Quelle  Conversions.thy

(*  Title:      HOL/Decision_Procs/Conversions.thy
  Author: Stefan Berghofer
*)

theory Conversions
imports Main
begin

ML ‹
 fun tactic_of_conv cv i st =
  if i > Thm.nprems_of st then Seq.empty
  else Seq.single (Conv.gconv_rule cv i st);
 
 fun binop_conv cv cv' = Conv.combination_conv (Conv.arg_conv cv) cv';
 ›

ML ‹
 fun err s ct =
  error (s ^ ": " ^ Syntax.string_of_term_global (Thm.theory_of_cterm ct) (Thm.term_of ct));
 ›

attribute_setup meta =
  ‹Scan.succeed (Thm.rule_attribute [] (K mk_meta_eq))›
  ‹convert equality to meta equality›

ML ‹
 fun strip_app ct = ct |> Drule.strip_comb |>> Thm.term_of |>> dest_Const_name;
 
 fun inst cTs cts th =
  Thm.instantiate' (map SOME cTs) (map SOME cts) th;
 
 fun transitive' eq eq' = Thm.transitive eq (eq' (Thm.rhs_of eq));
 
 fun type_of_eqn eqn = Thm.ctyp_of_cterm (Thm.dest_arg1 (Thm.cprop_of eqn));
 
 fun cong1 conv ct =
  Thm.combination (Thm.reflexive (Thm.dest_fun ct)) (conv (Thm.dest_arg ct));
 
 fun cong1' conv' conv ct =
  let val eqn = conv (Thm.dest_arg ct)
  in
  Thm.transitive
  (Thm.combination (Thm.reflexive (Thm.dest_fun ct)) eqn)
  (conv' (Thm.rhs_of eqn))
  end;
 
 fun cong2 conv1 conv2 ct =
  Thm.combination
  (Thm.combination
  (Thm.reflexive (Thm.dest_fun2 ct))
  (conv1 (Thm.dest_arg1 ct)))
  (conv2 (Thm.dest_arg ct));
 
 fun cong2' conv conv1 conv2 ct =
  let
  val eqn1 = conv1 (Thm.dest_arg1 ct);
  val eqn2 = conv2 (Thm.dest_arg ct)
  in
  Thm.transitive
  (Thm.combination
  (Thm.combination (Thm.reflexive (Thm.dest_fun2 ct)) eqn1)
  eqn2)
  (conv (Thm.rhs_of eqn1) (Thm.rhs_of eqn2))
  end;
 
 fun cong2'' conv eqn1 eqn2 =
  let val eqn3 = conv (Thm.rhs_of eqn1) (Thm.rhs_of eqn2)
  in
  Thm.transitive
  (Thm.combination
  (Thm.combination (Thm.reflexive (Thm.dest_fun2 (Thm.lhs_of eqn3))) eqn1)
  eqn2)
  eqn3
  end;
 
 fun args1 conv ct = conv (Thm.dest_arg ct);
 fun args2 conv ct = conv (Thm.dest_arg1 ct) (Thm.dest_arg ct);
 ›

ML ‹
 fun strip_numeral ct = (case strip_app ct of
  (🍋‹uminus›, [n]) => (case strip_app n of
  (🍋‹numeral›, [b]) => (🍋‹uminus›, [b])
  | _ => ("", []))
  | x => x);
 ›

lemma nat_minus1_eq: "nat (- 1) = 0"
  by simp

ML ‹
 fun nat_conv i = (case strip_app i of
  (🍋‹zero_class.zero›, []) => @{thm nat_0 [meta]}
  | (🍋‹one_class.one›, []) => @{thm nat_one_as_int [meta, symmetric]}
  | (🍋‹numeral›, [b]) => inst [] [b] @{thm nat_numeral [meta]}
  | (🍋‹uminus›, [b]) => (case strip_app b of
  (🍋‹one_class.one›, []) => @{thm nat_minus1_eq [meta]}
  | (🍋‹numeral›, [b']) => inst [] [b'] @{thm nat_neg_numeral [meta]}));
 ›

ML ‹
 fun add_num_conv b b' = (case (strip_app b, strip_app b') of
  ((🍋‹Num.One›, []), (🍋‹Num.One›, [])) =>
  @{thm add_num_simps(1) [meta]}
  | ((🍋‹Num.One›, []), (🍋‹Num.Bit0›, [n])) =>
  inst [] [n] @{thm add_num_simps(2) [meta]}
  | ((🍋‹Num.One›, []), (🍋‹Num.Bit1›, [n])) =>
  transitive'
  (inst [] [n] @{thm add_num_simps(3) [meta]})
  (cong1 (args2 add_num_conv))
  | ((🍋‹Num.Bit0›, [m]), (🍋‹Num.One›, [])) =>
  inst [] [m] @{thm add_num_simps(4) [meta]}
  | ((🍋‹Num.Bit0›, [m]), (🍋‹Num.Bit0›, [n])) =>
  transitive'
  (inst [] [m, n] @{thm add_num_simps(5) [meta]})
  (cong1 (args2 add_num_conv))
  | ((🍋‹Num.Bit0›, [m]), (🍋‹Num.Bit1›, [n])) =>
  transitive'
  (inst [] [m, n] @{thm add_num_simps(6) [meta]})
  (cong1 (args2 add_num_conv))
  | ((🍋‹Num.Bit1›, [m]), (🍋‹Num.One›, [])) =>
  transitive'
  (inst [] [m] @{thm add_num_simps(7) [meta]})
  (cong1 (args2 add_num_conv))
  | ((🍋‹Num.Bit1›, [m]), (🍋‹Num.Bit0›, [n])) =>
  transitive'
  (inst [] [m, n] @{thm add_num_simps(8) [meta]})
  (cong1 (args2 add_num_conv))
  | ((🍋‹Num.Bit1›, [m]), (🍋‹Num.Bit1›, [n])) =>
  transitive'
  (inst [] [m, n] @{thm add_num_simps(9) [meta]})
  (cong1 (cong2' add_num_conv (args2 add_num_conv) Thm.reflexive)));
 ›

ML ‹
 fun BitM_conv m = (case strip_app m of
  (🍋‹Num.One›, []) => @{thm BitM.simps(1) [meta]}
  | (🍋‹Num.Bit0›, [n]) =>
  transitive'
  (inst [] [n] @{thm BitM.simps(2) [meta]})
  (cong1 (args1 BitM_conv))
  | (🍋‹Num.Bit1›, [n]) =>
  inst [] [n] @{thm BitM.simps(3) [meta]});
 ›

lemma dbl_neg_numeral:
  "Num.dbl (- Num.numeral k) = - Num.numeral (Num.Bit0 k)"
  by simp

ML ‹
 fun dbl_conv a =
  let
  val dbl_neg_numeral_a = inst [a] [] @{thm dbl_neg_numeral [meta]};
  val dbl_0_a = inst [a] [] @{thm dbl_simps(2) [meta]};
  val dbl_numeral_a = inst [a] [] @{thm dbl_simps(5) [meta]}
  in
  fn n =>
  case strip_numeral n of
  (🍋‹zero_class.zero›, []) => dbl_0_a
  | (🍋‹numeral›, [k]) => inst [] [k] dbl_numeral_a
  | (🍋‹uminus›, [k]) => inst [] [k] dbl_neg_numeral_a
  end;
 ›

lemma dbl_inc_neg_numeral:
  "Num.dbl_inc (- Num.numeral k) = - Num.numeral (Num.BitM k)"
  by simp

ML ‹
 fun dbl_inc_conv a =
  let
  val dbl_inc_neg_numeral_a = inst [a] [] @{thm dbl_inc_neg_numeral [meta]};
  val dbl_inc_0_a = inst [a] [] @{thm dbl_inc_simps(2) [folded numeral_One, meta]};
  val dbl_inc_numeral_a = inst [a] [] @{thm dbl_inc_simps(5) [meta]};
  in
  fn n =>
  case strip_numeral n of
  (🍋‹zero_class.zero›, []) => dbl_inc_0_a
  | (🍋‹numeral›, [k]) => inst [] [k] dbl_inc_numeral_a
  | (🍋‹uminus›, [k]) =>
  transitive'
  (inst [] [k] dbl_inc_neg_numeral_a)
  (cong1 (cong1 (args1 BitM_conv)))
  end;
 ›

lemma dbl_dec_neg_numeral:
  "Num.dbl_dec (- Num.numeral k) = - Num.numeral (Num.Bit1 k)"
  by simp

ML ‹
 fun dbl_dec_conv a =
  let
  val dbl_dec_neg_numeral_a = inst [a] [] @{thm dbl_dec_neg_numeral [meta]};
  val dbl_dec_0_a = inst [a] [] @{thm dbl_dec_simps(2) [folded numeral_One, meta]};
  val dbl_dec_numeral_a = inst [a] [] @{thm dbl_dec_simps(5) [meta]};
  in
  fn n =>
  case strip_numeral n of
  (🍋‹zero_class.zero›, []) => dbl_dec_0_a
  | (🍋‹uminus›, [k]) => inst [] [k] dbl_dec_neg_numeral_a
  | (🍋‹numeral›, [k]) =>
  transitive'
  (inst [] [k] dbl_dec_numeral_a)
  (cong1 (args1 BitM_conv))
  end;
 ›

ML ‹
 fun sub_conv a =
  let
  val [sub_One_One, sub_One_Bit0, sub_One_Bit1,
  sub_Bit0_One, sub_Bit1_One, sub_Bit0_Bit0,
  sub_Bit0_Bit1, sub_Bit1_Bit0, sub_Bit1_Bit1] =
  map (inst [a] []) @{thms sub_num_simps [meta]};
  val dbl_conv_a = dbl_conv a;
  val dbl_inc_conv_a = dbl_inc_conv a;
  val dbl_dec_conv_a = dbl_dec_conv a;
 
  fun conv m n = (case (strip_app m, strip_app n) of
  ((🍋‹Num.One›, []), (🍋‹Num.One›, [])) =>
  sub_One_One
  | ((🍋‹Num.One›, []), (🍋‹Num.Bit0›, [l])) =>
  transitive'
  (inst [] [l] sub_One_Bit0)
  (cong1 (cong1 (args1 BitM_conv)))
  | ((🍋‹Num.One›, []), (🍋‹Num.Bit1›, [l])) =>
  inst [] [l] sub_One_Bit1
  | ((🍋‹Num.Bit0›, [k]), (🍋‹Num.One›, [])) =>
  transitive'
  (inst [] [k] sub_Bit0_One)
  (cong1 (args1 BitM_conv))
  | ((🍋‹Num.Bit1›, [k]), (🍋‹Num.One›, [])) =>
  inst [] [k] sub_Bit1_One
  | ((🍋‹Num.Bit0›, [k]), (🍋‹Num.Bit0›, [l])) =>
  transitive'
  (inst [] [k, l] sub_Bit0_Bit0)
  (cong1' dbl_conv_a (args2 conv))
  | ((🍋‹Num.Bit0›, [k]), (🍋‹Num.Bit1›, [l])) =>
  transitive'
  (inst [] [k, l] sub_Bit0_Bit1)
  (cong1' dbl_dec_conv_a (args2 conv))
  | ((🍋‹Num.Bit1›, [k]), (🍋‹Num.Bit0›, [l])) =>
  transitive'
  (inst [] [k, l] sub_Bit1_Bit0)
  (cong1' dbl_inc_conv_a (args2 conv))
  | ((🍋‹Num.Bit1›, [k]), (🍋‹Num.Bit1›, [l])) =>
  transitive'
  (inst [] [k, l] sub_Bit1_Bit1)
  (cong1' dbl_conv_a (args2 conv)))
  in conv end;
 ›

ML ‹
 fun expand1 a =
  let val numeral_1_eq_1_a = inst [a] [] @{thm numeral_One [meta, symmetric]}
  in
  fn n =>
  case Thm.term_of n of
  🍋‹one_class.one _› => numeral_1_eq_1_a
  | 🍋‹uminus _ for 🍋‹one_class.one _›\› =>
  Thm.combination (Thm.reflexive (Thm.dest_fun n)) numeral_1_eq_1_a
  | 🍋‹zero_class.zero _› => Thm.reflexive n
  | 🍋‹numeral _ for _› => Thm.reflexive n
  | 🍋‹uminus _ for 🍋‹numeral _ for _›\› => Thm.reflexive n
  | _ => err "expand1" n
  end;
 
 fun norm1_eq a =
  let val numeral_1_eq_1_a = inst [a] [] @{thm numeral_One [meta]}
  in
  fn eq =>
  case Thm.term_of (Thm.rhs_of eq) of
  🍋‹Num.numeral _ for 🍋‹Num.One›\› => Thm.transitive eq numeral_1_eq_1_a
  | 🍋‹uminus _ for 🍋‹Num.numeral _ for 🍋‹Num.One›\›\› =>
  Thm.transitive eq
  (Thm.combination (Thm.reflexive (Thm.dest_fun (Thm.rhs_of eq)))
  numeral_1_eq_1_a)
  | _ => eq
  end;
 ›

ML ‹
 fun plus_conv f a =
  let
  val add_0_a = inst [a] [] @{thm add_0 [meta]};
  val add_0_right_a = inst [a] [] @{thm add_0_right [meta]};
  val numeral_plus_numeral_a = inst [a] [] @{thm numeral_plus_numeral [meta]};
  val expand1_a = expand1 a;
 
  fun conv m n = (case (strip_app m, strip_app n) of
  ((🍋‹zero_class.zero›, []), _) => inst [] [n] add_0_a
  | (_, (🍋‹zero_class.zero›, [])) => inst [] [m] add_0_right_a
  | ((🍋‹numeral›, [m]), (🍋‹numeral›, [n])) =>
  transitive'
  (inst [] [m, n] numeral_plus_numeral_a)
  (cong1 (args2 add_num_conv))
  | _ => cong2'' (f conv) (expand1_a m) (expand1_a n))
  in f conv end;
 
 val nat_plus_conv = plus_conv I 🍋‹nat›;
 ›

lemma neg_numeral_plus_neg_numeral:
  "- Num.numeral m + - Num.numeral n = (- Num.numeral (m + n) ::'a::neg_numeral)"
  by simp

ML ‹
 fun plus_neg_conv a =
  let
  val numeral_plus_neg_numeral_a =
  inst [a] [] @{thm add_neg_numeral_simps(1) [meta]};
  val neg_numeral_plus_numeral_a =
  inst [a] [] @{thm add_neg_numeral_simps(2) [meta]};
  val neg_numeral_plus_neg_numeral_a =
  inst [a] [] @{thm neg_numeral_plus_neg_numeral [meta]};
  val sub_conv_a = sub_conv a;
  in
  fn conv => fn m => fn n =>
  case (strip_numeral m, strip_numeral n) of
  ((🍋‹Num.numeral›, [m]), (🍋‹uminus›, [n])) =>
  Thm.transitive
  (inst [] [m, n] numeral_plus_neg_numeral_a)
  (sub_conv_a m n)
  | ((🍋‹uminus›, [m]), (🍋‹Num.numeral›, [n])) =>
  Thm.transitive
  (inst [] [m, n] neg_numeral_plus_numeral_a)
  (sub_conv_a n m)
  | ((🍋‹uminus›, [m]), (🍋‹uminus›, [n])) =>
  transitive'
  (inst [] [m, n] neg_numeral_plus_neg_numeral_a)
  (cong1 (cong1 (args2 add_num_conv)))
  | _ => conv m n
  end;
 
 fun plus_conv' a = norm1_eq a oo plus_conv (plus_neg_conv a) a;
 
 val int_plus_conv = plus_conv' 🍋‹int›;
 ›

lemma minus_one: "- 1 = - 1" by simp
lemma minus_numeral: "- numeral b = - numeral b" by simp

ML ‹
 fun uminus_conv a =
  let
  val minus_zero_a = inst [a] [] @{thm minus_zero [meta]};
  val minus_one_a = inst [a] [] @{thm minus_one [meta]};
  val minus_numeral_a = inst [a] [] @{thm minus_numeral [meta]};
  val minus_minus_a = inst [a] [] @{thm minus_minus [meta]}
  in
  fn n =>
  case strip_app n of
  (🍋‹zero_class.zero›, []) => minus_zero_a
  | (🍋‹one_class.one›, []) => minus_one_a
  | (🍋‹Num.numeral›, [m]) => inst [] [m] minus_numeral_a
  | (🍋‹uminus›, [m]) => inst [] [m] minus_minus_a
  end;
 
 val int_neg_conv = uminus_conv 🍋‹int›;
 ›

ML ‹
 fun minus_conv a =
  let
  val [numeral_minus_numeral_a, numeral_minus_neg_numeral_a,
  neg_numeral_minus_numeral_a, neg_numeral_minus_neg_numeral_a] =
  map (inst [a] []) @{thms diff_numeral_simps [meta]};
  val diff_0_a = inst [a] [] @{thm diff_0 [meta]};
  val diff_0_right_a = inst [a] [] @{thm diff_0_right [meta]};
  val sub_conv_a = sub_conv a;
  val uminus_conv_a = uminus_conv a;
  val expand1_a = expand1 a;
  val norm1_eq_a = norm1_eq a;
 
  fun conv m n = (case (strip_numeral m, strip_numeral n) of
  ((🍋‹zero_class.zero›, []), _) =>
  Thm.transitive (inst [] [n] diff_0_a) (uminus_conv_a n)
  | (_, (🍋‹zero_class.zero›, [])) => inst [] [m] diff_0_right_a
  | ((🍋‹Num.numeral›, [m]), (🍋‹Num.numeral›, [n])) =>
  Thm.transitive
  (inst [] [m, n] numeral_minus_numeral_a)
  (sub_conv_a m n)
  | ((🍋‹Num.numeral›, [m]), (🍋‹uminus›, [n])) =>
  transitive'
  (inst [] [m, n] numeral_minus_neg_numeral_a)
  (cong1 (args2 add_num_conv))
  | ((🍋‹uminus›, [m]), (🍋‹Num.numeral›, [n])) =>
  transitive'
  (inst [] [m, n] neg_numeral_minus_numeral_a)
  (cong1 (cong1 (args2 add_num_conv)))
  | ((🍋‹uminus›, [m]), (🍋‹uminus›, [n])) =>
  Thm.transitive
  (inst [] [m, n] neg_numeral_minus_neg_numeral_a)
  (sub_conv_a n m)
  | _ => cong2'' conv (expand1_a m) (expand1_a n))
  in norm1_eq_a oo conv end;
 
 val int_minus_conv = minus_conv 🍋‹int›;
 ›

ML ‹
 val int_numeral = Thm.apply 🍋‹numeral :: num ==> int›;
 
 val nat_minus_refl = Thm.reflexive 🍋‹minus :: nat ==> nat ==> nat›;
 
 val expand1_nat = expand1 🍋‹nat›;
 
 fun nat_minus_conv m n = (case (strip_app m, strip_app n) of
  ((🍋‹zero_class.zero›, []), _) =>
  inst [] [n] @{thm diff_0_eq_0 [meta]}
  | (_, (🍋‹zero_class.zero›, [])) =>
  inst [] [m] @{thm minus_nat.diff_0 [meta]}
  | ((🍋‹numeral›, [m]), (🍋‹numeral›, [n])) =>
  transitive'
  (inst [] [m, n] @{thm diff_nat_numeral [meta]})
  (cong1' nat_conv (args2 int_minus_conv))
  | _ => cong2'' nat_minus_conv (expand1_nat m) (expand1_nat n));
 ›

ML ‹
 fun mult_num_conv m n = (case (strip_app m, strip_app n) of
  (_, (🍋‹Num.One›, [])) =>
  inst [] [m] @{thm mult_num_simps(1) [meta]}
  | ((🍋‹Num.One›, []), _) =>
  inst [] [n] @{thm mult_num_simps(2) [meta]}
  | ((🍋‹Num.Bit0›, [m]), (🍋‹Num.Bit0›, [n])) =>
  transitive'
  (inst [] [m, n] @{thm mult_num_simps(3) [meta]})
  (cong1 (cong1 (args2 mult_num_conv)))
  | ((🍋‹Num.Bit0›, [m]), (🍋‹Num.Bit1›, [n'])) =>
  transitive'
  (inst [] [m, n'] @{thm mult_num_simps(4) [meta]})
  (cong1 (args2 mult_num_conv))
  | ((🍋‹Num.Bit1›, [m']), (🍋‹Num.Bit0›, [n])) =>
  transitive'
  (inst [] [m', n] @{thm mult_num_simps(5) [meta]})
  (cong1 (args2 mult_num_conv))
  | ((🍋‹Num.Bit1›, [m]), (🍋‹Num.Bit1›, [n])) =>
  transitive'
  (inst [] [m, n] @{thm mult_num_simps(6) [meta]})
  (cong1 (cong2' add_num_conv
  (args2 add_num_conv)
  (cong1 (args2 mult_num_conv)))));
 ›

ML ‹
 fun mult_conv f a =
  let
  val mult_zero_left_a = inst [a] [] @{thm mult_zero_left [meta]};
  val mult_zero_right_a = inst [a] [] @{thm mult_zero_right [meta]};
  val numeral_times_numeral_a = inst [a] [] @{thm numeral_times_numeral [meta]};
  val expand1_a = expand1 a;
  val norm1_eq_a = norm1_eq a;
 
  fun conv m n = (case (strip_app m, strip_app n) of
  ((🍋‹zero_class.zero›, []), _) => inst [] [n] mult_zero_left_a
  | (_, (🍋‹zero_class.zero›, [])) => inst [] [m] mult_zero_right_a
  | ((🍋‹numeral›, [m]), (🍋‹numeral›, [n])) =>
  transitive'
  (inst [] [m, n] numeral_times_numeral_a)
  (cong1 (args2 mult_num_conv))
  | _ => cong2'' (f conv) (expand1_a m) (expand1_a n))
  in norm1_eq_a oo f conv end;
 
 val nat_mult_conv = mult_conv I 🍋‹nat›;
 ›

ML ‹
 fun mult_neg_conv a =
  let
  val [neg_numeral_times_neg_numeral_a, neg_numeral_times_numeral_a,
  numeral_times_neg_numeral_a] =
  map (inst [a] []) @{thms mult_neg_numeral_simps [meta]};
  in
  fn conv => fn m => fn n =>
  case (strip_numeral m, strip_numeral n) of
  ((🍋‹uminus›, [m]), (🍋‹uminus›, [n])) =>
  transitive'
  (inst [] [m, n] neg_numeral_times_neg_numeral_a)
  (cong1 (args2 mult_num_conv))
  | ((🍋‹uminus›, [m]), (🍋‹numeral›, [n])) =>
  transitive'
  (inst [] [m, n] neg_numeral_times_numeral_a)
  (cong1 (cong1 (args2 mult_num_conv)))
  | ((🍋‹numeral›, [m]), (🍋‹uminus›, [n])) =>
  transitive'
  (inst [] [m, n] numeral_times_neg_numeral_a)
  (cong1 (cong1 (args2 mult_num_conv)))
  | _ => conv m n
  end;
 
 fun mult_conv' a = mult_conv (mult_neg_conv a) a;
 
 val int_mult_conv = mult_conv' 🍋‹int›;
 ›

ML ‹
 fun eq_num_conv m n = (case (strip_app m, strip_app n) of
  ((🍋‹Num.One›, []), (🍋‹Num.One›, [])) =>
  @{thm eq_num_simps(1) [meta]}
  | ((🍋‹Num.One›, []), (🍋‹Num.Bit0›, [n])) =>
  inst [] [n] @{thm eq_num_simps(2) [meta]}
  | ((🍋‹Num.One›, []), (🍋‹Num.Bit1›, [n])) =>
  inst [] [n] @{thm eq_num_simps(3) [meta]}
  | ((🍋‹Num.Bit0›, [m]), (🍋‹Num.One›, [])) =>
  inst [] [m] @{thm eq_num_simps(4) [meta]}
  | ((🍋‹Num.Bit1›, [m]), (🍋‹Num.One›, [])) =>
  inst [] [m] @{thm eq_num_simps(5) [meta]}
  | ((🍋‹Num.Bit0›, [m]), (🍋‹Num.Bit0›, [n])) =>
  Thm.transitive
  (inst [] [m, n] @{thm eq_num_simps(6) [meta]})
  (eq_num_conv m n)
  | ((🍋‹Num.Bit0›, [m]), (🍋‹Num.Bit1›, [n])) =>
  inst [] [m, n] @{thm eq_num_simps(7) [meta]}
  | ((🍋‹Num.Bit1›, [m]), (🍋‹Num.Bit0›, [n])) =>
  inst [] [m, n] @{thm eq_num_simps(8) [meta]}
  | ((🍋‹Num.Bit1›, [m]), (🍋‹Num.Bit1›, [n])) =>
  Thm.transitive
  (inst [] [m, n] @{thm eq_num_simps(9) [meta]})
  (eq_num_conv m n));
 ›

ML ‹
 fun eq_conv f a =
  let
  val zero_eq_zero_a = inst [a] [] @{thm refl [of 0, THEN Eq_TrueI]};
  val zero_neq_numeral_a =
  inst [a] [] @{thm zero_neq_numeral [THEN Eq_FalseI]};
  val numeral_neq_zero_a =
  inst [a] [] @{thm numeral_neq_zero [THEN Eq_FalseI]};
  val numeral_eq_iff_a = inst [a] [] @{thm numeral_eq_iff [meta]};
  val expand1_a = expand1 a;
 
  fun conv m n = (case (strip_app m, strip_app n) of
  ((🍋‹zero_class.zero›, []), (🍋‹zero_class.zero›, [])) =>
  zero_eq_zero_a
  | ((🍋‹zero_class.zero›, []), (🍋‹numeral›, [n])) =>
  inst [] [n] zero_neq_numeral_a
  | ((🍋‹numeral›, [m]), (🍋‹zero_class.zero›, [])) =>
  inst [] [m] numeral_neq_zero_a
  | ((🍋‹numeral›, [m]), (🍋‹numeral›, [n])) =>
  Thm.transitive
  (inst [] [m, n] numeral_eq_iff_a)
  (eq_num_conv m n)
  | _ => cong2'' (f conv) (expand1_a m) (expand1_a n))
  in f conv end;
 
 val nat_eq_conv = eq_conv I 🍋‹nat›;
 ›

ML ‹
 fun eq_neg_conv a =
  let
  val neg_numeral_neq_zero_a =
  inst [a] [] @{thm neg_numeral_neq_zero [THEN Eq_FalseI]};
  val zero_neq_neg_numeral_a =
  inst [a] [] @{thm zero_neq_neg_numeral [THEN Eq_FalseI]};
  val neg_numeral_neq_numeral_a =
  inst [a] [] @{thm neg_numeral_neq_numeral [THEN Eq_FalseI]};
  val numeral_neq_neg_numeral_a =
  inst [a] [] @{thm numeral_neq_neg_numeral [THEN Eq_FalseI]};
  val neg_numeral_eq_iff_a = inst [a] [] @{thm neg_numeral_eq_iff [meta]}
  in
  fn conv => fn m => fn n =>
  case (strip_numeral m, strip_numeral n) of
  ((🍋‹uminus›, [m]), (🍋‹zero_class.zero›, [])) =>
  inst [] [m] neg_numeral_neq_zero_a
  | ((🍋‹zero_class.zero›, []), (🍋‹uminus›, [n])) =>
  inst [] [n] zero_neq_neg_numeral_a
  | ((🍋‹Num.numeral›, [m]), (🍋‹uminus›, [n])) =>
  inst [] [m, n] numeral_neq_neg_numeral_a
  | ((🍋‹uminus›, [m]), (🍋‹Num.numeral›, [n])) =>
  inst [] [m, n] neg_numeral_neq_numeral_a
  | ((🍋‹uminus›, [m]), (🍋‹uminus›, [n])) =>
  Thm.transitive
  (inst [] [m, n] neg_numeral_eq_iff_a)
  (eq_num_conv m n)
  | _ => conv m n
  end;
 
 fun eq_conv' a = eq_conv (eq_neg_conv a) a;
 
 val int_eq_conv = eq_conv' 🍋‹int›;
 ›

ML ‹
 fun le_num_conv m n = (case (strip_app m, strip_app n) of
  ((🍋‹Num.One›, []), _) =>
  inst [] [n] @{thm le_num_simps(1) [meta]}
  | ((🍋‹Num.Bit0›, [m]), (🍋‹Num.One›, [])) =>
  inst [] [m] @{thm le_num_simps(2) [meta]}
  | ((🍋‹Num.Bit1›, [m]), (🍋‹Num.One›, [])) =>
  inst [] [m] @{thm le_num_simps(3) [meta]}
  | ((🍋‹Num.Bit0›, [m]), (🍋‹Num.Bit0›, [n])) =>
  Thm.transitive
  (inst [] [m, n] @{thm le_num_simps(4) [meta]})
  (le_num_conv m n)
  | ((🍋‹Num.Bit0›, [m]), (🍋‹Num.Bit1›, [n])) =>
  Thm.transitive
  (inst [] [m, n] @{thm le_num_simps(5) [meta]})
  (le_num_conv m n)
  | ((🍋‹Num.Bit1›, [m]), (🍋‹Num.Bit1›, [n])) =>
  Thm.transitive
  (inst [] [m, n] @{thm le_num_simps(6) [meta]})
  (le_num_conv m n)
  | ((🍋‹Num.Bit1›, [m]), (🍋‹Num.Bit0›, [n])) =>
  Thm.transitive
  (inst [] [m, n] @{thm le_num_simps(7) [meta]})
  (less_num_conv m n))
 
 and less_num_conv m n = (case (strip_app m, strip_app n) of
  (_, (🍋‹Num.One›, [])) =>
  inst [] [m] @{thm less_num_simps(1) [meta]}
  | ((🍋‹Num.One›, []), (🍋‹Num.Bit0›, [n])) =>
  inst [] [n] @{thm less_num_simps(2) [meta]}
  | ((🍋‹Num.One›, []), (🍋‹Num.Bit1›, [n])) =>
  inst [] [n] @{thm less_num_simps(3) [meta]}
  | ((🍋‹Num.Bit0›, [m]), (🍋‹Num.Bit0›, [n])) =>
  Thm.transitive
  (inst [] [m, n] @{thm less_num_simps(4) [meta]})
  (less_num_conv m n)
  | ((🍋‹Num.Bit0›, [m]), (🍋‹Num.Bit1›, [n])) =>
  Thm.transitive
  (inst [] [m, n] @{thm less_num_simps(5) [meta]})
  (le_num_conv m n)
  | ((🍋‹Num.Bit1›, [m]), (🍋‹Num.Bit1›, [n])) =>
  Thm.transitive
  (inst [] [m, n] @{thm less_num_simps(6) [meta]})
  (less_num_conv m n)
  | ((🍋‹Num.Bit1›, [m]), (🍋‹Num.Bit0›, [n])) =>
  Thm.transitive
  (inst [] [m, n] @{thm less_num_simps(7) [meta]})
  (less_num_conv m n));
 ›

ML ‹
 fun le_conv f a =
  let
  val zero_le_zero_a = inst [a] [] @{thm order_refl [of 0, THEN Eq_TrueI]};
  val zero_le_numeral_a =
  inst [a] [] @{thm zero_le_numeral [THEN Eq_TrueI]};
  val not_numeral_le_zero_a =
  inst [a] [] @{thm not_numeral_le_zero [THEN Eq_FalseI]};
  val numeral_le_iff_a = inst [a] [] @{thm numeral_le_iff [meta]};
  val expand1_a = expand1 a;
 
  fun conv m n = (case (strip_app m, strip_app n) of
  ((🍋‹zero_class.zero›, []), (🍋‹zero_class.zero›, [])) =>
  zero_le_zero_a
  | ((🍋‹zero_class.zero›, []), (🍋‹numeral›, [n])) =>
  inst [] [n] zero_le_numeral_a
  | ((🍋‹numeral›, [m]), (🍋‹zero_class.zero›, [])) =>
  inst [] [m] not_numeral_le_zero_a
  | ((🍋‹numeral›, [m]), (🍋‹numeral›, [n])) =>
  Thm.transitive
  (inst [] [m, n] numeral_le_iff_a)
  (le_num_conv m n)
  | _ => cong2'' (f conv) (expand1_a m) (expand1_a n))
  in f conv end;
 
 val nat_le_conv = le_conv I 🍋‹nat›;
 ›

ML ‹
 fun le_neg_conv a =
  let
  val neg_numeral_le_zero_a =
  inst [a] [] @{thm neg_numeral_le_zero [THEN Eq_TrueI]};
  val not_zero_le_neg_numeral_a =
  inst [a] [] @{thm not_zero_le_neg_numeral [THEN Eq_FalseI]};
  val neg_numeral_le_numeral_a =
  inst [a] [] @{thm neg_numeral_le_numeral [THEN Eq_TrueI]};
  val not_numeral_le_neg_numeral_a =
  inst [a] [] @{thm not_numeral_le_neg_numeral [THEN Eq_FalseI]};
  val neg_numeral_le_iff_a = inst [a] [] @{thm neg_numeral_le_iff [meta]}
  in
  fn conv => fn m => fn n =>
  case (strip_numeral m, strip_numeral n) of
  ((🍋‹uminus›, [m]), (🍋‹zero_class.zero›, [])) =>
  inst [] [m] neg_numeral_le_zero_a
  | ((🍋‹zero_class.zero›, []), (🍋‹uminus›, [n])) =>
  inst [] [n] not_zero_le_neg_numeral_a
  | ((🍋‹Num.numeral›, [m]), (🍋‹uminus›, [n])) =>
  inst [] [m, n] not_numeral_le_neg_numeral_a
  | ((🍋‹uminus›, [m]), (🍋‹Num.numeral›, [n])) =>
  inst [] [m, n] neg_numeral_le_numeral_a
  | ((🍋‹uminus›, [m]), (🍋‹uminus›, [n])) =>
  Thm.transitive
  (inst [] [m, n] neg_numeral_le_iff_a)
  (le_num_conv n m)
  | _ => conv m n
  end;
 
 fun le_conv' a = le_conv (le_neg_conv a) a;
 
 val int_le_conv = le_conv' 🍋‹int›;
 ›

ML ‹
 fun less_conv f a =
  let
  val not_zero_less_zero_a = inst [a] [] @{thm less_irrefl [of 0, THEN Eq_FalseI]};
  val zero_less_numeral_a =
  inst [a] [] @{thm zero_less_numeral [THEN Eq_TrueI]};
  val not_numeral_less_zero_a =
  inst [a] [] @{thm not_numeral_less_zero [THEN Eq_FalseI]};
  val numeral_less_iff_a = inst [a] [] @{thm numeral_less_iff [meta]};
  val expand1_a = expand1 a;
 
  fun conv m n = (case (strip_app m, strip_app n) of
  ((🍋‹zero_class.zero›, []), (🍋‹zero_class.zero›, [])) =>
  not_zero_less_zero_a
  | ((🍋‹zero_class.zero›, []), (🍋‹numeral›, [n])) =>
  inst [] [n] zero_less_numeral_a
  | ((🍋‹numeral›, [m]), (🍋‹zero_class.zero›, [])) =>
  inst [] [m] not_numeral_less_zero_a
  | ((🍋‹numeral›, [m]), (🍋‹numeral›, [n])) =>
  Thm.transitive
  (inst [] [m, n] numeral_less_iff_a)
  (less_num_conv m n)
  | _ => cong2'' (f conv) (expand1_a m) (expand1_a n))
  in f conv end;
 
 val nat_less_conv = less_conv I 🍋‹nat›;
 ›

ML ‹
 fun less_neg_conv a =
  let
  val neg_numeral_less_zero_a =
  inst [a] [] @{thm neg_numeral_less_zero [THEN Eq_TrueI]};
  val not_zero_less_neg_numeral_a =
  inst [a] [] @{thm not_zero_less_neg_numeral [THEN Eq_FalseI]};
  val neg_numeral_less_numeral_a =
  inst [a] [] @{thm neg_numeral_less_numeral [THEN Eq_TrueI]};
  val not_numeral_less_neg_numeral_a =
  inst [a] [] @{thm not_numeral_less_neg_numeral [THEN Eq_FalseI]};
  val neg_numeral_less_iff_a = inst [a] [] @{thm neg_numeral_less_iff [meta]}
  in
  fn conv => fn m => fn n =>
  case (strip_numeral m, strip_numeral n) of
  ((🍋‹uminus›, [m]), (🍋‹zero_class.zero›, [])) =>
  inst [] [m] neg_numeral_less_zero_a
  | ((🍋‹zero_class.zero›, []), (🍋‹uminus›, [n])) =>
  inst [] [n] not_zero_less_neg_numeral_a
  | ((🍋‹Num.numeral›, [m]), (🍋‹uminus›, [n])) =>
  inst [] [m, n] not_numeral_less_neg_numeral_a
  | ((🍋‹uminus›, [m]), (🍋‹Num.numeral›, [n])) =>
  inst [] [m, n] neg_numeral_less_numeral_a
  | ((🍋‹uminus›, [m]), (🍋‹uminus›, [n])) =>
  Thm.transitive
  (inst [] [m, n] neg_numeral_less_iff_a)
  (less_num_conv n m)
  | _ => conv m n
  end;
 
 fun less_conv' a = less_conv (less_neg_conv a) a;
 
 val int_less_conv = less_conv' 🍋‹int›;
 ›

ML ‹
 fun If_conv a =
  let
  val if_True = inst [a] [] @{thm if_True [meta]};
  val if_False = inst [a] [] @{thm if_False [meta]}
  in
  fn p => fn x => fn y => fn ct =>
  case strip_app ct of
  (🍋‹If›, [cb, cx, cy]) =>
  let
  val p_eq = p cb
  val eq = Thm.combination (Thm.reflexive (Thm.dest_fun (Thm.dest_fun2 ct))) p_eq
  in
  case Thm.term_of (Thm.rhs_of p_eq) of
  🍋‹True› =>
  let
  val x_eq = x cx;
  val cx = Thm.rhs_of x_eq;
  in
  Thm.transitive
  (Thm.combination
  (Thm.combination eq x_eq)
  (Thm.reflexive cy))
  (inst [] [cx, cy] if_True)
  end
  | 🍋‹False› =>
  let
  val y_eq = y cy;
  val cy = Thm.rhs_of y_eq;
  in
  Thm.transitive
  (Thm.combination
  (Thm.combination eq (Thm.reflexive cx))
  y_eq)
  (inst [] [cx, cy] if_False)
  end
  | _ => err "If_conv" (Thm.rhs_of p_eq)
  end
  end;
 ›

ML ‹
 fun drop_conv a =
  let
  val drop_0_a = inst [a] [] @{thm drop_0 [meta]};
  val drop_Cons_a = inst [a] [] @{thm drop_Cons' [meta]};
  val If_conv_a = If_conv (type_of_eqn drop_0_a);
 
  fun conv n ys = (case Thm.term_of n of
  🍋‹zero_class.zero _› => inst [] [ys] drop_0_a
  | _ => (case strip_app ys of
  (🍋‹Cons›, [x, xs]) =>
  transitive'
  (inst [] [n, x, xs] drop_Cons_a)
  (If_conv_a (args2 nat_eq_conv)
  Thm.reflexive
  (cong2' conv (args2 nat_minus_conv) Thm.reflexive))))
  in conv end;
 ›

ML ‹
 fun nth_conv a =
  let
  val nth_Cons_a = inst [a] [] @{thm nth_Cons' [meta]};
  val If_conv_a = If_conv a;
 
  fun conv ys n = (case strip_app ys of
  (🍋‹Cons›, [x, xs]) =>
  transitive'
  (inst [] [x, xs, n] nth_Cons_a)
  (If_conv_a (args2 nat_eq_conv)
  Thm.reflexive
  (cong2' conv Thm.reflexive (args2 nat_minus_conv))))
  in conv end;
 ›

end