src/ZF/int_arith.ML
author wenzelm
Mon Feb 11 21:32:12 2008 +0100 (2008-02-11)
changeset 26059 b67a225b50fd
parent 26056 6a0801279f4c
child 26190 cf51a23c0cd0
permissions -rw-r--r--
removed unnecessary theory qualifiers;
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(*  Title:      ZF/int_arith.ML
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    ID:         $Id$
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    Author:     Larry Paulson
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    Copyright   2000  University of Cambridge
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Simprocs for linear arithmetic.
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*)
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(** To simplify inequalities involving integer negation and literals,
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    such as -x = #3
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**)
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Addsimps [inst "y" "integ_of(?w)" @{thm zminus_equation},
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          inst "x" "integ_of(?w)" @{thm equation_zminus}];
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AddIffs [inst "y" "integ_of(?w)" @{thm zminus_zless},
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         inst "x" "integ_of(?w)" @{thm zless_zminus}];
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AddIffs [inst "y" "integ_of(?w)" @{thm zminus_zle},
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         inst "x" "integ_of(?w)" @{thm zle_zminus}];
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Addsimps [inst "s" "integ_of(?w)" @{thm Let_def}];
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(*** Simprocs for numeric literals ***)
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(** Combining of literal coefficients in sums of products **)
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Goal "(x $< y) <-> (x$-y $< #0)";
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by (simp_tac (simpset() addsimps @{thms zcompare_rls}) 1);
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qed "zless_iff_zdiff_zless_0";
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Goal "[| x: int; y: int |] ==> (x = y) <-> (x$-y = #0)";
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by (asm_simp_tac (simpset() addsimps @{thms zcompare_rls}) 1);
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qed "eq_iff_zdiff_eq_0";
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Goal "(x $<= y) <-> (x$-y $<= #0)";
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by (asm_simp_tac (simpset() addsimps @{thms zcompare_rls}) 1);
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qed "zle_iff_zdiff_zle_0";
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(** For combine_numerals **)
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Goal "i$*u $+ (j$*u $+ k) = (i$+j)$*u $+ k";
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by (simp_tac (simpset() addsimps [@{thm zadd_zmult_distrib}]@ @{thms zadd_ac}) 1);
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qed "left_zadd_zmult_distrib";
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(** For cancel_numerals **)
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val rel_iff_rel_0_rls = map (inst "y" "?u$+?v")
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                          [zless_iff_zdiff_zless_0, eq_iff_zdiff_eq_0,
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                           zle_iff_zdiff_zle_0] @
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                        map (inst "y" "n")
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                          [zless_iff_zdiff_zless_0, eq_iff_zdiff_eq_0,
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                           zle_iff_zdiff_zle_0];
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Goal "(i$*u $+ m = j$*u $+ n) <-> ((i$-j)$*u $+ m = intify(n))";
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by (simp_tac (simpset() addsimps [@{thm zdiff_def}, @{thm zadd_zmult_distrib}]) 1);
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by (simp_tac (simpset() addsimps @{thms zcompare_rls}) 1);
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by (simp_tac (simpset() addsimps @{thms zadd_ac}) 1);
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qed "eq_add_iff1";
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Goal "(i$*u $+ m = j$*u $+ n) <-> (intify(m) = (j$-i)$*u $+ n)";
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by (simp_tac (simpset() addsimps [@{thm zdiff_def}, @{thm zadd_zmult_distrib}]) 1);
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by (simp_tac (simpset() addsimps @{thms zcompare_rls}) 1);
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by (simp_tac (simpset() addsimps @{thms zadd_ac}) 1);
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qed "eq_add_iff2";
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Goal "(i$*u $+ m $< j$*u $+ n) <-> ((i$-j)$*u $+ m $< n)";
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by (asm_simp_tac (simpset() addsimps [@{thm zdiff_def}, @{thm zadd_zmult_distrib}]@
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                                     @{thms zadd_ac} @ rel_iff_rel_0_rls) 1);
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qed "less_add_iff1";
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Goal "(i$*u $+ m $< j$*u $+ n) <-> (m $< (j$-i)$*u $+ n)";
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by (asm_simp_tac (simpset() addsimps [@{thm zdiff_def}, @{thm zadd_zmult_distrib}]@
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                                     @{thms zadd_ac} @ rel_iff_rel_0_rls) 1);
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qed "less_add_iff2";
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Goal "(i$*u $+ m $<= j$*u $+ n) <-> ((i$-j)$*u $+ m $<= n)";
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by (simp_tac (simpset() addsimps [@{thm zdiff_def}, @{thm zadd_zmult_distrib}]) 1);
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by (simp_tac (simpset() addsimps @{thms zcompare_rls}) 1);
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by (simp_tac (simpset() addsimps @{thms zadd_ac}) 1);
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qed "le_add_iff1";
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Goal "(i$*u $+ m $<= j$*u $+ n) <-> (m $<= (j$-i)$*u $+ n)";
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by (simp_tac (simpset() addsimps [@{thm zdiff_def}, @{thm zadd_zmult_distrib}]) 1);
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by (simp_tac (simpset() addsimps @{thms zcompare_rls}) 1);
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by (simp_tac (simpset() addsimps @{thms zadd_ac}) 1);
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qed "le_add_iff2";
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structure Int_Numeral_Simprocs =
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struct
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(*Utilities*)
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val integ_of_const = Const (@{const_name "Bin.integ_of"}, iT --> iT);
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fun mk_numeral n = integ_of_const $ NumeralSyntax.mk_bin n;
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(*Decodes a binary INTEGER*)
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fun dest_numeral (Const(@{const_name "Bin.integ_of"}, _) $ w) =
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     (NumeralSyntax.dest_bin w
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      handle Match => raise TERM("Int_Numeral_Simprocs.dest_numeral:1", [w]))
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  | dest_numeral t =  raise TERM("Int_Numeral_Simprocs.dest_numeral:2", [t]);
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fun find_first_numeral past (t::terms) =
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        ((dest_numeral t, rev past @ terms)
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         handle TERM _ => find_first_numeral (t::past) terms)
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  | find_first_numeral past [] = raise TERM("find_first_numeral", []);
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val zero = mk_numeral 0;
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val mk_plus = FOLogic.mk_binop @{const_name "zadd"};
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val iT = Ind_Syntax.iT;
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val zminus_const = Const (@{const_name "zminus"}, iT --> iT);
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(*Thus mk_sum[t] yields t+#0; longer sums don't have a trailing zero*)
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fun mk_sum []        = zero
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  | mk_sum [t,u]     = mk_plus (t, u)
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  | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
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(*this version ALWAYS includes a trailing zero*)
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fun long_mk_sum []        = zero
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  | long_mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
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val dest_plus = FOLogic.dest_bin @{const_name "zadd"} iT;
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(*decompose additions AND subtractions as a sum*)
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fun dest_summing (pos, Const (@{const_name "zadd"}, _) $ t $ u, ts) =
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        dest_summing (pos, t, dest_summing (pos, u, ts))
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  | dest_summing (pos, Const (@{const_name "zdiff"}, _) $ t $ u, ts) =
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        dest_summing (pos, t, dest_summing (not pos, u, ts))
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  | dest_summing (pos, t, ts) =
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        if pos then t::ts else zminus_const$t :: ts;
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fun dest_sum t = dest_summing (true, t, []);
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val mk_diff = FOLogic.mk_binop @{const_name "zdiff"};
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val dest_diff = FOLogic.dest_bin @{const_name "zdiff"} iT;
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val one = mk_numeral 1;
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val mk_times = FOLogic.mk_binop @{const_name "zmult"};
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fun mk_prod [] = one
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  | mk_prod [t] = t
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  | mk_prod (t :: ts) = if t = one then mk_prod ts
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                        else mk_times (t, mk_prod ts);
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val dest_times = FOLogic.dest_bin @{const_name "zmult"} iT;
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fun dest_prod t =
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      let val (t,u) = dest_times t
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      in  dest_prod t @ dest_prod u  end
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      handle TERM _ => [t];
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(*DON'T do the obvious simplifications; that would create special cases*)
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fun mk_coeff (k, t) = mk_times (mk_numeral k, t);
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(*Express t as a product of (possibly) a numeral with other sorted terms*)
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fun dest_coeff sign (Const (@{const_name "zminus"}, _) $ t) = dest_coeff (~sign) t
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  | dest_coeff sign t =
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    let val ts = sort Term.term_ord (dest_prod t)
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        val (n, ts') = find_first_numeral [] ts
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                          handle TERM _ => (1, ts)
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    in (sign*n, mk_prod ts') end;
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(*Find first coefficient-term THAT MATCHES u*)
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fun find_first_coeff past u [] = raise TERM("find_first_coeff", [])
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  | find_first_coeff past u (t::terms) =
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        let val (n,u') = dest_coeff 1 t
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        in  if u aconv u' then (n, rev past @ terms)
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                          else find_first_coeff (t::past) u terms
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        end
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        handle TERM _ => find_first_coeff (t::past) u terms;
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(*Simplify #1*n and n*#1 to n*)
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val add_0s = [@{thm zadd_0_intify}, @{thm zadd_0_right_intify}];
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val mult_1s = [@{thm zmult_1_intify}, @{thm zmult_1_right_intify},
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               @{thm zmult_minus1}, @{thm zmult_minus1_right}];
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val tc_rules = [@{thm integ_of_type}, @{thm intify_in_int},
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                @{thm int_of_type}, @{thm zadd_type}, @{thm zdiff_type}, @{thm zmult_type}] @ 
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               @{thms bin.intros};
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val intifys = [@{thm intify_ident}, @{thm zadd_intify1}, @{thm zadd_intify2},
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               @{thm zdiff_intify1}, @{thm zdiff_intify2}, @{thm zmult_intify1}, @{thm zmult_intify2},
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               @{thm zless_intify1}, @{thm zless_intify2}, @{thm zle_intify1}, @{thm zle_intify2}];
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(*To perform binary arithmetic*)
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val bin_simps = [@{thm add_integ_of_left}] @ @{thms bin_arith_simps} @ @{thms bin_rel_simps};
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(*To evaluate binary negations of coefficients*)
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val zminus_simps = @{thms NCons_simps} @
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                   [@{thm integ_of_minus} RS sym,
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                    @{thm bin_minus_1}, @{thm bin_minus_0}, @{thm bin_minus_Pls}, @{thm bin_minus_Min},
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                    @{thm bin_pred_1}, @{thm bin_pred_0}, @{thm bin_pred_Pls}, @{thm bin_pred_Min}];
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(*To let us treat subtraction as addition*)
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val diff_simps = [@{thm zdiff_def}, @{thm zminus_zadd_distrib}, @{thm zminus_zminus}];
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(*push the unary minus down: - x * y = x * - y *)
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val int_minus_mult_eq_1_to_2 =
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    [@{thm zmult_zminus}, @{thm zmult_zminus_right} RS sym] MRS trans |> standard;
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(*to extract again any uncancelled minuses*)
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val int_minus_from_mult_simps =
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    [@{thm zminus_zminus}, @{thm zmult_zminus}, @{thm zmult_zminus_right}];
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(*combine unary minus with numeric literals, however nested within a product*)
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val int_mult_minus_simps =
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    [@{thm zmult_assoc}, @{thm zmult_zminus} RS sym, int_minus_mult_eq_1_to_2];
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fun prep_simproc (name, pats, proc) =
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  Simplifier.simproc (the_context ()) name pats proc;
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structure CancelNumeralsCommon =
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  struct
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  val mk_sum            = (fn T:typ => mk_sum)
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  val dest_sum          = dest_sum
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  val mk_coeff          = mk_coeff
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  val dest_coeff        = dest_coeff 1
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  val find_first_coeff  = find_first_coeff []
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  fun trans_tac _       = ArithData.gen_trans_tac iff_trans
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  val norm_ss1 = ZF_ss addsimps add_0s @ mult_1s @ diff_simps @ zminus_simps @ @{thms zadd_ac}
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  val norm_ss2 = ZF_ss addsimps bin_simps @ int_mult_minus_simps @ intifys
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  val norm_ss3 = ZF_ss addsimps int_minus_from_mult_simps @ @{thms zadd_ac} @ @{thms zmult_ac} @ tc_rules @ intifys
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  fun norm_tac ss =
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    ALLGOALS (asm_simp_tac (Simplifier.inherit_context ss norm_ss1))
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    THEN ALLGOALS (asm_simp_tac (Simplifier.inherit_context ss norm_ss2))
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    THEN ALLGOALS (asm_simp_tac (Simplifier.inherit_context ss norm_ss3))
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  val numeral_simp_ss = ZF_ss addsimps add_0s @ bin_simps @ tc_rules @ intifys
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  fun numeral_simp_tac ss =
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    ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
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    THEN ALLGOALS (SIMPSET' (fn simpset => asm_simp_tac (Simplifier.inherit_context ss simpset)))
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  val simplify_meta_eq  = ArithData.simplify_meta_eq (add_0s @ mult_1s)
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  end;
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structure EqCancelNumerals = CancelNumeralsFun
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 (open CancelNumeralsCommon
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  val prove_conv = ArithData.prove_conv "inteq_cancel_numerals"
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  val mk_bal   = FOLogic.mk_eq
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  val dest_bal = FOLogic.dest_eq
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  val bal_add1 = eq_add_iff1 RS iff_trans
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  val bal_add2 = eq_add_iff2 RS iff_trans
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);
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structure LessCancelNumerals = CancelNumeralsFun
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 (open CancelNumeralsCommon
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  val prove_conv = ArithData.prove_conv "intless_cancel_numerals"
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  val mk_bal   = FOLogic.mk_binrel @{const_name "zless"}
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  val dest_bal = FOLogic.dest_bin @{const_name "zless"} iT
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  val bal_add1 = less_add_iff1 RS iff_trans
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  val bal_add2 = less_add_iff2 RS iff_trans
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);
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structure LeCancelNumerals = CancelNumeralsFun
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 (open CancelNumeralsCommon
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  val prove_conv = ArithData.prove_conv "intle_cancel_numerals"
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  val mk_bal   = FOLogic.mk_binrel @{const_name "zle"}
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  val dest_bal = FOLogic.dest_bin @{const_name "zle"} iT
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  val bal_add1 = le_add_iff1 RS iff_trans
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  val bal_add2 = le_add_iff2 RS iff_trans
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);
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val cancel_numerals =
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  map prep_simproc
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   [("inteq_cancel_numerals",
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     ["l $+ m = n", "l = m $+ n",
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      "l $- m = n", "l = m $- n",
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      "l $* m = n", "l = m $* n"],
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     K EqCancelNumerals.proc),
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    ("intless_cancel_numerals",
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     ["l $+ m $< n", "l $< m $+ n",
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      "l $- m $< n", "l $< m $- n",
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      "l $* m $< n", "l $< m $* n"],
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     K LessCancelNumerals.proc),
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    ("intle_cancel_numerals",
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     ["l $+ m $<= n", "l $<= m $+ n",
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      "l $- m $<= n", "l $<= m $- n",
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      "l $* m $<= n", "l $<= m $* n"],
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     K LeCancelNumerals.proc)];
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(*version without the hyps argument*)
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fun prove_conv_nohyps name tacs sg = ArithData.prove_conv name tacs sg [];
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structure CombineNumeralsData =
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  struct
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  type coeff            = int
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  val iszero            = (fn x => x = 0)
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  val add               = op + 
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  val mk_sum            = (fn T:typ => long_mk_sum) (*to work for #2*x $+ #3*x *)
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  val dest_sum          = dest_sum
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  val mk_coeff          = mk_coeff
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  val dest_coeff        = dest_coeff 1
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  val left_distrib      = left_zadd_zmult_distrib RS trans
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  val prove_conv        = prove_conv_nohyps "int_combine_numerals"
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  fun trans_tac _       = ArithData.gen_trans_tac trans
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  val norm_ss1 = ZF_ss addsimps add_0s @ mult_1s @ diff_simps @ zminus_simps @ @{thms zadd_ac} @ intifys
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  val norm_ss2 = ZF_ss addsimps bin_simps @ int_mult_minus_simps @ intifys
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  val norm_ss3 = ZF_ss addsimps int_minus_from_mult_simps @ @{thms zadd_ac} @ @{thms zmult_ac} @ tc_rules @ intifys
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  fun norm_tac ss =
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    ALLGOALS (asm_simp_tac (Simplifier.inherit_context ss norm_ss1))
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    THEN ALLGOALS (asm_simp_tac (Simplifier.inherit_context ss norm_ss2))
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    THEN ALLGOALS (asm_simp_tac (Simplifier.inherit_context ss norm_ss3))
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  val numeral_simp_ss = ZF_ss addsimps add_0s @ bin_simps @ tc_rules @ intifys
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  fun numeral_simp_tac ss =
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    ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
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  val simplify_meta_eq  = ArithData.simplify_meta_eq (add_0s @ mult_1s)
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  end;
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structure CombineNumerals = CombineNumeralsFun(CombineNumeralsData);
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val combine_numerals =
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  prep_simproc ("int_combine_numerals", ["i $+ j", "i $- j"], K CombineNumerals.proc);
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(** Constant folding for integer multiplication **)
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(*The trick is to regard products as sums, e.g. #3 $* x $* #4 as
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  the "sum" of #3, x, #4; the literals are then multiplied*)
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structure CombineNumeralsProdData =
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  struct
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  type coeff            = int
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  val iszero            = (fn x => x = 0)
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  val add               = op *
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  val mk_sum            = (fn T:typ => mk_prod)
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  val dest_sum          = dest_prod
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  fun mk_coeff(k,t) = if t=one then mk_numeral k
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                      else raise TERM("mk_coeff", [])
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  fun dest_coeff t = (dest_numeral t, one)  (*We ONLY want pure numerals.*)
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  val left_distrib      = @{thm zmult_assoc} RS sym RS trans
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  val prove_conv        = prove_conv_nohyps "int_combine_numerals_prod"
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  fun trans_tac _       = ArithData.gen_trans_tac trans
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val norm_ss1 = ZF_ss addsimps mult_1s @ diff_simps @ zminus_simps
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  val norm_ss2 = ZF_ss addsimps [@{thm zmult_zminus_right} RS sym] @
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    bin_simps @ @{thms zmult_ac} @ tc_rules @ intifys
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  fun norm_tac ss =
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    ALLGOALS (asm_simp_tac (Simplifier.inherit_context ss norm_ss1))
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    THEN ALLGOALS (asm_simp_tac (Simplifier.inherit_context ss norm_ss2))
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  val numeral_simp_ss = ZF_ss addsimps bin_simps @ tc_rules @ intifys
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  fun numeral_simp_tac ss =
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    ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
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  val simplify_meta_eq  = ArithData.simplify_meta_eq (mult_1s);
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  end;
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structure CombineNumeralsProd = CombineNumeralsFun(CombineNumeralsProdData);
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val combine_numerals_prod =
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  prep_simproc ("int_combine_numerals_prod", ["i $* j"], K CombineNumeralsProd.proc);
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end;
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Addsimprocs Int_Numeral_Simprocs.cancel_numerals;
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Addsimprocs [Int_Numeral_Simprocs.combine_numerals,
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             Int_Numeral_Simprocs.combine_numerals_prod];
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(*examples:*)
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(*
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print_depth 22;
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set timing;
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set trace_simp;
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fun test s = (Goal s; by (Asm_simp_tac 1));
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val sg = #sign (rep_thm (topthm()));
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val t = FOLogic.dest_Trueprop (Logic.strip_assums_concl(getgoal 1));
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val (t,_) = FOLogic.dest_eq t;
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(*combine_numerals_prod (products of separate literals) *)
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test "#5 $* x $* #3 = y";
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test "y2 $+ ?x42 = y $+ y2";
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test "oo : int ==> l $+ (l $+ #2) $+ oo = oo";
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test "#9$*x $+ y = x$*#23 $+ z";
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test "y $+ x = x $+ z";
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test "x : int ==> x $+ y $+ z = x $+ z";
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test "x : int ==> y $+ (z $+ x) = z $+ x";
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test "z : int ==> x $+ y $+ z = (z $+ y) $+ (x $+ w)";
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test "z : int ==> x$*y $+ z = (z $+ y) $+ (y$*x $+ w)";
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test "#-3 $* x $+ y $<= x $* #2 $+ z";
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test "y $+ x $<= x $+ z";
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test "x $+ y $+ z $<= x $+ z";
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test "y $+ (z $+ x) $< z $+ x";
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test "x $+ y $+ z $< (z $+ y) $+ (x $+ w)";
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test "x$*y $+ z $< (z $+ y) $+ (y$*x $+ w)";
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test "l $+ #2 $+ #2 $+ #2 $+ (l $+ #2) $+ (oo $+ #2) = uu";
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test "u : int ==> #2 $* u = u";
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test "(i $+ j $+ #12 $+ k) $- #15 = y";
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test "(i $+ j $+ #12 $+ k) $- #5 = y";
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test "y $- b $< b";
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test "y $- (#3 $* b $+ c) $< b $- #2 $* c";
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test "(#2 $* x $- (u $* v) $+ y) $- v $* #3 $* u = w";
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test "(#2 $* x $* u $* v $+ (u $* v) $* #4 $+ y) $- v $* u $* #4 = w";
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test "(#2 $* x $* u $* v $+ (u $* v) $* #4 $+ y) $- v $* u = w";
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test "u $* v $- (x $* u $* v $+ (u $* v) $* #4 $+ y) = w";
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test "(i $+ j $+ #12 $+ k) = u $+ #15 $+ y";
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test "(i $+ j $* #2 $+ #12 $+ k) = j $+ #5 $+ y";
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test "#2 $* y $+ #3 $* z $+ #6 $* w $+ #2 $* y $+ #3 $* z $+ #2 $* u = #2 $* y' $+ #3 $* z' $+ #6 $* w' $+ #2 $* y' $+ #3 $* z' $+ u $+ vv";
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test "a $+ $-(b$+c) $+ b = d";
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test "a $+ $-(b$+c) $- b = d";
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(*negative numerals*)
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test "(i $+ j $+ #-2 $+ k) $- (u $+ #5 $+ y) = zz";
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test "(i $+ j $+ #-3 $+ k) $< u $+ #5 $+ y";
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test "(i $+ j $+ #3 $+ k) $< u $+ #-6 $+ y";
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test "(i $+ j $+ #-12 $+ k) $- #15 = y";
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test "(i $+ j $+ #12 $+ k) $- #-15 = y";
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test "(i $+ j $+ #-12 $+ k) $- #-15 = y";
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(*Multiplying separated numerals*)
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Goal "#6 $* ($# x $* #2) =  uu";
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Goal "#4 $* ($# x $* $# x) $* (#2 $* $# x) =  uu";
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*)
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