genltl.cc

// -*- coding: utf-8 -*-
// Copyright (C) 2012, 2013, 2015, 2016, 2017 Laboratoire de Recherche et
// Développement de l'Epita (LRDE).
//
// This file is part of Spot, a model checking library.
//
// Spot is free software; you can redistribute it and/or modify it
// under the terms of the GNU General Public License as published by
// the Free Software Foundation; either version 3 of the License, or
// (at your option) any later version.
//
// Spot is distributed in the hope that it will be useful, but WITHOUT
// ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
// or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public
// License for more details.
//
// You should have received a copy of the GNU General Public License
// along with this program.  If not, see <http://www.gnu.org/licenses/>.

// Families defined here come from the following papers:
//
// @InProceedings{cichon.09.depcos,
//   author = {Jacek Cicho{\'n} and Adam Czubak and Andrzej Jasi{\'n}ski},
//   title = {Minimal {B\"uchi} Automata for Certain Classes of {LTL} Formulas},
//   booktitle = {Proceedings of the Fourth International Conference on
//                Dependability of Computer Systems},
//   pages = {17--24},
//   year = 2009,
//   publisher = {IEEE Computer Society},
// }
//
// @InProceedings{geldenhuys.06.spin,
//   author = {Jaco Geldenhuys and Henri Hansen},
//   title = {Larger Automata and Less Work for LTL Model Checking},
//   booktitle = {Proceedings of the 13th International SPIN Workshop},
//   year = {2006},
//   pages = {53--70},
//   series = {Lecture Notes in Computer Science},
//   volume = {3925},
//   publisher = {Springer}
// }
//
// @InProceedings{gastin.01.cav,
//   author = {Paul Gastin and Denis Oddoux},
//   title = {Fast {LTL} to {B\"u}chi Automata Translation},
//   booktitle        = {Proceedings of the 13th International Conference on
//                   Computer Aided Verification (CAV'01)},
//   pages = {53--65},
//   year = 2001,
//   editor = {G. Berry and H. Comon and A. Finkel},
//   volume = {2102},
//   series = {Lecture Notes in Computer Science},
//   address = {Paris, France},
//   publisher = {Springer-Verlag}
// }
//
// @InProceedings{rozier.07.spin,
//   author = {Kristin Y. Rozier and Moshe Y. Vardi},
//   title = {LTL Satisfiability Checking},
//   booktitle = {Proceedings of the 12th International SPIN Workshop on
//                   Model Checking of Software (SPIN'07)},
//   pages = {149--167},
//   year = {2007},
//   volume = {4595},
//   series = {Lecture Notes in Computer Science},
//   publisher = {Springer-Verlag}
// }
//
// @InProceedings{dwyer.98.fmsp,
//   author = {Matthew B. Dwyer and George S. Avrunin and James C. Corbett},
//   title         = {Property Specification Patterns for Finite-state
//                   Verification},
//   booktitle     = {Proceedings of the 2nd Workshop on Formal Methods in
//                   Software Practice (FMSP'98)},
//   publisher     = {ACM Press},
//   address       = {New York},
//   editor        = {Mark Ardis},
//   month         = mar,
//   year          = {1998},
//   pages         = {7--15}
// }
//
// @InProceedings{etessami.00.concur,
//   author = {Kousha Etessami and Gerard J. Holzmann},
//   title = {Optimizing {B\"u}chi Automata},
//   booktitle = {Proceedings of the 11th International Conference on
//                Concurrency Theory (Concur'00)},
//   pages = {153--167},
//   year = {2000},
//   editor = {C. Palamidessi},
//   volume = {1877},
//   series = {Lecture Notes in Computer Science},
//   address = {Pennsylvania, USA},
//   publisher = {Springer-Verlag}
// }
//
// @InProceedings{somenzi.00.cav,
//   author = {Fabio Somenzi and Roderick Bloem},
//   title = {Efficient {B\"u}chi Automata for {LTL} Formul{\ae}},
//   booktitle = {Proceedings of the 12th International Conference on
//                Computer Aided Verification (CAV'00)},
//   pages = {247--263},
//   year = {2000},
//   volume = {1855},
//   series = {Lecture Notes in Computer Science},
//   address = {Chicago, Illinois, USA},
//   publisher = {Springer-Verlag}
// }
//
// @InProceedings{tabakov.10.rv,
//   author = {Deian Tabakov and Moshe Y. Vardi},
//   title = {Optimized Temporal Monitors for {SystemC}},
//   booktitle = {Proceedings of the 1st International Conference on Runtime
// 		  Verification (RV'10)},
//   pages = {436--451},
//   year = 2010,
//   volume = {6418},
//   series = {Lecture Notes in Computer Science},
//   month = nov,
//   publisher = {Springer}
// }
//
// @InProceedings{kupferman.10.mochart,
//   author = {Orna Kupferman and And Rosenberg},
//   title = {The Blow-Up in Translating LTL do Deterministic Automata},
//   booktitle = {Proceedings of the 6th International Workshop on Model
//                Checking and Artificial Intelligence (MoChArt 2010)},
//   pages = {85--94},
//   year = 2011,
//   volume = {6572},
//   series = {Lecture Notes in Artificial Intelligence},
//   month = nov,
//   publisher = {Springer}
// }
//
// @techreport{holevek.04.tr,
//    title = {Verification Results in {Liberouter} Project},
//    author = {J. Hole\v{c}ek and T. Kratochv\'ila and V. \v{R}eh\'ak
//              and D. \v{S}afr\'anek and P. \v{S}ime\v{c}ek},
//    month = {September},
//    year = 2004,
//    number = 03,
//    institution = {CESNET}
// }
//
// @InProceedings{pelanek.07.spin,
//   author = {Radek Pel\'{a}nek},
//   title = {{BEEM}: benchmarks for explicit model checkers},
//   booktitle = {Proceedings of the 14th international SPIN conference on
//     Model checking software},
//   year = 2007,
//   pages = {263--267},
//   numpages = {5},
//   volume = {4595},
//   series = {Lecture Notes in Computer Science},
//   publisher = {Springer-Verlag}
// }


#include "common_sys.hh"

#include <iostream>
#include <fstream>
#include <argp.h>
#include <cstdlib>
#include "error.h"
#include <vector>

#include "common_setup.hh"
#include "common_output.hh"
#include "common_range.hh"
#include "common_cout.hh"

#include <cassert>
#include <iostream>
#include <sstream>
#include <set>
#include <string>
#include <cmath>
#include <cstdlib>
#include <cstring>
#include <spot/tl/formula.hh>
#include <spot/tl/relabel.hh>
#include <spot/tl/parse.hh>
#include <spot/tl/exclusive.hh>

using namespace spot;

const char argp_program_doc[] ="\
Generate temporal logic formulas from predefined patterns.";

enum {
  FIRST_CLASS = 256,
  OPT_AND_F = FIRST_CLASS,
  OPT_AND_FG,
  OPT_AND_GF,
  OPT_CCJ_ALPHA,
  OPT_CCJ_BETA,
  OPT_CCJ_BETA_PRIME,
  OPT_DAC_PATTERNS,
  OPT_EH_PATTERNS,
  OPT_GH_Q,
  OPT_GH_R,
  OPT_GO_THETA,
  OPT_HKRSS_PATTERNS,
  OPT_KR_N,
  OPT_KR_NLOGN,
  OPT_KV_PSI,
  OPT_OR_FG,
  OPT_OR_G,
  OPT_OR_GF,
  OPT_P_PATTERNS,
  OPT_R_LEFT,
  OPT_R_RIGHT,
  OPT_RV_COUNTER,
  OPT_RV_COUNTER_CARRY,
  OPT_RV_COUNTER_CARRY_LINEAR,
  OPT_RV_COUNTER_LINEAR,
  OPT_SB_PATTERNS,
  OPT_TV_F1,
  OPT_TV_F2,
  OPT_TV_G1,
  OPT_TV_G2,
  OPT_TV_UU,
  OPT_U_LEFT,
  OPT_U_RIGHT,
  LAST_CLASS,
  OPT_POSITIVE,
  OPT_NEGATIVE,
};

const char* const class_name[LAST_CLASS - FIRST_CLASS] =
  {
    "and-f",
    "and-fg",
    "and-gf",
    "ccj-alpha",
    "ccj-beta",
    "ccj-beta-prime",
    "dac-patterns",
    "eh-patterns",
    "gh-q",
    "gh-r",
    "go-theta",
    "hkrss-patterns",
    "kr-n",
    "kr-nlogn",
    "kv-psi",
    "or-fg",
    "or-g",
    "or-gf",
    "p-patterns",
    "r-left",
    "r-right",
    "rv-counter",
    "rv-counter-carry",
    "rv-counter-carry-linear",
    "rv-counter-linear",
    "sb-patterns",
    "tv-f1",
    "tv-f2",
    "tv-g1",
    "tv-g2",
    "tv-uu",
    "u-left",
    "u-right",
  };


#define OPT_ALIAS(o) { #o, 0, nullptr, OPTION_ALIAS, nullptr, 0 }

static const argp_option options[] =
  {
    /**************************************************/
    // Keep this alphabetically sorted (expect for aliases).
    { nullptr, 0, nullptr, 0, "Pattern selection:", 1},
    // J. Geldenhuys and H. Hansen (Spin'06): Larger automata and less
    // work for LTL model checking.
    { "and-f", OPT_AND_F, "RANGE", 0, "F(p1)&F(p2)&...&F(pn)", 0 },
    OPT_ALIAS(gh-e),
    { "and-fg", OPT_AND_FG, "RANGE", 0, "FG(p1)&FG(p2)&...&FG(pn)", 0 },
    { "and-gf", OPT_AND_GF, "RANGE", 0, "GF(p1)&GF(p2)&...&GF(pn)", 0 },
    OPT_ALIAS(ccj-phi),
    OPT_ALIAS(gh-c2),
    { "ccj-alpha", OPT_CCJ_ALPHA, "RANGE", 0,
      "F(p1&F(p2&F(p3&...F(pn)))) & F(q1&F(q2&F(q3&...F(qn))))", 0 },
    { "ccj-beta", OPT_CCJ_BETA, "RANGE", 0,
      "F(p&X(p&X(p&...X(p)))) & F(q&X(q&X(q&...X(q))))", 0 },
    { "ccj-beta-prime", OPT_CCJ_BETA_PRIME, "RANGE", 0,
      "F(p&(Xp)&(XXp)&...(X...X(p))) & F(q&(Xq)&(XXq)&...(X...X(q)))", 0 },
    { "dac-patterns", OPT_DAC_PATTERNS, "RANGE", OPTION_ARG_OPTIONAL,
      "Dwyer et al. [FMSP'98] Spec. Patterns for LTL "
      "(range should be included in 1..55)", 0 },
    OPT_ALIAS(spec-patterns),
    { "eh-patterns", OPT_EH_PATTERNS, "RANGE", OPTION_ARG_OPTIONAL,
      "Etessami and Holzmann [Concur'00] patterns "
      "(range should be included in 1..12)", 0 },
    { "gh-q", OPT_GH_Q, "RANGE", 0,
      "(F(p1)|G(p2))&(F(p2)|G(p3))&...&(F(pn)|G(p{n+1}))", 0 },
    { "gh-r", OPT_GH_R, "RANGE", 0,
      "(GF(p1)|FG(p2))&(GF(p2)|FG(p3))&... &(GF(pn)|FG(p{n+1}))", 0 },
    { "go-theta", OPT_GO_THETA, "RANGE", 0,
      "!((GF(p1)&GF(p2)&...&GF(pn)) -> G(q->F(r)))", 0 },
    { "hkrss-patterns", OPT_HKRSS_PATTERNS, "RANGE", OPTION_ARG_OPTIONAL,
      "Holeček et al. patterns from the Liberouter project "
      "(range should be included in 1..55)", 0 },
    OPT_ALIAS(liberouter-patterns),
    { "kr-n", OPT_KR_N, "RANGE", 0,
      "linear formula with doubly exponential DBA", 0 },
    { "kr-nlogn", OPT_KR_NLOGN, "RANGE", 0,
      "quasilinear formula with doubly exponential DBA", 0 },
    { "kv-psi", OPT_KV_PSI, "RANGE", 0,
      "quadratic formula with doubly exponential DBA", 0 },
    OPT_ALIAS(kr-n2),
    { "or-fg", OPT_OR_FG, "RANGE", 0, "FG(p1)|FG(p2)|...|FG(pn)", 0 },
    OPT_ALIAS(ccj-xi),
    { "or-g", OPT_OR_G, "RANGE", 0, "G(p1)|G(p2)|...|G(pn)", 0 },
    OPT_ALIAS(gh-s),
    { "or-gf", OPT_OR_GF, "RANGE", 0, "GF(p1)|GF(p2)|...|GF(pn)", 0 },
    OPT_ALIAS(gh-c1),
    { "p-patterns", OPT_P_PATTERNS, "RANGE", OPTION_ARG_OPTIONAL,
      "Pelánek [Spin'07] patterns from BEEM "
      "(range should be included in 1..20)", 0 },
    OPT_ALIAS(beem-patterns),
    OPT_ALIAS(p),
    { "r-left", OPT_R_LEFT, "RANGE", 0, "(((p1 R p2) R p3) ... R pn)", 0 },
    { "r-right", OPT_R_RIGHT, "RANGE", 0, "(p1 R (p2 R (... R pn)))", 0 },
    { "rv-counter", OPT_RV_COUNTER, "RANGE", 0,
      "n-bit counter", 0 },
    { "rv-counter-carry", OPT_RV_COUNTER_CARRY, "RANGE", 0,
      "n-bit counter w/ carry", 0 },
    { "rv-counter-carry-linear", OPT_RV_COUNTER_CARRY_LINEAR, "RANGE", 0,
      "n-bit counter w/ carry (linear size)", 0 },
    { "rv-counter-linear", OPT_RV_COUNTER_LINEAR, "RANGE", 0,
      "n-bit counter (linear size)", 0 },
    { "sb-patterns", OPT_SB_PATTERNS, "RANGE", OPTION_ARG_OPTIONAL,
      "Somenzi and Bloem [CAV'00] patterns "
      "(range should be included in 1..27)", 0 },
    { "tv-f1", OPT_TV_F1, "RANGE", 0, "G(p -> (q | Xq | ... | XX...Xq)", 0 },
    { "tv-f2", OPT_TV_F2, "RANGE", 0, "G(p -> (q | X(q | X(... | Xq)))", 0 },
    { "tv-g1", OPT_TV_G1, "RANGE", 0, "G(p -> (q & Xq & ... & XX...Xq)", 0 },
    { "tv-g2", OPT_TV_G2, "RANGE", 0, "G(p -> (q & X(q & X(... & Xq)))", 0 },
    { "tv-uu", OPT_TV_UU, "RANGE", 0,
      "G(p1 -> (p1 U (p2 & (p2 U (p3 & (p3 U ...))))))", 0 },
    { "u-left", OPT_U_LEFT, "RANGE", 0, "(((p1 U p2) U p3) ... U pn)", 0 },
    OPT_ALIAS(gh-u),
    { "u-right", OPT_U_RIGHT, "RANGE", 0, "(p1 U (p2 U (... U pn)))", 0 },
    OPT_ALIAS(gh-u2),
    OPT_ALIAS(go-phi),
    RANGE_DOC,
  /**************************************************/
    { nullptr, 0, nullptr, 0, "Output options:", -20 },
    { "negative", OPT_NEGATIVE, nullptr, 0,
      "output the negated versions of all formulas", 0 },
    OPT_ALIAS(negated),
    { "positive", OPT_POSITIVE, nullptr, 0,
      "output the positive versions of all formulas (done by default, unless"
      " --negative is specified without --positive)", 0 },
    { nullptr, 0, nullptr, 0, "The FORMAT string passed to --format may use "
      "the following interpreted sequences:", -19 },
    { "%f", 0, nullptr, OPTION_DOC | OPTION_NO_USAGE,
      "the formula (in the selected syntax)", 0 },
    { "%F", 0, nullptr, OPTION_DOC | OPTION_NO_USAGE,
      "the name of the pattern", 0 },
    { "%L", 0, nullptr, OPTION_DOC | OPTION_NO_USAGE,
      "the argument of the pattern", 0 },
    { "%%", 0, nullptr, OPTION_DOC | OPTION_NO_USAGE,
      "a single %", 0 },
    COMMON_LTL_OUTPUT_SPECS,
    { nullptr, 0, nullptr, 0, "Miscellaneous options:", -1 },
    { nullptr, 0, nullptr, 0, nullptr, 0 }
  };

struct job
{
  int pattern;
  struct range range;
};

typedef std::vector<job> jobs_t;
static jobs_t jobs;
bool opt_positive = false;
bool opt_negative = false;

const struct argp_child children[] =
  {
    { &output_argp, 0, nullptr, -20 },
    { &misc_argp, 0, nullptr, -1 },
    { nullptr, 0, nullptr, 0 }
  };

static void
enqueue_job(int pattern, const char* range_str)
{
  job j;
  j.pattern = pattern;
  j.range = parse_range(range_str);
  jobs.push_back(j);
}

static void
enqueue_job(int pattern, int min, int max)
{
  job j;
  j.pattern = pattern;
  j.range = {min, max};
  jobs.push_back(j);
}

static int
parse_opt(int key, char* arg, struct argp_state*)
{
  // This switch is alphabetically-ordered.
  switch (key)
    {
    case OPT_AND_F:
    case OPT_AND_FG:
    case OPT_AND_GF:
    case OPT_CCJ_ALPHA:
    case OPT_CCJ_BETA:
    case OPT_CCJ_BETA_PRIME:
    case OPT_GH_Q:
    case OPT_GH_R:
    case OPT_GO_THETA:
    case OPT_KR_N:
    case OPT_KR_NLOGN:
    case OPT_KV_PSI:
    case OPT_OR_FG:
    case OPT_OR_G:
    case OPT_OR_GF:
    case OPT_R_LEFT:
    case OPT_R_RIGHT:
    case OPT_RV_COUNTER:
    case OPT_RV_COUNTER_CARRY:
    case OPT_RV_COUNTER_CARRY_LINEAR:
    case OPT_RV_COUNTER_LINEAR:
    case OPT_TV_F1:
    case OPT_TV_F2:
    case OPT_TV_G1:
    case OPT_TV_G2:
    case OPT_TV_UU:
    case OPT_U_LEFT:
    case OPT_U_RIGHT:
      enqueue_job(key, arg);
      break;
    case OPT_DAC_PATTERNS:
    case OPT_HKRSS_PATTERNS:
      if (arg)
        enqueue_job(key, arg);
      else
        enqueue_job(key, 1, 55);
      break;
    case OPT_EH_PATTERNS:
      if (arg)
        enqueue_job(key, arg);
      else
        enqueue_job(key, 1, 12);
      break;
    case OPT_P_PATTERNS:
      if (arg)
        enqueue_job(key, arg);
      else
        enqueue_job(key, 1, 20);
      break;
    case OPT_SB_PATTERNS:
      if (arg)
        enqueue_job(key, arg);
      else
        enqueue_job(key, 1, 27);
      break;
    case OPT_POSITIVE:
      opt_positive = true;
      break;
    case OPT_NEGATIVE:
      opt_negative = true;
      break;
    default:
      return ARGP_ERR_UNKNOWN;
    }
  return 0;
}

#define G_(x) formula::G(x)
#define F_(x) formula::F(x)
#define X_(x) formula::X(x)
#define Not_(x) formula::Not(x)

#define Implies_(x, y) formula::Implies((x), (y))
#define Equiv_(x, y) formula::Equiv((x), (y))
#define And_(x, y) formula::And({(x), (y)})
#define Or_(x, y) formula::Or({(x), (y)})
#define U_(x, y) formula::U((x), (y))

// F(p_1 & F(p_2 & F(p_3 & ... F(p_n))))
static formula
E_n(std::string name, int n)
{
  if (n <= 0)
    return formula::tt();

  formula result = nullptr;

  for (; n > 0; --n)
    {
      std::ostringstream p;
      p << name << n;
      formula f = formula::ap(p.str());
      if (result)
        result = And_(f, result);
      else
        result = f;
      result = F_(result);
    }
  return result;
}

// p & X(p & X(p & ... X(p)))
static formula
phi_n(std::string name, int n,
      op oper = op::And)
{
  if (n <= 0)
    return formula::tt();

  formula result = nullptr;
  formula p = formula::ap(name);
  for (; n > 0; --n)
    {
      if (result)
        result = formula::multop(oper, {p, X_(result)});
      else
        result = p;
    }
  return result;
}

static formula
N_n(std::string name, int n)
{
  return formula::F(phi_n(name, n));
}

// p & X(p) & XX(p) & XXX(p) & ... X^n(p)
static formula
phi_prime_n(std::string name, int n,
            op oper = op::And)
{
  if (n <= 0)
    return formula::tt();

  formula result = nullptr;
  formula p = formula::ap(name);
  for (; n > 0; --n)
    {
      if (result)
        {
          p = X_(p);
          result = formula::multop(oper, {result, p});
        }
      else
        {
          result = p;
        }
    }
  return result;
}

static formula
N_prime_n(std::string name, int n)
{
  return F_(phi_prime_n(name, n));
}


// GF(p_1) & GF(p_2) & ... & GF(p_n)   if conj == true
// GF(p_1) | GF(p_2) | ... | GF(p_n)   if conj == false
static formula
GF_n(std::string name, int n, bool conj = true)
{
  if (n <= 0)
    return conj ? formula::tt() : formula::ff();

  formula result = nullptr;

  op o = conj ? op::And : op::Or;

  for (int i = 1; i <= n; ++i)
    {
      std::ostringstream p;
      p << name << i;
      formula f = G_(F_(formula::ap(p.str())));

      if (result)
        result = formula::multop(o, {f, result});
      else
        result = f;
    }
  return result;
}

// FG(p_1) | FG(p_2) | ... | FG(p_n)   if conj == false
// FG(p_1) & FG(p_2) & ... & FG(p_n)   if conj == true
static formula
FG_n(std::string name, int n, bool conj = false)
{
  if (n <= 0)
    return conj ? formula::tt() : formula::ff();

  formula result = nullptr;

  op o = conj ? op::And : op::Or;

  for (int i = 1; i <= n; ++i)
    {
      std::ostringstream p;
      p << name << i;
      formula f = F_(G_(formula::ap(p.str())));

      if (result)
        result = formula::multop(o, {f, result});
      else
        result = f;
    }
  return result;
}

//  (((p1 OP p2) OP p3)...OP pn)   if right_assoc == false
//  (p1 OP (p2 OP (p3 OP (... pn)  if right_assoc == true
static formula
bin_n(std::string name, int n, op o, bool right_assoc = false)
{
  if (n <= 0)
    n = 1;

  formula result = nullptr;

  for (int i = 1; i <= n; ++i)
    {
      std::ostringstream p;
      p << name << (right_assoc ? (n + 1 - i) : i);
      formula f = formula::ap(p.str());
      if (!result)
        result = f;
      else if (right_assoc)
        result = formula::binop(o, f, result);
      else
        result = formula::binop(o, result, f);
    }
  return result;
}

// (GF(p1)|FG(p2))&(GF(p2)|FG(p3))&...&(GF(pn)|FG(p{n+1}))"
static formula
R_n(std::string name, int n)
{
  if (n <= 0)
    return formula::tt();

  formula pi;

  {
    std::ostringstream p;
    p << name << 1;
    pi = formula::ap(p.str());
  }

  formula result = nullptr;

  for (int i = 1; i <= n; ++i)
    {
      formula gf = G_(F_(pi));
      std::ostringstream p;
      p << name << i + 1;
      pi = formula::ap(p.str());

      formula fg = F_(G_(pi));

      formula f = Or_(gf, fg);

      if (result)
        result = And_(f, result);
      else
        result = f;
    }
  return result;
}

// (F(p1)|G(p2))&(F(p2)|G(p3))&...&(F(pn)|G(p{n+1}))"
static formula
Q_n(std::string name, int n)
{
  if (n <= 0)
    return formula::tt();

  formula pi;

  {
    std::ostringstream p;
    p << name << 1;
    pi = formula::ap(p.str());
  }

  formula result = nullptr;

  for (int i = 1; i <= n; ++i)
    {
      formula f = F_(pi);

      std::ostringstream p;
      p << name << i + 1;
      pi = formula::ap(p.str());

      formula g = G_(pi);

      f = Or_(f, g);

      if (result)
        result = And_(f, result);
      else
        result = f;
    }
  return result;
}

//  OP(p1) | OP(p2) | ... | OP(Pn) if conj == false
//  OP(p1) & OP(p2) & ... & OP(Pn) if conj == true
static formula
combunop_n(std::string name, int n, op o, bool conj = false)
{
  if (n <= 0)
    return conj ? formula::tt() : formula::ff();

  formula result = nullptr;

  op cop = conj ? op::And : op::Or;

  for (int i = 1; i <= n; ++i)
    {
      std::ostringstream p;
      p << name << i;
      formula f = formula::unop(o, formula::ap(p.str()));

      if (result)
        result = formula::multop(cop, {f, result});
      else
        result = f;
    }
  return result;
}

// !((GF(p1)&GF(p2)&...&GF(pn))->G(q -> F(r)))
// From "Fast LTL to Büchi Automata Translation" [gastin.01.cav]
static formula
fair_response(std::string p, std::string q, std::string r, int n)
{
  formula fair = GF_n(p, n);
  formula resp = G_(Implies_(formula::ap(q), F_(formula::ap(r))));
  return Not_(Implies_(fair, resp));
}


// Builds X(X(...X(p))) with n occurrences of X.
static formula
X_n(formula p, int n)
{
  assert(n >= 0);
  formula res = p;
  while (n--)
    res = X_(res);
  return res;
}

// Based on LTLcounter.pl from Kristin Rozier.
// http://shemesh.larc.nasa.gov/people/kyr/benchmarking_scripts/
static formula
ltl_counter(std::string bit, std::string marker, int n, bool linear)
{
  formula b = formula::ap(bit);
  formula neg_b = Not_(b);
  formula m = formula::ap(marker);
  formula neg_m = Not_(m);

  std::vector<formula> res(4);

  // The marker starts with "1", followed by n-1 "0", then "1" again,
  // n-1 "0", etc.
  if (!linear)
    {
      // G(m -> X(!m)&XX(!m)&XXX(m))          [if n = 3]
      std::vector<formula> v(n);
      for (int i = 0; i + 1 < n; ++i)
        v[i] = X_n(neg_m, i + 1);
      v[n - 1] = X_n(m, n);
      res[0] = And_(m, G_(Implies_(m, formula::And(std::move(v)))));
    }
  else
    {
      // G(m -> X(!m & X(!m X(m))))          [if n = 3]
      formula p = m;
      for (int i = n - 1; i > 0; --i)
        p = And_(neg_m, X_(p));
      res[0] = And_(m, G_(Implies_(m, X_(p))));
    }

  // All bits are initially zero.
  if (!linear)
    {
      // !b & X(!b) & XX(!b)    [if n = 3]
      std::vector<formula> v2(n);
      for (int i = 0; i < n; ++i)
        v2[i] = X_n(neg_b, i);
      res[1] = formula::And(std::move(v2));
    }
  else
    {
      // !b & X(!b & X(!b))     [if n = 3]
      formula p = neg_b;
      for (int i = n - 1; i > 0; --i)
        p = And_(neg_b, X_(p));
      res[1] = p;
    }

#define AndX_(x, y) (linear ? X_(And_((x), (y))) : And_(X_(x), X_(y)))

  // If the least significant bit is 0, it will be 1 at the next time,
  // and other bits stay the same.
  formula Xnm1_b = X_n(b, n - 1);
  formula Xn_b = X_(Xnm1_b);
  res[2] = G_(Implies_(And_(m, neg_b),
                       AndX_(Xnm1_b, U_(And_(Not_(m), Equiv_(b, Xn_b)), m))));

  // From the least significant bit to the first 0, all the bits
  // are flipped on the next value.  Remaining bits are identical.
  formula Xnm1_negb = X_n(neg_b, n - 1);
  formula Xn_negb = X_(Xnm1_negb);
  res[3] = G_(Implies_(And_(m, b),
                       AndX_(Xnm1_negb,
                             U_(And_(And_(b, neg_m), Xn_negb),
                                Or_(m, And_(And_(neg_m, neg_b),
                                            AndX_(Xnm1_b,
                                                  U_(And_(neg_m,
                                                          Equiv_(b, Xn_b)),
                                                     m))))))));
  return formula::And(std::move(res));
}

static formula
ltl_counter_carry(std::string bit, std::string marker,
                  std::string carry, int n, bool linear)
{
  formula b = formula::ap(bit);
  formula neg_b = Not_(b);
  formula m = formula::ap(marker);
  formula neg_m = Not_(m);
  formula c = formula::ap(carry);
  formula neg_c = Not_(c);

  std::vector<formula> res(6);

  // The marker starts with "1", followed by n-1 "0", then "1" again,
  // n-1 "0", etc.
  if (!linear)
    {
      // G(m -> X(!m)&XX(!m)&XXX(m))          [if n = 3]
      std::vector<formula> v(n);
      for (int i = 0; i + 1 < n; ++i)
        v[i] = X_n(neg_m, i + 1);
      v[n - 1] = X_n(m, n);
      res[0] = And_(m, G_(Implies_(m, formula::And(std::move(v)))));
    }
  else
    {
      // G(m -> X(!m & X(!m X(m))))          [if n = 3]
      formula p = m;
      for (int i = n - 1; i > 0; --i)
        p = And_(neg_m, X_(p));
      res[0] = And_(m, G_(Implies_(m, X_(p))));
    }

  // All bits are initially zero.
  if (!linear)
    {
      // !b & X(!b) & XX(!b)    [if n = 3]
      std::vector<formula> v2(n);
      for (int i = 0; i < n; ++i)
        v2[i] = X_n(neg_b, i);
      res[1] = formula::And(std::move(v2));
    }
  else
    {
      // !b & X(!b & X(!b))     [if n = 3]
      formula p = neg_b;
      for (int i = n - 1; i > 0; --i)
        p = And_(neg_b, X_(p));
      res[1] = p;
    }

  formula Xn_b = X_n(b, n);
  formula Xn_negb = X_n(neg_b, n);

  // If m is 1 and b is 0 then c is 0 and n steps later b is 1.
  res[2] = G_(Implies_(And_(m, neg_b), And_(neg_c, Xn_b)));

  // If m is 1 and b is 1 then c is 1 and n steps later b is 0.
  res[3] = G_(Implies_(And_(m, b), And_(c, Xn_negb)));

  if (!linear)
    {
      // If there's no carry, then all of the bits stay the same n steps later.
      res[4] = G_(Implies_(And_(neg_c, X_(neg_m)),
                           And_(X_(Not_(c)), Equiv_(X_(b), X_(Xn_b)))));

      // If there's a carry, then add one: flip the bits of b and
      // adjust the carry.
      res[5] = G_(Implies_(c, And_(Implies_(X_(neg_b),
                                            And_(X_(neg_c), X_(Xn_b))),
                                   Implies_(X_(b),
                                            And_(X_(c), X_(Xn_negb))))));
    }
  else
    {
      // If there's no carry, then all of the bits stay the same n steps later.
      res[4] = G_(Implies_(And_(neg_c, X_(neg_m)),
                           X_(And_(Not_(c), Equiv_(b, Xn_b)))));
      // If there's a carry, then add one: flip the bits of b and
      // adjust the carry.
      res[5] = G_(Implies_(c, X_(And_(Implies_(neg_b, And_(neg_c, Xn_b)),
                                      Implies_(b, And_(c, Xn_negb))))));
    }
  return formula::And(std::move(res));
}

static formula
tv_f1(std::string p, std::string q, int n)
{
  return G_(Implies_(formula::ap(p), phi_prime_n(q, n, op::Or)));
}

static formula
tv_f2(std::string p, std::string q, int n)
{
  return G_(Implies_(formula::ap(p), phi_n(q, n, op::Or)));
}

static formula
tv_g1(std::string p, std::string q, int n)
{
  return G_(Implies_(formula::ap(p), phi_prime_n(q, n)));
}

static formula
tv_g2(std::string p, std::string q, int n)
{
  return G_(Implies_(formula::ap(p), phi_n(q, n)));
}

static formula
tv_uu(std::string name, int n)
{
  std::ostringstream p;
  p << name << n + 1;
  formula q = formula::ap(p.str());
  formula f = q;

  for (int i = n; i > 0; --i)
    {
      p.str("");
      p << name << i;
      q = formula::ap(p.str());
      f = U_(q, f);
      if (i > 1)
        f = And_(q, f);
    }
  return G_(Implies_(q,  f));
}

static void
bad_number(const char* pattern, int n, int max = 0)
{
  std::ostringstream err;
  err << "no pattern " << n << " for " << pattern
      << ", supported range is 1..";
  if (max)
    err << max;
  throw std::runtime_error(err.str());
}

static formula
dac_pattern(int n)
{
  static const char* formulas[] = {
    "[](!p0)",
    "<>p2 -> (!p0 U p2)",
    "[](p1 -> [](!p0))",
    "[]((p1 & !p2 & <>p2) -> (!p0 U p2))",
    "[](p1 & !p2 -> (!p0 W p2))",

    "<>(p0)",
    "!p2 W (p0 & !p2)",
    "[](!p1) | <>(p1 & <>p0)",
    "[](p1 & !p2 -> (!p2 W (p0 & !p2)))",
    "[](p1 & !p2 -> (!p2 U (p0 & !p2)))",

    "(!p0 W (p0 W (!p0 W (p0 W []!p0))))",
    "<>p2 -> ((!p0 & !p2) U (p2 | ((p0 & !p2) U (p2 |"
    " ((!p0 & !p2) U (p2 | ((p0 & !p2) U (p2 | (!p0 U p2)))))))))",
    "<>p1 -> (!p1 U (p1 & (!p0 W (p0 W (!p0 W (p0 W []!p0))))))",
    "[]((p1 & <>p2) -> ((!p0 & !p2) U (p2 | ((p0 & !p2) U (p2 |"
    "((!p0 & !p2) U (p2 | ((p0 & !p2) U (p2 | (!p0 U p2))))))))))",
    "[](p1 -> ((!p0 & !p2) U (p2 | ((p0 & !p2) U (p2 | ((!p0 & !p2) U"
    "(p2 | ((p0 & !p2) U (p2 | (!p0 W p2) | []p0)))))))))",

    "[](p0)",
    "<>p2 -> (p0 U p2)",
    "[](p1 -> [](p0))",
    "[]((p1 & !p2 & <>p2) -> (p0 U p2))",
    "[](p1 & !p2 -> (p0 W p2))",

    "!p0 W p3",
    "<>p2 -> (!p0 U (p3 | p2))",
    "[]!p1 | <>(p1 & (!p0 W p3))",
    "[]((p1 & !p2 & <>p2) -> (!p0 U (p3 | p2)))",
    "[](p1 & !p2 -> (!p0 W (p3 | p2)))",

    "[](p0 -> <>p3)",
    "<>p2 -> (p0 -> (!p2 U (p3 & !p2))) U p2",
    "[](p1 -> [](p0 -> <>p3))",
    "[]((p1 & !p2 & <>p2) -> ((p0 -> (!p2 U (p3 & !p2))) U p2))",
    "[](p1 & !p2 -> ((p0 -> (!p2 U (p3 & !p2))) W p2))",

    "<>p0 -> (!p0 U (p3 & !p0 & X(!p0 U p4)))",
    "<>p2 -> (!p0 U (p2 | (p3 & !p0 & X(!p0 U p4))))",
    "([]!p1) | (!p1 U (p1 & <>p0 -> (!p0 U (p3 & !p0 & X(!p0 U p4)))))",
    "[]((p1 & <>p2) -> (!p0 U (p2 | (p3 & !p0 & X(!p0 U p4)))))",
    "[](p1 -> (<>p0 -> (!p0 U (p2 | (p3 & !p0 & X(!p0 U p4))))))",

    "(<>(p3 & X<>p4)) -> ((!p3) U p0)",
    "<>p2 -> ((!(p3 & (!p2) & X(!p2 U (p4 & !p2)))) U (p2 | p0))",
    "([]!p1) | ((!p1) U (p1 & ((<>(p3 & X<>p4)) -> ((!p3) U p0))))",
    "[]((p1 & <>p2) -> ((!(p3 & (!p2) & X(!p2 U (p4 & !p2)))) U (p2 | p0)))",
    "[](p1 -> (!(p3 & (!p2) & X(!p2 U (p4 & !p2))) U (p2 | p0) |"
    " [](!(p3 & X<>p4))))",

    "[] (p3 & X<> p4 -> X(<>(p4 & <> p0)))",
    "<>p2 -> (p3 & X(!p2 U p4) -> X(!p2 U (p4 & <> p0))) U p2",
    "[] (p1 -> [] (p3 & X<> p4 -> X(!p4 U (p4 & <> p0))))",
    "[] ((p1 & <>p2) -> (p3 & X(!p2 U p4) -> X(!p2 U (p4 & <> p0))) U p2)",
    "[] (p1 -> (p3 & X(!p2 U p4) -> X(!p2 U (p4 & <> p0))) U (p2 |"
    "[] (p3 & X(!p2 U p4) -> X(!p2 U (p4 & <> p0)))))",

    "[] (p0 -> <>(p3 & X<>p4))",
    "<>p2 -> (p0 -> (!p2 U (p3 & !p2 & X(!p2 U p4)))) U p2",
    "[] (p1 -> [] (p0 -> (p3 & X<> p4)))",
    "[] ((p1 & <>p2) -> (p0 -> (!p2 U (p3 & !p2 & X(!p2 U p4)))) U p2)",
    "[] (p1 -> (p0 -> (!p2 U (p3 & !p2 & X(!p2 U p4)))) U (p2 | []"
    "(p0 -> (p3 & X<> p4))))",

    "[] (p0 -> <>(p3 & !p5 & X(!p5 U p4)))",
    "<>p2 -> (p0 -> (!p2 U (p3 & !p2 & !p5 & X((!p2 & !p5) U p4)))) U p2",
    "[] (p1 -> [] (p0 -> (p3 & !p5 & X(!p5 U p4))))",
    "[] ((p1 & <>p2) -> (p0 -> (!p2 U (p3 & !p2 & !p5 & X((!p2 & !p5) U"
    " p4)))) U p2)",
    "[] (p1 -> (p0 -> (!p2 U (p3 & !p2 & !p5 & X((!p2 & !p5) U p4)))) U (p2 |"
    "[] (p0 -> (p3 & !p5 & X(!p5 U p4)))))",
  };

  constexpr unsigned max = (sizeof formulas)/(sizeof *formulas);
  if (n < 1 || (unsigned) n > max)
    bad_number("--dac-patterns", n, max);
  return spot::relabel(parse_formula(formulas[n - 1]), Pnn);
}

static formula
hkrss_pattern(int n)
{
  static const char* formulas[] = {
    "G(Fp0 & F!p0)",
    "GFp0 & GF!p0",
    "GF(!(p1 <-> Xp1) | !(p0 <-> Xp0))",
    "GF(!(p1 <-> Xp1) | !(p0 <-> Xp0) | !(p2 <-> Xp2) | !(p3 <-> Xp3))",
    "G!p0",
    "G((p0 -> F!p0) & (!p0 -> Fp0))",
    "G(p0 -> F(p0 & p1))",
    "G(p0 -> F((!p0 & p1 & p2 & p3) -> Fp4))",
    "G(p0 -> F!p1)",
    "G(p0 -> Fp1)",
    "G(p0 -> F(p1 -> Fp2))",
    "G(p0 -> F((p1 & p2) -> Fp3))",
    "G((p0 -> Fp1) & (p2 -> Fp3) & (p4 -> Fp5) & (p6 -> Fp7))",
    "G(!p0 & !p1)",
    "G!(p0 & p1)",
    "G(p0 -> p1)",
    "G((p0 -> !p1) & (p1 -> !p0))",
    "G(!p0 -> (p1 <-> !p2))",
    "G((!p0 & (p1 | p2 | p3)) -> p4)",
    "G((p0 & p1) -> (p2 | !(p3 & p4)))",
    "G((!p0 & p1 & !p2 & !p3 & !p4) -> F(!p5 & !p6 & !p7 & !p8))",
    "G((p0 & p1 & !p2 & !p3 & !p4) -> F(p5 & !p6 & !p7 & !p8))",
    "G(!p0 -> !(p1 & p2 & p3 & p4 & p5))",
    "G(!p0 -> ((p1 & p2 & p3 & p4) -> !p5))",
    "G((p0 & p1) -> (p2 | p3 | !(p4 & p5)))",
    "G((!p0 & (p1 | p2 | p3 | p4)) -> (!p5 <-> p6))",
    "G((p0 & p1) -> (p2 | p3 | p4 | !(p5 & p6)))",
    "G((p0 & p1) -> (p2 | p3 | p4 | p5 | !(p6 & p7)))",
    "G((p0 & p1 & !p2 & Xp2) -> X(p3 | X(!p1 | p3)))",
    "G((p0 & p1 & !p2 & Xp2) -> X(X!p1 | (p2 U (!p2 U (p2 U (!p1 | p3))))))",
    "G(p0 & p1 & !p2 & Xp2) -> X(X!p1 | (p2 U (!p2 U (p2 U (!p1 | p3)))))",
    "G(p0 -> (p1 U (!p1 U (!p2 | p3))))",
    "G(p0 -> (p1 U (!p1 U (p2 | p3))))",
    "G((!p0 & p1) -> Xp2)",
    "G(p0 -> X(p0 | p1))",
    ("G((!(p1 <-> Xp1) | !(p0 <-> Xp0) | !(p2 <-> Xp2) | !(p3 <-> Xp3)) -> "
     "(X!p4 & X(!(!(p1 <-> Xp1) | !(p0 <-> Xp0) | !(p2 <-> Xp2) | "
     "!(p3 <-> Xp3)) U p4)))"),
    "G((p0 & !p1 & Xp1 & Xp0) -> (p2 -> Xp3))",
    "G(p0 -> X(!p0 U p1))",
    "G((!p0 & Xp0) -> X((p0 U p1) | Gp0))",
    "G((!p0 & Xp0) -> X(p0 U (p0 & !p1 & X(p0 & p1))))",
    ("G((!p0 & Xp0) -> X(p0 U (p0 & !p1 & X(p0 & p1 & (p0 U (p0 & !p1 & "
     "X(p0 & p1)))))))"),
    ("G((p0 & X!p0) -> X(!p0 U (!p0 & !p1 & X(!p0 & p1 & (!p0 U (!p0 & !p1 "
     "& X(!p0 & p1)))))))"),
    ("G((p0 & X!p0) -> X(!p0 U (!p0 & !p1 & X(!p0 & p1 & (!p0 U (!p0 & !p1 & "
     "X(!p0 & p1 & (!p0 U (!p0 & !p1 & X(!p0 & p1))))))))))"),
    "G((!p0 & Xp0) -> X(!(!p0 & Xp0) U (!p1 & Xp1)))",
    ("G(!p0 | X(!p0 | X(!p0 | X(!p0 | X(!p0 | X(!p0 | X(!p0 | X(!p0 | X(!p0 | "
     "X(!p0 | X(!p0 | X!p0)))))))))))"),
    "G((Xp0 -> p0) -> (p1 <-> Xp1))",
    "G((Xp0 -> p0) -> ((p1 -> Xp1) & (!p1 -> X!p1)))",
    ("!p0 U G!((p1 & p2) | (p3 & p4) | (p2 & p3) | (p2 & p4) | (p1 & p4) | "
     "(p1 & p3))"),
    "!p0 U p1",
    "(p0 U p1) | Gp0",
    "p0 & XG!p0",
    "XG(p0 -> (G!p1 | (!Xp1 U p2)))",
    "XG((p0 & !p1) -> (G!p1 | (!p1 U p2)))",
    "XG((p0 & p1) -> ((p1 U p2) | Gp1))",
    "Xp0 & G((!p0 & Xp0) -> XXp0)",
  };

  constexpr unsigned max = (sizeof formulas)/(sizeof *formulas);
  if (n < 1 || (unsigned) n > max)
    bad_number("--hkrss-patterns", n, max);
  return spot::relabel(parse_formula(formulas[n - 1]), Pnn);
}

static formula
eh_pattern(int n)
{
  static const char* formulas[] = {
    "p0 U (p1 & G(p2))",
    "p0 U (p1 & X(p2 U p3))",
    "p0 U (p1 & X(p2 & (F(p3 & X(F(p4 & X(F(p5 & X(F(p6))))))))))",
    "F(p0 & X(G(p1)))",
    "F(p0 & X(p1 & X(F(p2))))",
    "F(p0 & X(p1 U p2))",
    "(F(G(p0))) | (G(F(p1)))",
    "G(p0 -> (p1 U p2))",
    "G(p0 & X(F(p1 & X(F(p2 & X(F(p3)))))))",
    "(G(F(p0))) & (G(F(p1))) & (G(F(p2))) & (G(F(p3))) & (G(F(p4)))",
    "(p0 U (p1 U p2)) | (p1 U (p2 U p0)) | (p2 U (p0 U p1))",
    "G(p0 -> (p1 U ((G(p2)) | (G(p3)))))",
  };

  constexpr unsigned max = (sizeof formulas)/(sizeof *formulas);
  if (n < 1 || (unsigned) n > max)
    bad_number("--eh-patterns", n, max);
  return spot::relabel(parse_formula(formulas[n - 1]), Pnn);
}

static formula
p_pattern(int n)
{
  static const char* formulas[] = {
    "G(p0 -> Fp1)",
    "(GFp1 & GFp0) -> GFp2",
    "G(p0 -> (p1 & (p2 U p3)))",
    "F(p0 | p1)",
    "GF(p0 | p1)",
    "(p0 U p1) -> ((p2 U p3) | Gp2)",
    "G(p0 -> (!p1 U (p1 U (!p1 & (p2 R !p1)))))",
    "G(p0 -> (p1 R !p2))",
    "G(!p0 -> Fp0)",
    "G(p0 -> F(p1 | p2))",
    "!(!(p0 | p1) U p2) & G(p3 -> !(!(p0 | p1) U p2))",
    "G!p0 -> G!p1",
    "G(p0 -> (G!p1 | (!p2 U p1)))",
    "G(p0 -> (p1 R (p1 | !p2)))",
    "G((p0 & p1) -> (!p1 R (p0 | !p1)))",
    "G(p0 -> F(p1 & p2))",
    "G(p0 -> (!p1 U (p1 U (p1 & p2))))",
    "G(p0 -> (!p1 U (p1 U (!p1 U (p1 U (p1 & p2))))))",
    "GFp0 -> GFp1",
    "GF(p0 | p1) & GF(p1 | p2)",
  };

  constexpr unsigned max = (sizeof formulas)/(sizeof *formulas);
  if (n < 1 || (unsigned) n > max)
    bad_number("--p-patterns", n, max);
  return spot::relabel(parse_formula(formulas[n - 1]), Pnn);
}

static formula
sb_pattern(int n)
{
  static const char* formulas[] = {
    "p0 U p1",
    "p0 U (p1 U p2)",
    "!(p0 U (p1 U p2))",
    "G(F(p0)) -> G(F(p1))",
    "(F(p0)) U (G(p1))",
    "(G(p0)) U p1",
    "!((F(F(p0))) <-> (F(p)))",
    "!((G(F(p0))) -> (G(F(p))))",
    "!((G(F(p0))) <-> (G(F(p))))",
    "p0 R (p0 | p1)",
    "(Xp0 U Xp1) | !X(p0 U p1)",
    "(Xp0 U p1) | !X(p0 U (p0 & p1))",
    "G(p0 -> F(p1)) & (((X(p0)) U p1) | !X(p0 U (p0 & p1)))",
    "G(p0 -> F(p1)) & (((X(p0)) U X(p1)) | !X(p0 U p1))",
    "G(p0 -> F(p1))",
    "!G(p0 -> X(p1 R p2))",
    "!(F(G(p0)) | F(G(p1)))",
    "G(F(p0) & F(p1))",
    "F(p0) & F(!p0)",
    "(X(p1) & p2) R X(((p3 U p0) R p2) U (p3 R p2))",
    "(G(p1 | G(F(p0))) & G(p2 | G(F(!p0)))) | G(p1) | G(p2)",
    "(G(p1 | F(G(p0))) & G(p2 | F(G(!p0)))) | G(p1) | G(p2)",
    "!((G(p1 | G(F(p0))) & G(p2 | G(F(!p0)))) | G(p1) | G(p2))",
    "!((G(p1 | F(G(p0))) & G(p2 | F(G(!p0)))) | G(p1) | G(p2))",
    "(G(p1 | X(G p0))) & (G (p2 | X(G !p0)))",
    "G(p1 | (Xp0 & X!p0))",
    // p0 U p0 can't be represented other than as p0 in Spot
    "(p0 U p0) | (p1 U p0)",
  };

  constexpr unsigned max = (sizeof formulas)/(sizeof *formulas);
  if (n < 1 || (unsigned) n > max)
    bad_number("--sb-patterns", n, max);
  return spot::relabel(parse_formula(formulas[n - 1]), Pnn);
}

static formula
X_n_kv_exp(formula a, int n, formula b)
{
  formula f = X_n(a, n);
  return And_(f, G_(Implies_(b, f)));
}

static formula
kv_exp(int n, std::string a, std::string b, std::string c, std::string d)
{
  formula fa = formula::ap(a);
  formula fb = formula::ap(b);
  formula fc = formula::ap(c);
  formula fd = formula::ap(d);

  exclusive_ap m;
  m.add_group({ fa, fb, fc, fd });

  formula xn = X_(G_(fc));
  for (int i = 0; i < n; i++)
    xn = X_(And_(Or_(fa, fb), xn));
  formula f1 = U_(Not_(fd), And_(fd, xn));

  formula f_and = nullptr;
  for (int i = 1; i <= n; i++)
    f_and = And_(f_and, Or_(X_n_kv_exp(fa, i, fd), X_n_kv_exp(fb, i, fd)));

  formula f2 = F_(And_(fc, And_(f_and, X_n(fc, n + 1))));

  return m.constrain(And_(f1, f2));
}

static formula
bit_ni(unsigned i, unsigned j, formula fbin[2])
{
  return fbin[((1u << j) & (i - 1)) != 0];
}

static formula
binary_ki(int k, unsigned i, formula fbin[2])
{
  formula res = bit_ni(i, k - 1, fbin);
  for (int j = k - 1; j >= 1; j--)
    res = And_(bit_ni(i, j - 1, fbin), X_(res));
  return res;
}

static formula
kr1_exp_1(int k, formula fc, formula fd, formula fbin[2])
{
    return And_(fc, X_(Or_(binary_ki(k, 1, fbin), fd)));
}

static formula
kr1_exp_2(int n, int k, formula fa, formula fb, formula fbin[2])
{
  formula res = formula::tt();
  for (int i = 1; i <= n - 1; i++)
    res = And_(res,
               Implies_(binary_ki(k, i, fbin),
                        X_n(And_(Or_(fa, fb),
                            X_(binary_ki(k, i + 1, fbin))), k)));

  return G_(res);
}

static formula
kr1_exp_3(int n, int k, formula fa, formula fb, formula fc, formula fd,
          formula fbin[2])
{
  return G_(Implies_(binary_ki(k, n, fbin),
                     X_n(And_(Or_(fa, fb),
                              X_(And_(fc,
                                      X_(Or_(binary_ki(