodT (type_of_pattern Ts l, type_of_pattern Ts r)

 fun term_of_pattern _ (TBound n) = Bound n

   | term_of_pattern Ts (TPair (l, r)) =

     let

       val (lt, rt) = pairself (term_of_pattern Ts) (l, r)

       val (lT, rT) = pairself (curry fastype_of1 Ts) (lt, rt) 

     in

       HOLogic.pair_const lT rT $ lt $ rt

     end;

 fun bounds_of_pattern (TBound i) = [i]

   | bounds_of_pattern (TPair (l, r)) = union (op =) (bounds_of_pattern l) (bounds_of_pattern r)

 (* formulas *)

 datatype formula = Atom of (pattern * term) | Int of formula * formula | Un of formula * formula

 fun mk_atom (Const (@{const_name "Set.member"}, _) $ x $ s) = (mk_pattern x, Atom (mk_pattern x, s))

   | mk_atom (Const (@{const_name "HOL.Not"}, _) $ (Const (@{const_name "Set.member"}, _) $ x $ s)) =

       (mk_pattern x, Atom (mk_pattern x, mk_Compl s))

 fun can_merge (pats1, pats2) =

   let

     fun check pat1 pat2 = (pat1 = pat2)

       orelse (inter (op =) (bounds_of_pattern pat1) (bounds_of_pattern pat2) = [])

   in

     forall (fn pat1 => forall (fn pat2 => check pat1 pat2) pats2) pats1 

   end

 fun merge_patterns (pats1, pats2) =

   if can_merge (pats1, pats2) then

     union (op =) pats1 pats2

   else raise Fail "merge_patterns: variable groups overlap"

 fun merge oper (pats1, sp1) (pats2, sp2) = (merge_patterns (pats1, pats2), oper (sp1, sp2))

 fun mk_formula (@{const HOL.conj} $ t1 $ t2) = merge Int (mk_formula t1) (mk_formula t2)

   | mk_formula (@{const HOL.disj} $ t1 $ t2) = merge Un (mk_formula t1) (mk_formula t2)

   | mk_formula t = apfst single (mk_atom t)

 fun strip_Int (Int (fm1, fm2)) = fm1 :: (strip_Int fm2) 

   | strip_Int fm = [fm]

 (* term construction *)

 fun reorder_bounds pats t =

   let

     val bounds = maps bounds_of_pattern pats

     val bperm = bounds ~~ ((length bounds - 1) downto 0)

       |> sort (fn (i,j) => int_ord (fst i, fst j)) |> map snd

   in

     subst_bounds (map Bound bperm, t)

   end;

 fun mk_split_abs vs (Bound i) t = let val (x, T) = nth vs i in Abs (x, T, t) end

   | mk_split_abs vs (Const ("Product_Type.Pair", _) $ u $ v) t =

       HOLogic.mk_split (mk_split_abs vs u (mk_split_abs vs v t))

   | mk_split_abs _ t _ = raise TERM ("mk_split_abs: bad term", [t]);

 fun mk_pointfree_expr t =

   let

     val (vs, t'') = strip_ex (dest_Collect t)

     val Ts = map snd (rev vs)

     fun mk_mem_UNIV n = HOLogic.mk_mem (Bound n, HOLogic.mk_UNIV (nth Ts n))

     fun lookup (pat', t) pat = if pat = pat' then t else HOLogic.mk_UNIV (type_of_pattern Ts pat)

     val conjs = HOLogic.dest_conj t''

     val is_the_eq =

       the_default false o (try (fn eq => fst (HOLogic.dest_eq eq) = Bound (length vs)))

     val SOME eq = find_first is_the_eq conjs

     val f = snd (HOLogic.dest_eq eq)

     val conjs' = filter_out (fn t => eq = t) conjs

     val unused_bounds = subtract (op =) (distinct (op =) (maps loose_bnos conjs'))

       (0 upto (length vs - 1))

     val (pats, fm) =

       mk_formula (foldr1 HOLogic.mk_conj (conjs' @ map mk_mem_UNIV unused_bounds))

     fun mk_set (Atom pt) = (case map (lookup pt) pats of [t'] => t' | ts => foldr1 mk_sigma ts)

       | mk_set (Un (f1, f2)) = mk_sup (mk_set f1, mk_set f2)

       | mk_set (Int (f1, f2)) = mk_inf (mk_set f1, mk_set f2)

     val pat = foldr1 (mk_prod1 Ts) (map (term_of_pattern Ts) pats)

     val t = mk_split_abs (rev vs) pat (reorder_bounds pats f)

   in

     (fm, mk_image t (mk_set fm))

   end;

 val rewrite_term = try mk_pointfree_expr

 (* proof tactic *)

 val prod_case_distrib = @{lemma "(prod_case g x) z = prod_case (% x y. (g x y) z) x" by (simp add: prod_case_beta)}

 (* FIXME: one of many clones *)

 fun Trueprop_conv cv ct =

   (case Thm.term_of ct of

     Const (@{const_name Trueprop}, _) $ _ => Conv.arg_conv cv ct

   | _ => raise CTERM ("Trueprop_conv", [ct]))

 (* FIXME: another clone *)

 fun eq_conv cv1 cv2 ct =

   (case Thm.term_of ct of

     Const (@{const_name HOL.eq}, _) $ _ $ _ => Conv.combination_conv (Conv.arg_conv cv1) cv2 ct

   | _ => raise CTERM ("eq_conv", [ct]))

 val elim_Collect_tac = dtac @{thm iffD1[OF mem_Collect_eq]}

   THEN' (REPEAT_DETERM o (eresolve_tac @{thms exE}))

   THEN' TRY o etac @{thm conjE}

   THEN' hyp_subst_tac;

 fun intro_image_tac ctxt = rtac @{thm image_eqI}

     THEN' (REPEAT_DETERM1 o

       (rtac @{thm refl}

       ORELSE' rtac

         @{thm arg_cong2[OF refl, where f="op =", OF prod.cases, THEN iffD2]}

       ORELSE' CONVERSION (Conv.params_conv ~1 (K (Conv.concl_conv ~1

         (Trueprop_conv (eq_conv Conv.all_conv (Conv.rewr_conv (mk_meta_eq prod_case_distrib)))))) ctxt)))

 val elim_image_tac = etac @{thm imageE}

   THEN' (TRY o REPEAT_DETERM1 o Splitter.split_asm_tac @{thms prod.split_asm})

   THEN' hyp_subst_tac

 fun tac1_of_formula (Int (fm1, fm2)) =

     TRY o etac @{thm conjE}

     THEN' rtac @{thm IntI}

     THEN' (fn i => tac1_of_formula fm2 (i + 1))

     THEN' tac1_of_formula fm1

   | tac1_of_formula (Un (fm1, fm2)) =

     etac @{thm disjE} THEN' rtac @{thm UnI1}

     THEN' tac1_of_formula fm1

     THEN' rtac @{thm UnI2}

     THEN' tac1_of_formula fm2

   | tac1_of_formula (Atom _) =

     (REPEAT_DETERM1 o (rtac @{thm SigmaI}

       ORELSE' rtac @{thm UNIV_I}

       ORELSE' rtac @{thm iffD2[OF Compl_iff]}

       ORELSE' atac))

 fun tac2_of_formula (Int (fm1, fm2)) =

     TRY o etac @{thm IntE}

     THEN' TRY o rtac @{thm conjI}

     THEN' (fn i => tac2_of_formula fm2 (i + 1))

     THEN' tac2_of_formula fm1

   | tac2_of_formula (Un (fm1, fm2)) =

     etac @{thm UnE} THEN' rtac @{thm disjI1}

     THEN' tac2_of_formula fm1

     THEN' rtac @{thm disjI2}

     THEN' tac2_of_formula fm2

   | tac2_of_formula (Atom _) =

     TRY o REPEAT_DETERM1 o

       (dtac @{thm iffD1[OF mem_Sigma_iff]}

        ORELSE' etac @{thm conjE}

        ORELSE' etac @{thm ComplE}

        ORELSE' atac)

 fun tac ctxt fm =

   let

     val subset_tac1 = rtac @{thm subsetI}

       THEN' elim_Collect_tac

       THEN' (intro_image_tac ctxt)

       THEN' (tac1_of_formula fm)

     val subset_tac2 = rtac @{thm subsetI}

       THEN' elim_image_tac

       THEN' rtac @{thm iffD2[OF mem_Collect_eq]}

       THEN' REPEAT_DETERM1 o resolve_tac @{thms exI}

       THEN' (TRY o REPEAT_ALL_NEW (rtac @{thm conjI}))

       THEN' (K (SOMEGOAL ((TRY o hyp_subst_tac) THEN' rtac @{thm refl})))

       THEN' (fn i => EVERY (rev (map_index (fn (j, f) =>

         REPEAT_DETERM (etac @{thm IntE} (i + j)) THEN tac2_of_formula f (i + j)) (strip_Int fm))))

   in

     rtac @{thm subset_antisym} THEN' subset_tac1 THEN' subset_tac2

   end;

 (* main simprocs *)

 val post_thms =

   map mk_meta_eq [@{thm Times_Un_distrib1[symmetric]},

   @{lemma "A \<times> B \<union> A \<times> C = A \<times> (B \<union> C)" by auto},

   @{lemma "(A \<times> B \<inter> C \<times> D) = (A \<inter> C) \<times> (B \<inter> D)" by auto}]

 fun conv ctxt t =

   let

     val ct = cterm_of (Proof_Context.theory_of ctxt) t

     val Bex_def = mk_meta_eq @{thm Bex_def}

     val unfold_eq = Conv.bottom_conv (K (Conv.try_conv (Conv.rewr_conv Bex_def))) ctxt ct

     val t' = term_of (Thm.rhs_of unfold_eq)

     fun mk_thm (fm, t'') = Goal.prove ctxt [] []

       (HOLogic.mk_Trueprop (HOLogic.mk_eq (t', t''))) (fn {context, ...} => tac context fm 1)

     fun unfold th = th RS ((unfold_eq RS meta_eq_to_obj_eq) RS @{thm trans})

     fun post th = Conv.fconv_rule (Trueprop_conv (eq_conv Conv.all_conv

       (Raw_Simplifier.rewrite true post_thms))) th

   in

     Option.map (post o unfold o mk_thm) (rewrite_term t')

   end;

 fun base_simproc ss redex =

   let

     val ctxt = Simplifier.the_context ss

     val set_compr = term_of redex

   in

     conv ctxt set_compr

     |> Option.map (fn thm => thm RS @{thm eq_reflection})

   end;

 fun instantiate_arg_cong ctxt pred =

   let

     val certify = cterm_of (Proof_Context.theory_of ctxt)

     val arg_cong = Thm.incr_indexes (maxidx_of_term pred + 1) @{thm arg_cong}

     val f $ _ = fst (HOLogic.dest_eq (HOLogic.dest_Trueprop (concl_of arg_cong)))

   in

     cterm_instantiate [(certify f, certify pred)] arg_cong

   end;

 fun simproc ss redex =

   let

     val ctxt = Simplifier.the_context ss

     val pred $ set_compr = term_of redex

     val arg_cong' = instantiate_arg_cong ctxt pred

   in

     conv ctxt set_compr

     |> Option.map (fn thm => thm RS arg_cong' RS @{thm eq_reflection})

   end;

 fun code_simproc ss redex =

   let

     val prep_thm = Raw_Simplifier.rewrite false @{thms eq_equal[symmetric]} redex

   in

     case base_simproc ss (Thm.rhs_of prep_thm) of

       SOME rewr_thm => SOME (transitive_thm OF [transitive_thm OF [prep_thm, rewr_thm],

         Raw_Simplifier.rewrite false @{thms eq_equal} (Thm.rhs_of rewr_thm)])

     | NONE => NONE

   end;

 end;