src/HOL/Tools/Nitpick/nitpick_peephole.ML
author blanchet
Thu Oct 22 14:51:47 2009 +0200 (2009-10-22)
changeset 33192 08a39a957ed7
child 33232 f93390060bbe
permissions -rw-r--r--
added Nitpick's theory and ML files to Isabelle/HOL;
the examples and the documentation are on their way.
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(*  Title:      HOL/Nitpick/Tools/nitpick_peephole.ML
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    Author:     Jasmin Blanchette, TU Muenchen
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    Copyright   2008, 2009
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Peephole optimizer for Nitpick.
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*)
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signature NITPICK_PEEPHOLE =
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sig
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  type formula = Kodkod.formula
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  type int_expr = Kodkod.int_expr
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  type rel_expr = Kodkod.rel_expr
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  type decl = Kodkod.decl
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  type expr_assign = Kodkod.expr_assign
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  type name_pool = {
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    rels: Kodkod.n_ary_index list,
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    vars: Kodkod.n_ary_index list,
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    formula_reg: int,
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    rel_reg: int}
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  val initial_pool : name_pool
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  val not3_rel : rel_expr
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  val suc_rel : rel_expr
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  val nat_add_rel : rel_expr
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  val int_add_rel : rel_expr
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  val nat_subtract_rel : rel_expr
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  val int_subtract_rel : rel_expr
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  val nat_multiply_rel : rel_expr
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  val int_multiply_rel : rel_expr
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  val nat_divide_rel : rel_expr
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  val int_divide_rel : rel_expr
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  val nat_modulo_rel : rel_expr
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  val int_modulo_rel : rel_expr
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  val nat_less_rel : rel_expr
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  val int_less_rel : rel_expr
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  val gcd_rel : rel_expr
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  val lcm_rel : rel_expr
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  val norm_frac_rel : rel_expr
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  val atom_for_bool : int -> bool -> rel_expr
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  val formula_for_bool : bool -> formula
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  val atom_for_nat : int * int -> int -> int
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  val min_int_for_card : int -> int
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  val max_int_for_card : int -> int
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  val int_for_atom : int * int -> int -> int
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  val atom_for_int : int * int -> int -> int
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  val inline_rel_expr : rel_expr -> bool
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  val empty_n_ary_rel : int -> rel_expr
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  val num_seq : int -> int -> int_expr list
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  val s_and : formula -> formula -> formula
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  type kodkod_constrs = {
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    kk_all: decl list -> formula -> formula,
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    kk_exist: decl list -> formula -> formula,
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    kk_formula_let: expr_assign list -> formula -> formula,
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    kk_formula_if: formula -> formula -> formula -> formula,
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    kk_or: formula -> formula -> formula,
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    kk_not: formula -> formula,
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    kk_iff: formula -> formula -> formula,
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    kk_implies: formula -> formula -> formula,
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    kk_and: formula -> formula -> formula,
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    kk_subset: rel_expr -> rel_expr -> formula,
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    kk_rel_eq: rel_expr -> rel_expr -> formula,
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    kk_no: rel_expr -> formula,
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    kk_lone: rel_expr -> formula,
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    kk_one: rel_expr -> formula,
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    kk_some: rel_expr -> formula,
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    kk_rel_let: expr_assign list -> rel_expr -> rel_expr,
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    kk_rel_if: formula -> rel_expr -> rel_expr -> rel_expr,
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    kk_union: rel_expr -> rel_expr -> rel_expr,
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    kk_difference: rel_expr -> rel_expr -> rel_expr,
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    kk_override: rel_expr -> rel_expr -> rel_expr,
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    kk_intersect: rel_expr -> rel_expr -> rel_expr,
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    kk_product: rel_expr -> rel_expr -> rel_expr,
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    kk_join: rel_expr -> rel_expr -> rel_expr,
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    kk_closure: rel_expr -> rel_expr,
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    kk_reflexive_closure: rel_expr -> rel_expr,
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    kk_comprehension: decl list -> formula -> rel_expr,
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    kk_project: rel_expr -> int_expr list -> rel_expr,
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    kk_project_seq: rel_expr -> int -> int -> rel_expr,
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    kk_not3: rel_expr -> rel_expr,
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    kk_nat_less: rel_expr -> rel_expr -> rel_expr,
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    kk_int_less: rel_expr -> rel_expr -> rel_expr
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  }
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  val kodkod_constrs : bool -> int -> int -> int -> kodkod_constrs
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end;
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structure NitpickPeephole : NITPICK_PEEPHOLE =
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struct
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open Kodkod
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open NitpickUtil
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type name_pool = {
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  rels: n_ary_index list,
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  vars: n_ary_index list,
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  formula_reg: int,
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  rel_reg: int}
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(* If you add new built-in relations, make sure to increment the counters here
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   as well to avoid name clashes (which fortunately would be detected by
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   Kodkodi). *)
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val initial_pool =
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  {rels = [(2, 10), (3, 20), (4, 10)], vars = [], formula_reg = 10,
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   rel_reg = 10}
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val not3_rel = Rel (2, 0)
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val suc_rel = Rel (2, 1)
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val nat_add_rel = Rel (3, 0)
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val int_add_rel = Rel (3, 1)
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val nat_subtract_rel = Rel (3, 2)
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val int_subtract_rel = Rel (3, 3)
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val nat_multiply_rel = Rel (3, 4)
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val int_multiply_rel = Rel (3, 5)
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val nat_divide_rel = Rel (3, 6)
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val int_divide_rel = Rel (3, 7)
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val nat_modulo_rel = Rel (3, 8)
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val int_modulo_rel = Rel (3, 9)
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val nat_less_rel = Rel (3, 10)
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val int_less_rel = Rel (3, 11)
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val gcd_rel = Rel (3, 12)
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val lcm_rel = Rel (3, 13)
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val norm_frac_rel = Rel (4, 0)
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(* int -> bool -> rel_expr *)
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fun atom_for_bool j0 = Atom o Integer.add j0 o int_for_bool
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(* bool -> formula *)
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fun formula_for_bool b = if b then True else False
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(* int * int -> int -> int *)
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fun atom_for_nat (k, j0) n = if n < 0 orelse n >= k then ~1 else n + j0
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(* int -> int *)
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fun min_int_for_card k = ~k div 2 + 1
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fun max_int_for_card k = k div 2
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(* int * int -> int -> int *)
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fun int_for_atom (k, j0) j =
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  let val j = j - j0 in if j <= max_int_for_card k then j else j - k end
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fun atom_for_int (k, j0) n =
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  if n < min_int_for_card k orelse n > max_int_for_card k then ~1
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  else if n < 0 then n + k + j0
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  else n + j0
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(* rel_expr -> bool *)
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fun is_none_product (Product (r1, r2)) =
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    is_none_product r1 orelse is_none_product r2
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  | is_none_product None = true
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  | is_none_product _ = false
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(* rel_expr -> bool *)
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fun is_one_rel_expr (Atom _) = true
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  | is_one_rel_expr (AtomSeq (1, _)) = true
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  | is_one_rel_expr (Var _) = true
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  | is_one_rel_expr _ = false
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(* rel_expr -> bool *)
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fun inline_rel_expr (Product (r1, r2)) =
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    inline_rel_expr r1 andalso inline_rel_expr r2
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  | inline_rel_expr Iden = true
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  | inline_rel_expr Ints = true
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  | inline_rel_expr None = true
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  | inline_rel_expr Univ = true
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  | inline_rel_expr (Atom _) = true
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  | inline_rel_expr (AtomSeq _) = true
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  | inline_rel_expr (Rel _) = true
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  | inline_rel_expr (Var _) = true
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  | inline_rel_expr (RelReg _) = true
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  | inline_rel_expr _ = false
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(* rel_expr -> rel_expr -> bool option *)
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fun rel_expr_equal None (Atom _) = SOME false
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  | rel_expr_equal None (AtomSeq (k, _)) = SOME (k = 0)
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  | rel_expr_equal (Atom _) None = SOME false
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  | rel_expr_equal (AtomSeq (k, _)) None = SOME (k = 0)
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  | rel_expr_equal (Atom j1) (Atom j2) = SOME (j1 = j2)
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  | rel_expr_equal (Atom j) (AtomSeq (k, j0)) = SOME (j = j0 andalso k = 1)
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  | rel_expr_equal (AtomSeq (k, j0)) (Atom j) = SOME (j = j0 andalso k = 1)
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  | rel_expr_equal (AtomSeq x1) (AtomSeq x2) = SOME (x1 = x2)
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  | rel_expr_equal r1 r2 = if r1 = r2 then SOME true else NONE
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(* rel_expr -> rel_expr -> bool option *)
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fun rel_expr_intersects (Atom j1) (Atom j2) = SOME (j1 = j2)
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  | rel_expr_intersects (Atom j) (AtomSeq (k, j0)) = SOME (j < j0 + k)
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  | rel_expr_intersects (AtomSeq (k, j0)) (Atom j) = SOME (j < j0 + k)
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  | rel_expr_intersects (AtomSeq (k1, j01)) (AtomSeq (k2, j02)) =
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    SOME (k1 > 0 andalso k2 > 0 andalso j01 + k1 > j02 andalso j02 + k2 > j01)
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  | rel_expr_intersects r1 r2 =
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    if is_none_product r1 orelse is_none_product r2 then SOME false else NONE
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(* int -> rel_expr *)
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fun empty_n_ary_rel 0 = raise ARG ("NitpickPeephole.empty_n_ary_rel", "0")
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  | empty_n_ary_rel n = funpow (n - 1) (curry Product None) None
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(* decl -> rel_expr *)
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fun decl_one_set (DeclOne (_, r)) = r
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  | decl_one_set _ =
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    raise ARG ("NitpickPeephole.decl_one_set", "not \"DeclOne\"")
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(* int_expr -> bool *)
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fun is_Num (Num _) = true
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  | is_Num _ = false
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(* int_expr -> int *)
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fun dest_Num (Num k) = k
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  | dest_Num _ = raise ARG ("NitpickPeephole.dest_Num", "not \"Num\"")
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(* int -> int -> int_expr list *)
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fun num_seq j0 n = map Num (index_seq j0 n)
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(* rel_expr -> rel_expr -> bool *)
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fun occurs_in_union r (Union (r1, r2)) =
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    occurs_in_union r r1 orelse occurs_in_union r r2
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  | occurs_in_union r r' = (r = r')
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(* rel_expr -> rel_expr -> rel_expr *)
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fun s_and True f2 = f2
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  | s_and False _ = False
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  | s_and f1 True = f1
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  | s_and _ False = False
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  | s_and f1 f2 = And (f1, f2)
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type kodkod_constrs = {
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  kk_all: decl list -> formula -> formula,
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  kk_exist: decl list -> formula -> formula,
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  kk_formula_let: expr_assign list -> formula -> formula,
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  kk_formula_if: formula -> formula -> formula -> formula,
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  kk_or: formula -> formula -> formula,
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  kk_not: formula -> formula,
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  kk_iff: formula -> formula -> formula,
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  kk_implies: formula -> formula -> formula,
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  kk_and: formula -> formula -> formula,
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  kk_subset: rel_expr -> rel_expr -> formula,
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  kk_rel_eq: rel_expr -> rel_expr -> formula,
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  kk_no: rel_expr -> formula,
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  kk_lone: rel_expr -> formula,
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  kk_one: rel_expr -> formula,
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  kk_some: rel_expr -> formula,
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  kk_rel_let: expr_assign list -> rel_expr -> rel_expr,
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  kk_rel_if: formula -> rel_expr -> rel_expr -> rel_expr,
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  kk_union: rel_expr -> rel_expr -> rel_expr,
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  kk_difference: rel_expr -> rel_expr -> rel_expr,
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  kk_override: rel_expr -> rel_expr -> rel_expr,
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  kk_intersect: rel_expr -> rel_expr -> rel_expr,
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  kk_product: rel_expr -> rel_expr -> rel_expr,
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  kk_join: rel_expr -> rel_expr -> rel_expr,
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  kk_closure: rel_expr -> rel_expr,
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  kk_reflexive_closure: rel_expr -> rel_expr,
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  kk_comprehension: decl list -> formula -> rel_expr,
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  kk_project: rel_expr -> int_expr list -> rel_expr,
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  kk_project_seq: rel_expr -> int -> int -> rel_expr,
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  kk_not3: rel_expr -> rel_expr,
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  kk_nat_less: rel_expr -> rel_expr -> rel_expr,
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  kk_int_less: rel_expr -> rel_expr -> rel_expr
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}
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(* We assume throughout that Kodkod variables have a "one" constraint. This is
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   always the case if Kodkod's skolemization is disabled. *)
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(* bool -> int -> int -> int -> kodkod_constrs *)
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fun kodkod_constrs optim nat_card int_card main_j0 =
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  let
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    val false_atom = Atom main_j0
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    val true_atom = Atom (main_j0 + 1)
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    (* bool -> int *)
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    val from_bool = atom_for_bool main_j0
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    (* int -> Kodkod.rel_expr *)
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    fun from_nat n = Atom (n + main_j0)
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    val from_int = Atom o atom_for_int (int_card, main_j0)
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    (* int -> int *)
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    fun to_nat j = j - main_j0
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    val to_int = int_for_atom (int_card, main_j0)
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    (* decl list -> formula -> formula *)
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    fun s_all _ True = True
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      | s_all _ False = False
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      | s_all [] f = f
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      | s_all ds (All (ds', f)) = All (ds @ ds', f)
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      | s_all ds f = All (ds, f)
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    fun s_exist _ True = True
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      | s_exist _ False = False
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      | s_exist [] f = f
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      | s_exist ds (Exist (ds', f)) = Exist (ds @ ds', f)
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      | s_exist ds f = Exist (ds, f)
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    (* expr_assign list -> formula -> formula *)
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    fun s_formula_let _ True = True
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      | s_formula_let _ False = False
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      | s_formula_let assigns f = FormulaLet (assigns, f)
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    (* formula -> formula *)
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    fun s_not True = False
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      | s_not False = True
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      | s_not (All (ds, f)) = Exist (ds, s_not f)
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      | s_not (Exist (ds, f)) = All (ds, s_not f)
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      | s_not (Or (f1, f2)) = And (s_not f1, s_not f2)
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      | s_not (Implies (f1, f2)) = And (f1, s_not f2)
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      | s_not (And (f1, f2)) = Or (s_not f1, s_not f2)
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      | s_not (Not f) = f
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      | s_not (No r) = Some r
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      | s_not (Some r) = No r
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      | s_not f = Not f
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    (* formula -> formula -> formula *)
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    fun s_or True _ = True
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      | s_or False f2 = f2
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      | s_or _ True = True
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      | s_or f1 False = f1
blanchet@33192
   306
      | s_or f1 f2 = if f1 = f2 then f1 else Or (f1, f2)
blanchet@33192
   307
    fun s_iff True f2 = f2
blanchet@33192
   308
      | s_iff False f2 = s_not f2
blanchet@33192
   309
      | s_iff f1 True = f1
blanchet@33192
   310
      | s_iff f1 False = s_not f1
blanchet@33192
   311
      | s_iff f1 f2 = if f1 = f2 then True else Iff (f1, f2)
blanchet@33192
   312
    fun s_implies True f2 = f2
blanchet@33192
   313
      | s_implies False _ = True
blanchet@33192
   314
      | s_implies _ True = True
blanchet@33192
   315
      | s_implies f1 False = s_not f1
blanchet@33192
   316
      | s_implies f1 f2 = if f1 = f2 then True else Implies (f1, f2)
blanchet@33192
   317
blanchet@33192
   318
    (* formula -> formula -> formula -> formula *)
blanchet@33192
   319
    fun s_formula_if True f2 _ = f2
blanchet@33192
   320
      | s_formula_if False _ f3 = f3
blanchet@33192
   321
      | s_formula_if f1 True f3 = s_or f1 f3
blanchet@33192
   322
      | s_formula_if f1 False f3 = s_and (s_not f1) f3
blanchet@33192
   323
      | s_formula_if f1 f2 True = s_implies f1 f2
blanchet@33192
   324
      | s_formula_if f1 f2 False = s_and f1 f2
blanchet@33192
   325
      | s_formula_if f f1 f2 = FormulaIf (f, f1, f2)
blanchet@33192
   326
blanchet@33192
   327
    (* rel_expr -> int_expr list -> rel_expr *)
blanchet@33192
   328
    fun s_project r is =
blanchet@33192
   329
      (case r of
blanchet@33192
   330
         Project (r1, is') =>
blanchet@33192
   331
         if forall is_Num is then
blanchet@33192
   332
           s_project r1 (map (nth is' o dest_Num) is)
blanchet@33192
   333
         else
blanchet@33192
   334
           raise SAME ()
blanchet@33192
   335
       | _ => raise SAME ())
blanchet@33192
   336
      handle SAME () =>
blanchet@33192
   337
             let val n = length is in
blanchet@33192
   338
               if arity_of_rel_expr r = n andalso is = num_seq 0 n then r
blanchet@33192
   339
               else Project (r, is)
blanchet@33192
   340
             end
blanchet@33192
   341
blanchet@33192
   342
    (* rel_expr -> formula *)
blanchet@33192
   343
    fun s_no None = True
blanchet@33192
   344
      | s_no (Product (r1, r2)) = s_or (s_no r1) (s_no r2)
blanchet@33192
   345
      | s_no (Intersect (Closure (Kodkod.Rel x), Kodkod.Iden)) = Acyclic x
blanchet@33192
   346
      | s_no r = if is_one_rel_expr r then False else No r
blanchet@33192
   347
    fun s_lone None = True
blanchet@33192
   348
      | s_lone r = if is_one_rel_expr r then True else Lone r
blanchet@33192
   349
    fun s_one None = False
blanchet@33192
   350
      | s_one r =
blanchet@33192
   351
        if is_one_rel_expr r then
blanchet@33192
   352
          True
blanchet@33192
   353
        else if inline_rel_expr r then
blanchet@33192
   354
          case arity_of_rel_expr r of
blanchet@33192
   355
            1 => One r
blanchet@33192
   356
          | arity => foldl1 And (map (One o s_project r o single o Num)
blanchet@33192
   357
                                     (index_seq 0 arity))
blanchet@33192
   358
        else
blanchet@33192
   359
          One r
blanchet@33192
   360
    fun s_some None = False
blanchet@33192
   361
      | s_some (Atom _) = True
blanchet@33192
   362
      | s_some (Product (r1, r2)) = s_and (s_some r1) (s_some r2)
blanchet@33192
   363
      | s_some r = if is_one_rel_expr r then True else Some r
blanchet@33192
   364
blanchet@33192
   365
    (* rel_expr -> rel_expr *)
blanchet@33192
   366
    fun s_not3 (Atom j) = Atom (if j = main_j0 then j + 1 else j - 1)
blanchet@33192
   367
      | s_not3 (r as Join (r1, r2)) =
blanchet@33192
   368
        if r2 = not3_rel then r1 else Join (r, not3_rel)
blanchet@33192
   369
      | s_not3 r = Join (r, not3_rel)
blanchet@33192
   370
blanchet@33192
   371
    (* rel_expr -> rel_expr -> formula *)
blanchet@33192
   372
    fun s_rel_eq r1 r2 =
blanchet@33192
   373
      (case (r1, r2) of
blanchet@33192
   374
         (Join (r11, r12), _) =>
blanchet@33192
   375
         if r12 = not3_rel then s_rel_eq r11 (s_not3 r2) else raise SAME ()
blanchet@33192
   376
       | (_, Join (r21, r22)) =>
blanchet@33192
   377
         if r22 = not3_rel then s_rel_eq r21 (s_not3 r1) else raise SAME ()
blanchet@33192
   378
       | _ => raise SAME ())
blanchet@33192
   379
      handle SAME () =>
blanchet@33192
   380
             case rel_expr_equal r1 r2 of
blanchet@33192
   381
               SOME true => True
blanchet@33192
   382
             | SOME false => False
blanchet@33192
   383
             | NONE =>
blanchet@33192
   384
               case (r1, r2) of
blanchet@33192
   385
                 (_, RelIf (f, r21, r22)) =>
blanchet@33192
   386
                  if inline_rel_expr r1 then
blanchet@33192
   387
                    s_formula_if f (s_rel_eq r1 r21) (s_rel_eq r1 r22)
blanchet@33192
   388
                  else
blanchet@33192
   389
                    RelEq (r1, r2)
blanchet@33192
   390
               | (RelIf (f, r11, r12), _) =>
blanchet@33192
   391
                  if inline_rel_expr r2 then
blanchet@33192
   392
                    s_formula_if f (s_rel_eq r11 r2) (s_rel_eq r12 r2)
blanchet@33192
   393
                  else
blanchet@33192
   394
                    RelEq (r1, r2)
blanchet@33192
   395
               | (_, Kodkod.None) => s_no r1
blanchet@33192
   396
               | (Kodkod.None, _) => s_no r2
blanchet@33192
   397
               | _ => RelEq (r1, r2)
blanchet@33192
   398
    fun s_subset (Atom j1) (Atom j2) = formula_for_bool (j1 = j2)
blanchet@33192
   399
      | s_subset (Atom j) (AtomSeq (k, j0)) =
blanchet@33192
   400
        formula_for_bool (j >= j0 andalso j < j0 + k)
blanchet@33192
   401
      | s_subset (r1 as Union (r11, r12)) r2 =
blanchet@33192
   402
        s_and (s_subset r11 r2) (s_subset r12 r2)
blanchet@33192
   403
      | s_subset r1 (r2 as Union (r21, r22)) =
blanchet@33192
   404
        if is_one_rel_expr r1 then
blanchet@33192
   405
          s_or (s_subset r1 r21) (s_subset r1 r22)
blanchet@33192
   406
        else
blanchet@33192
   407
          if s_subset r1 r21 = True orelse s_subset r1 r22 = True
blanchet@33192
   408
             orelse r1 = r2 then
blanchet@33192
   409
            True
blanchet@33192
   410
          else
blanchet@33192
   411
            Subset (r1, r2)
blanchet@33192
   412
      | s_subset r1 r2 =
blanchet@33192
   413
        if r1 = r2 orelse is_none_product r1 then True
blanchet@33192
   414
        else if is_none_product r2 then s_no r1
blanchet@33192
   415
        else if forall is_one_rel_expr [r1, r2] then s_rel_eq r1 r2
blanchet@33192
   416
        else Subset (r1, r2)
blanchet@33192
   417
blanchet@33192
   418
    (* expr_assign list -> rel_expr -> rel_expr *)
blanchet@33192
   419
    fun s_rel_let [b as AssignRelReg (x', r')] (r as RelReg x) =
blanchet@33192
   420
        if x = x' then r' else RelLet ([b], r)
blanchet@33192
   421
      | s_rel_let bs r = RelLet (bs, r)
blanchet@33192
   422
blanchet@33192
   423
    (* formula -> rel_expr -> rel_expr -> rel_expr *)
blanchet@33192
   424
    fun s_rel_if f r1 r2 =
blanchet@33192
   425
      (case (f, r1, r2) of
blanchet@33192
   426
         (True, _, _) => r1
blanchet@33192
   427
       | (False, _, _) => r2
blanchet@33192
   428
       | (No r1', None, RelIf (One r2', r3', r4')) =>
blanchet@33192
   429
         if r1' = r2' andalso r2' = r3' then s_rel_if (Lone r1') r1' r4'
blanchet@33192
   430
         else raise SAME ()
blanchet@33192
   431
       | _ => raise SAME ())
blanchet@33192
   432
      handle SAME () => if r1 = r2 then r1 else RelIf (f, r1, r2)
blanchet@33192
   433
blanchet@33192
   434
    (* rel_expr -> rel_expr -> rel_expr *)
blanchet@33192
   435
    fun s_union r1 (Union (r21, r22)) = s_union (s_union r1 r21) r22
blanchet@33192
   436
      | s_union r1 r2 =
blanchet@33192
   437
        if is_none_product r1 then r2
blanchet@33192
   438
        else if is_none_product r2 then r1
blanchet@33192
   439
        else if r1 = r2 then r1
blanchet@33192
   440
        else if occurs_in_union r2 r1 then r1
blanchet@33192
   441
        else Union (r1, r2)
blanchet@33192
   442
    fun s_difference r1 r2 =
blanchet@33192
   443
      if is_none_product r1 orelse is_none_product r2 then r1
blanchet@33192
   444
      else if r1 = r2 then empty_n_ary_rel (arity_of_rel_expr r1)
blanchet@33192
   445
      else Difference (r1, r2)
blanchet@33192
   446
    fun s_override r1 r2 =
blanchet@33192
   447
      if is_none_product r2 then r1
blanchet@33192
   448
      else if is_none_product r1 then r2
blanchet@33192
   449
      else Override (r1, r2)
blanchet@33192
   450
    fun s_intersect r1 r2 =
blanchet@33192
   451
      case rel_expr_intersects r1 r2 of
blanchet@33192
   452
        SOME true => if r1 = r2 then r1 else Intersect (r1, r2)
blanchet@33192
   453
      | SOME false => empty_n_ary_rel (arity_of_rel_expr r1)
blanchet@33192
   454
      | NONE => if is_none_product r1 then r1
blanchet@33192
   455
                else if is_none_product r2 then r2
blanchet@33192
   456
                else Intersect (r1, r2)
blanchet@33192
   457
    fun s_product r1 r2 =
blanchet@33192
   458
      if is_none_product r1 then
blanchet@33192
   459
        Product (r1, empty_n_ary_rel (arity_of_rel_expr r2))
blanchet@33192
   460
      else if is_none_product r2 then
blanchet@33192
   461
        Product (empty_n_ary_rel (arity_of_rel_expr r1), r2)
blanchet@33192
   462
      else
blanchet@33192
   463
        Product (r1, r2)
blanchet@33192
   464
    fun s_join r1 (Product (Product (r211, r212), r22)) =
blanchet@33192
   465
        Product (s_join r1 (Product (r211, r212)), r22)
blanchet@33192
   466
      | s_join (Product (r11, Product (r121, r122))) r2 =
blanchet@33192
   467
        Product (r11, s_join (Product (r121, r122)) r2)
blanchet@33192
   468
      | s_join None r = empty_n_ary_rel (arity_of_rel_expr r - 1)
blanchet@33192
   469
      | s_join r None = empty_n_ary_rel (arity_of_rel_expr r - 1)
blanchet@33192
   470
      | s_join (Product (None, None)) r = empty_n_ary_rel (arity_of_rel_expr r)
blanchet@33192
   471
      | s_join r (Product (None, None)) = empty_n_ary_rel (arity_of_rel_expr r)
blanchet@33192
   472
      | s_join Iden r2 = r2
blanchet@33192
   473
      | s_join r1 Iden = r1
blanchet@33192
   474
      | s_join (Product (r1, r2)) Univ =
blanchet@33192
   475
        if arity_of_rel_expr r2 = 1 then r1
blanchet@33192
   476
        else Product (r1, s_join r2 Univ)
blanchet@33192
   477
      | s_join Univ (Product (r1, r2)) =
blanchet@33192
   478
        if arity_of_rel_expr r1 = 1 then r2
blanchet@33192
   479
        else Product (s_join Univ r1, r2)
blanchet@33192
   480
      | s_join r1 (r2 as Product (r21, r22)) =
blanchet@33192
   481
        if arity_of_rel_expr r1 = 1 then
blanchet@33192
   482
          case rel_expr_intersects r1 r21 of
blanchet@33192
   483
            SOME true => r22
blanchet@33192
   484
          | SOME false => empty_n_ary_rel (arity_of_rel_expr r2 - 1)
blanchet@33192
   485
          | NONE => Join (r1, r2)
blanchet@33192
   486
        else
blanchet@33192
   487
          Join (r1, r2)
blanchet@33192
   488
      | s_join (r1 as Product (r11, r12)) r2 =
blanchet@33192
   489
        if arity_of_rel_expr r2 = 1 then
blanchet@33192
   490
          case rel_expr_intersects r2 r12 of
blanchet@33192
   491
            SOME true => r11
blanchet@33192
   492
          | SOME false => empty_n_ary_rel (arity_of_rel_expr r1 - 1)
blanchet@33192
   493
          | NONE => Join (r1, r2)
blanchet@33192
   494
        else
blanchet@33192
   495
          Join (r1, r2)
blanchet@33192
   496
      | s_join r1 (r2 as RelIf (f, r21, r22)) =
blanchet@33192
   497
        if inline_rel_expr r1 then s_rel_if f (s_join r1 r21) (s_join r1 r22)
blanchet@33192
   498
        else Join (r1, r2)
blanchet@33192
   499
      | s_join (r1 as RelIf (f, r11, r12)) r2 =
blanchet@33192
   500
        if inline_rel_expr r2 then s_rel_if f (s_join r11 r2) (s_join r12 r2)
blanchet@33192
   501
        else Join (r1, r2)
blanchet@33192
   502
      | s_join (r1 as Atom j1) (r2 as Rel (2, j2)) =
blanchet@33192
   503
        if r2 = suc_rel then
blanchet@33192
   504
          let val n = to_nat j1 + 1 in
blanchet@33192
   505
            if n < nat_card then from_nat n else None
blanchet@33192
   506
          end
blanchet@33192
   507
        else
blanchet@33192
   508
          Join (r1, r2)
blanchet@33192
   509
      | s_join r1 (r2 as Project (r21, Num k :: is)) =
blanchet@33192
   510
        if k = arity_of_rel_expr r21 - 1 andalso arity_of_rel_expr r1 = 1 then
blanchet@33192
   511
          s_project (s_join r21 r1) is
blanchet@33192
   512
        else
blanchet@33192
   513
          Join (r1, r2)
blanchet@33192
   514
      | s_join r1 (Join (r21, r22 as Rel (3, j22))) =
blanchet@33192
   515
        ((if r22 = nat_add_rel then
blanchet@33192
   516
            case (r21, r1) of
blanchet@33192
   517
              (Atom j1, Atom j2) =>
blanchet@33192
   518
              let val n = to_nat j1 + to_nat j2 in
blanchet@33192
   519
                if n < nat_card then from_nat n else None
blanchet@33192
   520
              end
blanchet@33192
   521
            | (Atom j, r) =>
blanchet@33192
   522
              (case to_nat j of
blanchet@33192
   523
                 0 => r
blanchet@33192
   524
               | 1 => s_join r suc_rel
blanchet@33192
   525
               | _ => raise SAME ())
blanchet@33192
   526
            | (r, Atom j) =>
blanchet@33192
   527
              (case to_nat j of
blanchet@33192
   528
                 0 => r
blanchet@33192
   529
               | 1 => s_join r suc_rel
blanchet@33192
   530
               | _ => raise SAME ())
blanchet@33192
   531
            | _ => raise SAME ()
blanchet@33192
   532
          else if r22 = nat_subtract_rel then
blanchet@33192
   533
            case (r21, r1) of
blanchet@33192
   534
              (Atom j1, Atom j2) => from_nat (to_nat j1 nat_minus to_nat j2)
blanchet@33192
   535
            | _ => raise SAME ()
blanchet@33192
   536
          else if r22 = nat_multiply_rel then
blanchet@33192
   537
            case (r21, r1) of
blanchet@33192
   538
              (Atom j1, Atom j2) =>
blanchet@33192
   539
              let val n = to_nat j1 * to_nat j2 in
blanchet@33192
   540
                if n < nat_card then from_nat n else None
blanchet@33192
   541
              end
blanchet@33192
   542
            | (Atom j, r) =>
blanchet@33192
   543
              (case to_nat j of 0 => Atom j | 1 => r | _ => raise SAME ())
blanchet@33192
   544
            | (r, Atom j) =>
blanchet@33192
   545
              (case to_nat j of 0 => Atom j | 1 => r | _ => raise SAME ())
blanchet@33192
   546
            | _ => raise SAME ()
blanchet@33192
   547
          else
blanchet@33192
   548
            raise SAME ())
blanchet@33192
   549
         handle SAME () => List.foldr Join r22 [r1, r21])
blanchet@33192
   550
      | s_join r1 r2 = Join (r1, r2)
blanchet@33192
   551
blanchet@33192
   552
    (* rel_expr -> rel_expr *)
blanchet@33192
   553
    fun s_closure Iden = Iden
blanchet@33192
   554
      | s_closure r = if is_none_product r then r else Closure r
blanchet@33192
   555
    fun s_reflexive_closure Iden = Iden
blanchet@33192
   556
      | s_reflexive_closure r =
blanchet@33192
   557
        if is_none_product r then Iden else ReflexiveClosure r
blanchet@33192
   558
blanchet@33192
   559
    (* decl list -> formula -> rel_expr *)
blanchet@33192
   560
    fun s_comprehension ds False = empty_n_ary_rel (length ds)
blanchet@33192
   561
      | s_comprehension ds True = fold1 s_product (map decl_one_set ds)
blanchet@33192
   562
      | s_comprehension [d as DeclOne ((1, j1), r)]
blanchet@33192
   563
                        (f as RelEq (Var (1, j2), Atom j)) =
blanchet@33192
   564
        if j1 = j2 andalso rel_expr_intersects (Atom j) r = SOME true then
blanchet@33192
   565
          Atom j
blanchet@33192
   566
        else
blanchet@33192
   567
          Comprehension ([d], f)
blanchet@33192
   568
      | s_comprehension ds f = Comprehension (ds, f)
blanchet@33192
   569
blanchet@33192
   570
    (* rel_expr -> int -> int -> rel_expr *)
blanchet@33192
   571
    fun s_project_seq r =
blanchet@33192
   572
      let
blanchet@33192
   573
        (* int -> rel_expr -> int -> int -> rel_expr *)
blanchet@33192
   574
        fun aux arity r j0 n =
blanchet@33192
   575
          if j0 = 0 andalso arity = n then
blanchet@33192
   576
            r
blanchet@33192
   577
          else case r of
blanchet@33192
   578
            RelIf (f, r1, r2) =>
blanchet@33192
   579
            s_rel_if f (aux arity r1 j0 n) (aux arity r2 j0 n)
blanchet@33192
   580
          | Product (r1, r2) =>
blanchet@33192
   581
            let
blanchet@33192
   582
              val arity2 = arity_of_rel_expr r2
blanchet@33192
   583
              val arity1 = arity - arity2
blanchet@33192
   584
              val n1 = Int.min (arity1 nat_minus j0, n)
blanchet@33192
   585
              val n2 = n - n1
blanchet@33192
   586
              (* unit -> rel_expr *)
blanchet@33192
   587
              fun one () = aux arity1 r1 j0 n1
blanchet@33192
   588
              fun two () = aux arity2 r2 (j0 nat_minus arity1) n2
blanchet@33192
   589
            in
blanchet@33192
   590
              case (n1, n2) of
blanchet@33192
   591
                (0, _) => s_rel_if (s_some r1) (two ()) (empty_n_ary_rel n2)
blanchet@33192
   592
              | (_, 0) => s_rel_if (s_some r2) (one ()) (empty_n_ary_rel n1)
blanchet@33192
   593
              | _ => s_product (one ()) (two ())
blanchet@33192
   594
            end
blanchet@33192
   595
          | _ => s_project r (num_seq j0 n)
blanchet@33192
   596
      in aux (arity_of_rel_expr r) r end
blanchet@33192
   597
blanchet@33192
   598
    (* rel_expr -> rel_expr -> rel_expr *)
blanchet@33192
   599
    fun s_nat_subtract r1 r2 = fold s_join [r1, r2] nat_subtract_rel
blanchet@33192
   600
    fun s_nat_less (Atom j1) (Atom j2) = from_bool (j1 < j2)
blanchet@33192
   601
      | s_nat_less r1 r2 = fold s_join [r1, r2] nat_less_rel
blanchet@33192
   602
    fun s_int_less (Atom j1) (Atom j2) = from_bool (to_int j1 < to_int j2)
blanchet@33192
   603
      | s_int_less r1 r2 = fold s_join [r1, r2] int_less_rel
blanchet@33192
   604
blanchet@33192
   605
    (* rel_expr -> int -> int -> rel_expr *)
blanchet@33192
   606
    fun d_project_seq r j0 n = Project (r, num_seq j0 n)
blanchet@33192
   607
    (* rel_expr -> rel_expr *)
blanchet@33192
   608
    fun d_not3 r = Join (r, not3_rel)
blanchet@33192
   609
    (* rel_expr -> rel_expr -> rel_expr *)
blanchet@33192
   610
    fun d_nat_subtract r1 r2 = List.foldl Join nat_subtract_rel [r1, r2]
blanchet@33192
   611
    fun d_nat_less r1 r2 = List.foldl Join nat_less_rel [r1, r2]
blanchet@33192
   612
    fun d_int_less r1 r2 = List.foldl Join int_less_rel [r1, r2]
blanchet@33192
   613
  in
blanchet@33192
   614
    if optim then
blanchet@33192
   615
      {kk_all = s_all, kk_exist = s_exist, kk_formula_let = s_formula_let,
blanchet@33192
   616
       kk_formula_if = s_formula_if, kk_or = s_or, kk_not = s_not,
blanchet@33192
   617
       kk_iff = s_iff, kk_implies = s_implies, kk_and = s_and,
blanchet@33192
   618
       kk_subset = s_subset, kk_rel_eq = s_rel_eq, kk_no = s_no,
blanchet@33192
   619
       kk_lone = s_lone, kk_one = s_one, kk_some = s_some,
blanchet@33192
   620
       kk_rel_let = s_rel_let, kk_rel_if = s_rel_if, kk_union = s_union,
blanchet@33192
   621
       kk_difference = s_difference, kk_override = s_override,
blanchet@33192
   622
       kk_intersect = s_intersect, kk_product = s_product, kk_join = s_join,
blanchet@33192
   623
       kk_closure = s_closure, kk_reflexive_closure = s_reflexive_closure,
blanchet@33192
   624
       kk_comprehension = s_comprehension, kk_project = s_project,
blanchet@33192
   625
       kk_project_seq = s_project_seq, kk_not3 = s_not3,
blanchet@33192
   626
       kk_nat_less = s_nat_less, kk_int_less = s_int_less}
blanchet@33192
   627
    else
blanchet@33192
   628
      {kk_all = curry All, kk_exist = curry Exist,
blanchet@33192
   629
       kk_formula_let = curry FormulaLet, kk_formula_if = curry3 FormulaIf,
blanchet@33192
   630
       kk_or = curry Or,kk_not = Not, kk_iff = curry Iff, kk_implies = curry
blanchet@33192
   631
       Implies, kk_and = curry And, kk_subset = curry Subset, kk_rel_eq = curry
blanchet@33192
   632
       RelEq, kk_no = No, kk_lone = Lone, kk_one = One, kk_some = Some,
blanchet@33192
   633
       kk_rel_let = curry RelLet, kk_rel_if = curry3 RelIf, kk_union = curry
blanchet@33192
   634
       Union, kk_difference = curry Difference, kk_override = curry Override,
blanchet@33192
   635
       kk_intersect = curry Intersect, kk_product = curry Product,
blanchet@33192
   636
       kk_join = curry Join, kk_closure = Closure,
blanchet@33192
   637
       kk_reflexive_closure = ReflexiveClosure, kk_comprehension = curry
blanchet@33192
   638
       Comprehension, kk_project = curry Project,
blanchet@33192
   639
       kk_project_seq = d_project_seq, kk_not3 = d_not3,
blanchet@33192
   640
       kk_nat_less = d_nat_less, kk_int_less = d_int_less}
blanchet@33192
   641
  end
blanchet@33192
   642
blanchet@33192
   643
end;