author | wenzelm |
Thu, 16 Apr 2015 15:00:03 +0200 | |
changeset 60095 | 35f626b11422 |
parent 59748 | a1c35e6fe735 |
child 61144 | 5e94dfead1c2 |
permissions | -rw-r--r-- |
23146 | 1 |
(* Title: ZF/int_arith.ML |
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Author: Larry Paulson |
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Simprocs for linear arithmetic. |
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*) |
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signature INT_NUMERAL_SIMPROCS = |
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sig |
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val cancel_numerals: simproc list |
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val combine_numerals: simproc |
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val combine_numerals_prod: simproc |
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end |
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structure Int_Numeral_Simprocs: INT_NUMERAL_SIMPROCS = |
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struct |
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(* abstract syntax operations *) |
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fun mk_bit 0 = @{term "0"} |
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| mk_bit 1 = @{term "succ(0)"} |
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parents:
38715
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changeset
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| mk_bit _ = raise TERM ("mk_bit", []); |
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parents:
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fun dest_bit @{term "0"} = 0 |
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| dest_bit @{term "succ(0)"} = 1 |
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parents:
38715
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| dest_bit t = raise TERM ("dest_bit", [t]); |
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parents:
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fun mk_bin i = |
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let |
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fun term_of [] = @{term Pls} |
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| term_of [~1] = @{term Min} |
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| term_of (b :: bs) = @{term Bit} $ term_of bs $ mk_bit b; |
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in term_of (Numeral_Syntax.make_binary i) end; |
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fun dest_bin tm = |
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let |
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fun bin_of @{term Pls} = [] |
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| bin_of @{term Min} = [~1] |
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parents:
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| bin_of (@{term Bit} $ bs $ b) = dest_bit b :: bin_of bs |
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| bin_of _ = raise TERM ("dest_bin", [tm]); |
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in Numeral_Syntax.dest_binary (bin_of tm) end; |
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(*Utilities*) |
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fun mk_numeral i = @{const integ_of} $ mk_bin i; |
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fun dest_numeral (Const(@{const_name integ_of}, _) $ w) = dest_bin w |
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| dest_numeral t = raise TERM ("dest_numeral", [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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(*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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(*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 @{const zminus} $ t :: ts; |
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fun dest_sum t = dest_summing (true, t, []); |
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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"} @{typ i}; |
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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_Ord.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 @{thm 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*) |
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val int_minus_mult_eq_1_to_2 = @{lemma "$- w $* z = w $* $- z" by simp}; |
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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 @{thm sym}, int_minus_mult_eq_1_to_2]; |
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fun prep_simproc thy (name, pats, proc) = |
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Simplifier.simproc_global thy name pats proc; |
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structure CancelNumeralsCommon = |
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struct |
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val mk_sum = (fn _ : 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 ctxt = ArithData.gen_trans_tac ctxt @{thm iff_trans} |
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val norm_ss1 = |
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simpset_of (put_simpset ZF_ss @{context} |
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addsimps add_0s @ mult_1s @ diff_simps @ zminus_simps @ @{thms zadd_ac}) |
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val norm_ss2 = |
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simpset_of (put_simpset ZF_ss @{context} |
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addsimps bin_simps @ int_mult_minus_simps @ intifys) |
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val norm_ss3 = |
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