--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Decision_Procs/Conversions.thy Sun Jan 29 11:59:48 2017 +0100
@@ -0,0 +1,844 @@
+(* 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 (fn (ctxt, th) => (NONE, SOME (mk_meta_eq th))) *}
+ {* convert equality to meta equality *}
+
+ML {*
+fun mk_obj_eq th = th RS @{thm meta_eq_to_obj_eq};
+
+fun strip_app ct = ct |> Drule.strip_comb |>> Thm.term_of |>> dest_Const |>> fst;
+
+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
+ (@{const_name uminus}, [n]) => (case strip_app n of
+ (@{const_name numeral}, [b]) => (@{const_name uminus}, [b])
+ | _ => ("", []))
+ | x => x);
+*}
+
+lemma nat_minus1_eq: "nat (- 1) = 0"
+ by simp
+
+ML {*
+fun nat_conv i = (case strip_app i of
+ (@{const_name zero_class.zero}, []) => @{thm nat_0 [meta]}
+ | (@{const_name one_class.one}, []) => @{thm transfer_nat_int_numerals(2) [meta, symmetric]}
+ | (@{const_name numeral}, [b]) => inst [] [b] @{thm nat_numeral [meta]}
+ | (@{const_name uminus}, [b]) => (case strip_app b of
+ (@{const_name one_class.one}, []) => @{thm nat_minus1_eq [meta]}
+ | (@{const_name numeral}, [b']) => inst [] [b'] @{thm nat_neg_numeral [meta]}));
+*}
+
+ML {*
+fun add_num_conv b b' = (case (strip_app b, strip_app b') of
+ ((@{const_name Num.One}, []), (@{const_name Num.One}, [])) =>
+ @{thm add_num_simps(1) [meta]}
+ | ((@{const_name Num.One}, []), (@{const_name Num.Bit0}, [n])) =>
+ inst [] [n] @{thm add_num_simps(2) [meta]}
+ | ((@{const_name Num.One}, []), (@{const_name Num.Bit1}, [n])) =>
+ transitive'
+ (inst [] [n] @{thm add_num_simps(3) [meta]})
+ (cong1 (args2 add_num_conv))
+ | ((@{const_name Num.Bit0}, [m]), (@{const_name Num.One}, [])) =>
+ inst [] [m] @{thm add_num_simps(4) [meta]}
+ | ((@{const_name Num.Bit0}, [m]), (@{const_name Num.Bit0}, [n])) =>
+ transitive'
+ (inst [] [m, n] @{thm add_num_simps(5) [meta]})
+ (cong1 (args2 add_num_conv))
+ | ((@{const_name Num.Bit0}, [m]), (@{const_name Num.Bit1}, [n])) =>
+ transitive'
+ (inst [] [m, n] @{thm add_num_simps(6) [meta]})
+ (cong1 (args2 add_num_conv))
+ | ((@{const_name Num.Bit1}, [m]), (@{const_name Num.One}, [])) =>
+ transitive'
+ (inst [] [m] @{thm add_num_simps(7) [meta]})
+ (cong1 (args2 add_num_conv))
+ | ((@{const_name Num.Bit1}, [m]), (@{const_name Num.Bit0}, [n])) =>
+ transitive'
+ (inst [] [m, n] @{thm add_num_simps(8) [meta]})
+ (cong1 (args2 add_num_conv))
+ | ((@{const_name Num.Bit1}, [m]), (@{const_name 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
+ (@{const_name Num.One}, []) => @{thm BitM.simps(1) [meta]}
+ | (@{const_name Num.Bit0}, [n]) =>
+ transitive'
+ (inst [] [n] @{thm BitM.simps(2) [meta]})
+ (cong1 (args1 BitM_conv))
+ | (@{const_name 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
+ (@{const_name zero_class.zero}, []) => dbl_0_a
+ | (@{const_name numeral}, [k]) => inst [] [k] dbl_numeral_a
+ | (@{const_name 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
+ (@{const_name zero_class.zero}, []) => dbl_inc_0_a
+ | (@{const_name numeral}, [k]) => inst [] [k] dbl_inc_numeral_a
+ | (@{const_name 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
+ (@{const_name zero_class.zero}, []) => dbl_dec_0_a
+ | (@{const_name uminus}, [k]) => inst [] [k] dbl_dec_neg_numeral_a
+ | (@{const_name 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
+ ((@{const_name Num.One}, []), (@{const_name Num.One}, [])) =>
+ sub_One_One
+ | ((@{const_name Num.One}, []), (@{const_name Num.Bit0}, [l])) =>
+ transitive'
+ (inst [] [l] sub_One_Bit0)
+ (cong1 (cong1 (args1 BitM_conv)))
+ | ((@{const_name Num.One}, []), (@{const_name Num.Bit1}, [l])) =>
+ inst [] [l] sub_One_Bit1
+ | ((@{const_name Num.Bit0}, [k]), (@{const_name Num.One}, [])) =>
+ transitive'
+ (inst [] [k] sub_Bit0_One)
+ (cong1 (args1 BitM_conv))
+ | ((@{const_name Num.Bit1}, [k]), (@{const_name Num.One}, [])) =>
+ inst [] [k] sub_Bit1_One
+ | ((@{const_name Num.Bit0}, [k]), (@{const_name Num.Bit0}, [l])) =>
+ transitive'
+ (inst [] [k, l] sub_Bit0_Bit0)
+ (cong1' dbl_conv_a (args2 conv))
+ | ((@{const_name Num.Bit0}, [k]), (@{const_name Num.Bit1}, [l])) =>
+ transitive'
+ (inst [] [k, l] sub_Bit0_Bit1)
+ (cong1' dbl_dec_conv_a (args2 conv))
+ | ((@{const_name Num.Bit1}, [k]), (@{const_name Num.Bit0}, [l])) =>
+ transitive'
+ (inst [] [k, l] sub_Bit1_Bit0)
+ (cong1' dbl_inc_conv_a (args2 conv))
+ | ((@{const_name Num.Bit1}, [k]), (@{const_name 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
+ Const (@{const_name one_class.one}, _) => numeral_1_eq_1_a
+ | Const (@{const_name uminus}, _) $ Const (@{const_name one_class.one}, _) =>
+ Thm.combination (Thm.reflexive (Thm.dest_fun n)) numeral_1_eq_1_a
+ | Const (@{const_name zero_class.zero}, _) => Thm.reflexive n
+ | Const (@{const_name numeral}, _) $ _ => Thm.reflexive n
+ | Const (@{const_name uminus}, _) $
+ (Const (@{const_name numeral}, _) $ _) => 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
+ Const (@{const_name Num.numeral}, _) $ Const (@{const_name Num.One}, _) =>
+ Thm.transitive eq numeral_1_eq_1_a
+ | Const (@{const_name uminus}, _) $
+ (Const (@{const_name Num.numeral}, _) $ Const (@{const_name 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
+ ((@{const_name zero_class.zero}, []), _) => inst [] [n] add_0_a
+ | (_, (@{const_name zero_class.zero}, [])) => inst [] [m] add_0_right_a
+ | ((@{const_name numeral}, [m]), (@{const_name 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 @{ctyp 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
+ ((@{const_name Num.numeral}, [m]), (@{const_name uminus}, [n])) =>
+ Thm.transitive
+ (inst [] [m, n] numeral_plus_neg_numeral_a)
+ (sub_conv_a m n)
+ | ((@{const_name uminus}, [m]), (@{const_name Num.numeral}, [n])) =>
+ Thm.transitive
+ (inst [] [m, n] neg_numeral_plus_numeral_a)
+ (sub_conv_a n m)
+ | ((@{const_name uminus}, [m]), (@{const_name 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' @{ctyp 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
+ (@{const_name zero_class.zero}, []) => minus_zero_a
+ | (@{const_name one_class.one}, []) => minus_one_a
+ | (@{const_name Num.numeral}, [m]) => inst [] [m] minus_numeral_a
+ | (@{const_name uminus}, [m]) => inst [] [m] minus_minus_a
+ end;
+
+val int_neg_conv = uminus_conv @{ctyp 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
+ ((@{const_name zero_class.zero}, []), _) =>
+ Thm.transitive (inst [] [n] diff_0_a) (uminus_conv_a n)
+ | (_, (@{const_name zero_class.zero}, [])) => inst [] [m] diff_0_right_a
+ | ((@{const_name Num.numeral}, [m]), (@{const_name Num.numeral}, [n])) =>
+ Thm.transitive
+ (inst [] [m, n] numeral_minus_numeral_a)
+ (sub_conv_a m n)
+ | ((@{const_name Num.numeral}, [m]), (@{const_name uminus}, [n])) =>
+ transitive'
+ (inst [] [m, n] numeral_minus_neg_numeral_a)
+ (cong1 (args2 add_num_conv))
+ | ((@{const_name uminus}, [m]), (@{const_name Num.numeral}, [n])) =>
+ transitive'
+ (inst [] [m, n] neg_numeral_minus_numeral_a)
+ (cong1 (cong1 (args2 add_num_conv)))
+ | ((@{const_name uminus}, [m]), (@{const_name 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 @{ctyp int};
+*}
+
+ML {*
+val int_numeral = Thm.apply @{cterm "numeral :: num \<Rightarrow> int"};
+
+val nat_minus_refl = Thm.reflexive @{cterm "minus :: nat \<Rightarrow> nat \<Rightarrow> nat"};
+
+val expand1_nat = expand1 @{ctyp nat};
+
+fun nat_minus_conv m n = (case (strip_app m, strip_app n) of
+ ((@{const_name zero_class.zero}, []), _) =>
+ inst [] [n] @{thm diff_0_eq_0 [meta]}
+ | (_, (@{const_name zero_class.zero}, [])) =>
+ inst [] [m] @{thm minus_nat.diff_0 [meta]}
+ | ((@{const_name numeral}, [m]), (@{const_name 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
+ (_, (@{const_name Num.One}, [])) =>
+ inst [] [m] @{thm mult_num_simps(1) [meta]}
+ | ((@{const_name Num.One}, []), _) =>
+ inst [] [n] @{thm mult_num_simps(2) [meta]}
+ | ((@{const_name Num.Bit0}, [m]), (@{const_name Num.Bit0}, [n])) =>
+ transitive'
+ (inst [] [m, n] @{thm mult_num_simps(3) [meta]})
+ (cong1 (cong1 (args2 mult_num_conv)))
+ | ((@{const_name Num.Bit0}, [m]), (@{const_name Num.Bit1}, [n'])) =>
+ transitive'
+ (inst [] [m, n'] @{thm mult_num_simps(4) [meta]})
+ (cong1 (args2 mult_num_conv))
+ | ((@{const_name Num.Bit1}, [m']), (@{const_name Num.Bit0}, [n])) =>
+ transitive'
+ (inst [] [m', n] @{thm mult_num_simps(5) [meta]})
+ (cong1 (args2 mult_num_conv))
+ | ((@{const_name Num.Bit1}, [m]), (@{const_name 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
+ ((@{const_name zero_class.zero}, []), _) => inst [] [n] mult_zero_left_a
+ | (_, (@{const_name zero_class.zero}, [])) => inst [] [m] mult_zero_right_a
+ | ((@{const_name numeral}, [m]), (@{const_name 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 @{ctyp 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
+ ((@{const_name uminus}, [m]), (@{const_name uminus}, [n])) =>
+ transitive'
+ (inst [] [m, n] neg_numeral_times_neg_numeral_a)
+ (cong1 (args2 mult_num_conv))
+ | ((@{const_name uminus}, [m]), (@{const_name numeral}, [n])) =>
+ transitive'
+ (inst [] [m, n] neg_numeral_times_numeral_a)
+ (cong1 (cong1 (args2 mult_num_conv)))
+ | ((@{const_name numeral}, [m]), (@{const_name 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' @{ctyp int};
+*}
+
+ML {*
+fun eq_num_conv m n = (case (strip_app m, strip_app n) of
+ ((@{const_name Num.One}, []), (@{const_name Num.One}, [])) =>
+ @{thm eq_num_simps(1) [meta]}
+ | ((@{const_name Num.One}, []), (@{const_name Num.Bit0}, [n])) =>
+ inst [] [n] @{thm eq_num_simps(2) [meta]}
+ | ((@{const_name Num.One}, []), (@{const_name Num.Bit1}, [n])) =>
+ inst [] [n] @{thm eq_num_simps(3) [meta]}
+ | ((@{const_name Num.Bit0}, [m]), (@{const_name Num.One}, [])) =>
+ inst [] [m] @{thm eq_num_simps(4) [meta]}
+ | ((@{const_name Num.Bit1}, [m]), (@{const_name Num.One}, [])) =>
+ inst [] [m] @{thm eq_num_simps(5) [meta]}
+ | ((@{const_name Num.Bit0}, [m]), (@{const_name Num.Bit0}, [n])) =>
+ Thm.transitive
+ (inst [] [m, n] @{thm eq_num_simps(6) [meta]})
+ (eq_num_conv m n)
+ | ((@{const_name Num.Bit0}, [m]), (@{const_name Num.Bit1}, [n])) =>
+ inst [] [m, n] @{thm eq_num_simps(7) [meta]}
+ | ((@{const_name Num.Bit1}, [m]), (@{const_name Num.Bit0}, [n])) =>
+ inst [] [m, n] @{thm eq_num_simps(8) [meta]}
+ | ((@{const_name Num.Bit1}, [m]), (@{const_name 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
+ ((@{const_name zero_class.zero}, []), (@{const_name zero_class.zero}, [])) =>
+ zero_eq_zero_a
+ | ((@{const_name zero_class.zero}, []), (@{const_name numeral}, [n])) =>
+ inst [] [n] zero_neq_numeral_a
+ | ((@{const_name numeral}, [m]), (@{const_name zero_class.zero}, [])) =>
+ inst [] [m] numeral_neq_zero_a
+ | ((@{const_name numeral}, [m]), (@{const_name 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 @{ctyp 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
+ ((@{const_name uminus}, [m]), (@{const_name zero_class.zero}, [])) =>
+ inst [] [m] neg_numeral_neq_zero_a
+ | ((@{const_name zero_class.zero}, []), (@{const_name uminus}, [n])) =>
+ inst [] [n] zero_neq_neg_numeral_a
+ | ((@{const_name Num.numeral}, [m]), (@{const_name uminus}, [n])) =>
+ inst [] [m, n] numeral_neq_neg_numeral_a
+ | ((@{const_name uminus}, [m]), (@{const_name Num.numeral}, [n])) =>
+ inst [] [m, n] neg_numeral_neq_numeral_a
+ | ((@{const_name uminus}, [m]), (@{const_name 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' @{ctyp int};
+*}
+
+ML {*
+fun le_num_conv m n = (case (strip_app m, strip_app n) of
+ ((@{const_name Num.One}, []), _) =>
+ inst [] [n] @{thm le_num_simps(1) [meta]}
+ | ((@{const_name Num.Bit0}, [m]), (@{const_name Num.One}, [])) =>
+ inst [] [m] @{thm le_num_simps(2) [meta]}
+ | ((@{const_name Num.Bit1}, [m]), (@{const_name Num.One}, [])) =>
+ inst [] [m] @{thm le_num_simps(3) [meta]}
+ | ((@{const_name Num.Bit0}, [m]), (@{const_name Num.Bit0}, [n])) =>
+ Thm.transitive
+ (inst [] [m, n] @{thm le_num_simps(4) [meta]})
+ (le_num_conv m n)
+ | ((@{const_name Num.Bit0}, [m]), (@{const_name Num.Bit1}, [n])) =>
+ Thm.transitive
+ (inst [] [m, n] @{thm le_num_simps(5) [meta]})
+ (le_num_conv m n)
+ | ((@{const_name Num.Bit1}, [m]), (@{const_name Num.Bit1}, [n])) =>
+ Thm.transitive
+ (inst [] [m, n] @{thm le_num_simps(6) [meta]})
+ (le_num_conv m n)
+ | ((@{const_name Num.Bit1}, [m]), (@{const_name 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
+ (_, (@{const_name Num.One}, [])) =>
+ inst [] [m] @{thm less_num_simps(1) [meta]}
+ | ((@{const_name Num.One}, []), (@{const_name Num.Bit0}, [n])) =>
+ inst [] [n] @{thm less_num_simps(2) [meta]}
+ | ((@{const_name Num.One}, []), (@{const_name Num.Bit1}, [n])) =>
+ inst [] [n] @{thm less_num_simps(3) [meta]}
+ | ((@{const_name Num.Bit0}, [m]), (@{const_name Num.Bit0}, [n])) =>
+ Thm.transitive
+ (inst [] [m, n] @{thm less_num_simps(4) [meta]})
+ (less_num_conv m n)
+ | ((@{const_name Num.Bit0}, [m]), (@{const_name Num.Bit1}, [n])) =>
+ Thm.transitive
+ (inst [] [m, n] @{thm less_num_simps(5) [meta]})
+ (le_num_conv m n)
+ | ((@{const_name Num.Bit1}, [m]), (@{const_name Num.Bit1}, [n])) =>
+ Thm.transitive
+ (inst [] [m, n] @{thm less_num_simps(6) [meta]})
+ (less_num_conv m n)
+ | ((@{const_name Num.Bit1}, [m]), (@{const_name 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
+ ((@{const_name zero_class.zero}, []), (@{const_name zero_class.zero}, [])) =>
+ zero_le_zero_a
+ | ((@{const_name zero_class.zero}, []), (@{const_name numeral}, [n])) =>
+ inst [] [n] zero_le_numeral_a
+ | ((@{const_name numeral}, [m]), (@{const_name zero_class.zero}, [])) =>
+ inst [] [m] not_numeral_le_zero_a
+ | ((@{const_name numeral}, [m]), (@{const_name 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 @{ctyp 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
+ ((@{const_name uminus}, [m]), (@{const_name zero_class.zero}, [])) =>
+ inst [] [m] neg_numeral_le_zero_a
+ | ((@{const_name zero_class.zero}, []), (@{const_name uminus}, [n])) =>
+ inst [] [n] not_zero_le_neg_numeral_a
+ | ((@{const_name Num.numeral}, [m]), (@{const_name uminus}, [n])) =>
+ inst [] [m, n] not_numeral_le_neg_numeral_a
+ | ((@{const_name uminus}, [m]), (@{const_name Num.numeral}, [n])) =>
+ inst [] [m, n] neg_numeral_le_numeral_a
+ | ((@{const_name uminus}, [m]), (@{const_name 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' @{ctyp 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
+ ((@{const_name zero_class.zero}, []), (@{const_name zero_class.zero}, [])) =>
+ not_zero_less_zero_a
+ | ((@{const_name zero_class.zero}, []), (@{const_name numeral}, [n])) =>
+ inst [] [n] zero_less_numeral_a
+ | ((@{const_name numeral}, [m]), (@{const_name zero_class.zero}, [])) =>
+ inst [] [m] not_numeral_less_zero_a
+ | ((@{const_name numeral}, [m]), (@{const_name 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 @{ctyp 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
+ ((@{const_name uminus}, [m]), (@{const_name zero_class.zero}, [])) =>
+ inst [] [m] neg_numeral_less_zero_a
+ | ((@{const_name zero_class.zero}, []), (@{const_name uminus}, [n])) =>
+ inst [] [n] not_zero_less_neg_numeral_a
+ | ((@{const_name Num.numeral}, [m]), (@{const_name uminus}, [n])) =>
+ inst [] [m, n] not_numeral_less_neg_numeral_a
+ | ((@{const_name uminus}, [m]), (@{const_name Num.numeral}, [n])) =>
+ inst [] [m, n] neg_numeral_less_numeral_a
+ | ((@{const_name uminus}, [m]), (@{const_name 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' @{ctyp 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
+ (@{const_name 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
+ Const (@{const_name 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
+ | Const (@{const_name 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
+ Const (@{const_name zero_class.zero}, _) => inst [] [ys] drop_0_a
+ | _ => (case strip_app ys of
+ (@{const_name 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
+ (@{const_name 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