--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Tools/numeral_simprocs.ML Fri May 08 09:48:07 2009 +0200
@@ -0,0 +1,786 @@
+(* Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+ Copyright 2000 University of Cambridge
+
+Simprocs for the integer numerals.
+*)
+
+(*To quote from Provers/Arith/cancel_numeral_factor.ML:
+
+Cancels common coefficients in balanced expressions:
+
+ u*#m ~~ u'*#m' == #n*u ~~ #n'*u'
+
+where ~~ is an appropriate balancing operation (e.g. =, <=, <, div, /)
+and d = gcd(m,m') and n=m/d and n'=m'/d.
+*)
+
+signature NUMERAL_SIMPROCS =
+sig
+ val mk_sum: typ -> term list -> term
+ val dest_sum: term -> term list
+
+ val assoc_fold_simproc: simproc
+ val combine_numerals: simproc
+ val cancel_numerals: simproc list
+ val cancel_factors: simproc list
+ val cancel_numeral_factors: simproc list
+ val field_combine_numerals: simproc
+ val field_cancel_numeral_factors: simproc list
+ val num_ss: simpset
+end;
+
+structure Numeral_Simprocs : NUMERAL_SIMPROCS =
+struct
+
+fun mk_number T n = HOLogic.number_of_const T $ HOLogic.mk_numeral n;
+
+fun find_first_numeral past (t::terms) =
+ ((snd (HOLogic.dest_number t), rev past @ terms)
+ handle TERM _ => find_first_numeral (t::past) terms)
+ | find_first_numeral past [] = raise TERM("find_first_numeral", []);
+
+val mk_plus = HOLogic.mk_binop @{const_name HOL.plus};
+
+fun mk_minus t =
+ let val T = Term.fastype_of t
+ in Const (@{const_name HOL.uminus}, T --> T) $ t end;
+
+(*Thus mk_sum[t] yields t+0; longer sums don't have a trailing zero*)
+fun mk_sum T [] = mk_number T 0
+ | mk_sum T [t,u] = mk_plus (t, u)
+ | mk_sum T (t :: ts) = mk_plus (t, mk_sum T ts);
+
+(*this version ALWAYS includes a trailing zero*)
+fun long_mk_sum T [] = mk_number T 0
+ | long_mk_sum T (t :: ts) = mk_plus (t, mk_sum T ts);
+
+val dest_plus = HOLogic.dest_bin @{const_name HOL.plus} Term.dummyT;
+
+(*decompose additions AND subtractions as a sum*)
+fun dest_summing (pos, Const (@{const_name HOL.plus}, _) $ t $ u, ts) =
+ dest_summing (pos, t, dest_summing (pos, u, ts))
+ | dest_summing (pos, Const (@{const_name HOL.minus}, _) $ t $ u, ts) =
+ dest_summing (pos, t, dest_summing (not pos, u, ts))
+ | dest_summing (pos, t, ts) =
+ if pos then t::ts else mk_minus t :: ts;
+
+fun dest_sum t = dest_summing (true, t, []);
+
+val mk_diff = HOLogic.mk_binop @{const_name HOL.minus};
+val dest_diff = HOLogic.dest_bin @{const_name HOL.minus} Term.dummyT;
+
+val mk_times = HOLogic.mk_binop @{const_name HOL.times};
+
+fun one_of T = Const(@{const_name HOL.one},T);
+
+(* build product with trailing 1 rather than Numeral 1 in order to avoid the
+ unnecessary restriction to type class number_ring
+ which is not required for cancellation of common factors in divisions.
+*)
+fun mk_prod T =
+ let val one = one_of T
+ fun mk [] = one
+ | mk [t] = t
+ | mk (t :: ts) = if t = one then mk ts else mk_times (t, mk ts)
+ in mk end;
+
+(*This version ALWAYS includes a trailing one*)
+fun long_mk_prod T [] = one_of T
+ | long_mk_prod T (t :: ts) = mk_times (t, mk_prod T ts);
+
+val dest_times = HOLogic.dest_bin @{const_name HOL.times} Term.dummyT;
+
+fun dest_prod t =
+ let val (t,u) = dest_times t
+ in dest_prod t @ dest_prod u end
+ handle TERM _ => [t];
+
+(*DON'T do the obvious simplifications; that would create special cases*)
+fun mk_coeff (k, t) = mk_times (mk_number (Term.fastype_of t) k, t);
+
+(*Express t as a product of (possibly) a numeral with other sorted terms*)
+fun dest_coeff sign (Const (@{const_name HOL.uminus}, _) $ t) = dest_coeff (~sign) t
+ | dest_coeff sign t =
+ let val ts = sort TermOrd.term_ord (dest_prod t)
+ val (n, ts') = find_first_numeral [] ts
+ handle TERM _ => (1, ts)
+ in (sign*n, mk_prod (Term.fastype_of t) ts') end;
+
+(*Find first coefficient-term THAT MATCHES u*)
+fun find_first_coeff past u [] = raise TERM("find_first_coeff", [])
+ | find_first_coeff past u (t::terms) =
+ let val (n,u') = dest_coeff 1 t
+ in if u aconv u' then (n, rev past @ terms)
+ else find_first_coeff (t::past) u terms
+ end
+ handle TERM _ => find_first_coeff (t::past) u terms;
+
+(*Fractions as pairs of ints. Can't use Rat.rat because the representation
+ needs to preserve negative values in the denominator.*)
+fun mk_frac (p, q) = if q = 0 then raise Div else (p, q);
+
+(*Don't reduce fractions; sums must be proved by rule add_frac_eq.
+ Fractions are reduced later by the cancel_numeral_factor simproc.*)
+fun add_frac ((p1, q1), (p2, q2)) = (p1 * q2 + p2 * q1, q1 * q2);
+
+val mk_divide = HOLogic.mk_binop @{const_name HOL.divide};
+
+(*Build term (p / q) * t*)
+fun mk_fcoeff ((p, q), t) =
+ let val T = Term.fastype_of t
+ in mk_times (mk_divide (mk_number T p, mk_number T q), t) end;
+
+(*Express t as a product of a fraction with other sorted terms*)
+fun dest_fcoeff sign (Const (@{const_name HOL.uminus}, _) $ t) = dest_fcoeff (~sign) t
+ | dest_fcoeff sign (Const (@{const_name HOL.divide}, _) $ t $ u) =
+ let val (p, t') = dest_coeff sign t
+ val (q, u') = dest_coeff 1 u
+ in (mk_frac (p, q), mk_divide (t', u')) end
+ | dest_fcoeff sign t =
+ let val (p, t') = dest_coeff sign t
+ val T = Term.fastype_of t
+ in (mk_frac (p, 1), mk_divide (t', one_of T)) end;
+
+
+(** New term ordering so that AC-rewriting brings numerals to the front **)
+
+(*Order integers by absolute value and then by sign. The standard integer
+ ordering is not well-founded.*)
+fun num_ord (i,j) =
+ (case int_ord (abs i, abs j) of
+ EQUAL => int_ord (Int.sign i, Int.sign j)
+ | ord => ord);
+
+(*This resembles TermOrd.term_ord, but it puts binary numerals before other
+ non-atomic terms.*)
+local open Term
+in
+fun numterm_ord (Abs (_, T, t), Abs(_, U, u)) =
+ (case numterm_ord (t, u) of EQUAL => TermOrd.typ_ord (T, U) | ord => ord)
+ | numterm_ord
+ (Const(@{const_name Int.number_of}, _) $ v, Const(@{const_name Int.number_of}, _) $ w) =
+ num_ord (HOLogic.dest_numeral v, HOLogic.dest_numeral w)
+ | numterm_ord (Const(@{const_name Int.number_of}, _) $ _, _) = LESS
+ | numterm_ord (_, Const(@{const_name Int.number_of}, _) $ _) = GREATER
+ | numterm_ord (t, u) =
+ (case int_ord (size_of_term t, size_of_term u) of
+ EQUAL =>
+ let val (f, ts) = strip_comb t and (g, us) = strip_comb u in
+ (case TermOrd.hd_ord (f, g) of EQUAL => numterms_ord (ts, us) | ord => ord)
+ end
+ | ord => ord)
+and numterms_ord (ts, us) = list_ord numterm_ord (ts, us)
+end;
+
+fun numtermless tu = (numterm_ord tu = LESS);
+
+val num_ss = HOL_ss settermless numtermless;
+
+(*Maps 0 to Numeral0 and 1 to Numeral1 so that arithmetic isn't complicated by the abstract 0 and 1.*)
+val numeral_syms = [@{thm numeral_0_eq_0} RS sym, @{thm numeral_1_eq_1} RS sym];
+
+(*Simplify Numeral0+n, n+Numeral0, Numeral1*n, n*Numeral1, 1*x, x*1, x/1 *)
+val add_0s = @{thms add_0s};
+val mult_1s = @{thms mult_1s mult_1_left mult_1_right divide_1};
+
+(*Simplify inverse Numeral1, a/Numeral1*)
+val inverse_1s = [@{thm inverse_numeral_1}];
+val divide_1s = [@{thm divide_numeral_1}];
+
+(*To perform binary arithmetic. The "left" rewriting handles patterns
+ created by the Numeral_Simprocs, such as 3 * (5 * x). *)
+val simps = [@{thm numeral_0_eq_0} RS sym, @{thm numeral_1_eq_1} RS sym,
+ @{thm add_number_of_left}, @{thm mult_number_of_left}] @
+ @{thms arith_simps} @ @{thms rel_simps};
+
+(*Binary arithmetic BUT NOT ADDITION since it may collapse adjacent terms
+ during re-arrangement*)
+val non_add_simps =
+ subtract Thm.eq_thm [@{thm add_number_of_left}, @{thm number_of_add} RS sym] simps;
+
+(*To evaluate binary negations of coefficients*)
+val minus_simps = [@{thm numeral_m1_eq_minus_1} RS sym, @{thm number_of_minus} RS sym] @
+ @{thms minus_bin_simps} @ @{thms pred_bin_simps};
+
+(*To let us treat subtraction as addition*)
+val diff_simps = [@{thm diff_minus}, @{thm minus_add_distrib}, @{thm minus_minus}];
+
+(*To let us treat division as multiplication*)
+val divide_simps = [@{thm divide_inverse}, @{thm inverse_mult_distrib}, @{thm inverse_inverse_eq}];
+
+(*push the unary minus down: - x * y = x * - y *)
+val minus_mult_eq_1_to_2 =
+ [@{thm mult_minus_left}, @{thm minus_mult_right}] MRS trans |> standard;
+
+(*to extract again any uncancelled minuses*)
+val minus_from_mult_simps =
+ [@{thm minus_minus}, @{thm mult_minus_left}, @{thm mult_minus_right}];
+
+(*combine unary minus with numeric literals, however nested within a product*)
+val mult_minus_simps =
+ [@{thm mult_assoc}, @{thm minus_mult_left}, minus_mult_eq_1_to_2];
+
+val norm_ss1 = num_ss addsimps numeral_syms @ add_0s @ mult_1s @
+ diff_simps @ minus_simps @ @{thms add_ac}
+val norm_ss2 = num_ss addsimps non_add_simps @ mult_minus_simps
+val norm_ss3 = num_ss addsimps minus_from_mult_simps @ @{thms add_ac} @ @{thms mult_ac}
+
+structure CancelNumeralsCommon =
+ struct
+ val mk_sum = mk_sum
+ val dest_sum = dest_sum
+ val mk_coeff = mk_coeff
+ val dest_coeff = dest_coeff 1
+ val find_first_coeff = find_first_coeff []
+ val trans_tac = K Arith_Data.trans_tac
+
+ fun norm_tac ss =
+ ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss1))
+ THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss2))
+ THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss3))
+
+ val numeral_simp_ss = HOL_ss addsimps add_0s @ simps
+ fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
+ val simplify_meta_eq = Arith_Data.simplify_meta_eq (add_0s @ mult_1s)
+ end;
+
+
+structure EqCancelNumerals = CancelNumeralsFun
+ (open CancelNumeralsCommon
+ val prove_conv = Arith_Data.prove_conv
+ val mk_bal = HOLogic.mk_eq
+ val dest_bal = HOLogic.dest_bin "op =" Term.dummyT
+ val bal_add1 = @{thm eq_add_iff1} RS trans
+ val bal_add2 = @{thm eq_add_iff2} RS trans
+);
+
+structure LessCancelNumerals = CancelNumeralsFun
+ (open CancelNumeralsCommon
+ val prove_conv = Arith_Data.prove_conv
+ val mk_bal = HOLogic.mk_binrel @{const_name HOL.less}
+ val dest_bal = HOLogic.dest_bin @{const_name HOL.less} Term.dummyT
+ val bal_add1 = @{thm less_add_iff1} RS trans
+ val bal_add2 = @{thm less_add_iff2} RS trans
+);
+
+structure LeCancelNumerals = CancelNumeralsFun
+ (open CancelNumeralsCommon
+ val prove_conv = Arith_Data.prove_conv
+ val mk_bal = HOLogic.mk_binrel @{const_name HOL.less_eq}
+ val dest_bal = HOLogic.dest_bin @{const_name HOL.less_eq} Term.dummyT
+ val bal_add1 = @{thm le_add_iff1} RS trans
+ val bal_add2 = @{thm le_add_iff2} RS trans
+);
+
+val cancel_numerals =
+ map Arith_Data.prep_simproc
+ [("inteq_cancel_numerals",
+ ["(l::'a::number_ring) + m = n",
+ "(l::'a::number_ring) = m + n",
+ "(l::'a::number_ring) - m = n",
+ "(l::'a::number_ring) = m - n",
+ "(l::'a::number_ring) * m = n",
+ "(l::'a::number_ring) = m * n"],
+ K EqCancelNumerals.proc),
+ ("intless_cancel_numerals",
+ ["(l::'a::{ordered_idom,number_ring}) + m < n",
+ "(l::'a::{ordered_idom,number_ring}) < m + n",
+ "(l::'a::{ordered_idom,number_ring}) - m < n",
+ "(l::'a::{ordered_idom,number_ring}) < m - n",
+ "(l::'a::{ordered_idom,number_ring}) * m < n",
+ "(l::'a::{ordered_idom,number_ring}) < m * n"],
+ K LessCancelNumerals.proc),
+ ("intle_cancel_numerals",
+ ["(l::'a::{ordered_idom,number_ring}) + m <= n",
+ "(l::'a::{ordered_idom,number_ring}) <= m + n",
+ "(l::'a::{ordered_idom,number_ring}) - m <= n",
+ "(l::'a::{ordered_idom,number_ring}) <= m - n",
+ "(l::'a::{ordered_idom,number_ring}) * m <= n",
+ "(l::'a::{ordered_idom,number_ring}) <= m * n"],
+ K LeCancelNumerals.proc)];
+
+structure CombineNumeralsData =
+ struct
+ type coeff = int
+ val iszero = (fn x => x = 0)
+ val add = op +
+ val mk_sum = long_mk_sum (*to work for e.g. 2*x + 3*x *)
+ val dest_sum = dest_sum
+ val mk_coeff = mk_coeff
+ val dest_coeff = dest_coeff 1
+ val left_distrib = @{thm combine_common_factor} RS trans
+ val prove_conv = Arith_Data.prove_conv_nohyps
+ val trans_tac = K Arith_Data.trans_tac
+
+ fun norm_tac ss =
+ ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss1))
+ THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss2))
+ THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss3))
+
+ val numeral_simp_ss = HOL_ss addsimps add_0s @ simps
+ fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
+ val simplify_meta_eq = Arith_Data.simplify_meta_eq (add_0s @ mult_1s)
+ end;
+
+structure CombineNumerals = CombineNumeralsFun(CombineNumeralsData);
+
+(*Version for fields, where coefficients can be fractions*)
+structure FieldCombineNumeralsData =
+ struct
+ type coeff = int * int
+ val iszero = (fn (p, q) => p = 0)
+ val add = add_frac
+ val mk_sum = long_mk_sum
+ val dest_sum = dest_sum
+ val mk_coeff = mk_fcoeff
+ val dest_coeff = dest_fcoeff 1
+ val left_distrib = @{thm combine_common_factor} RS trans
+ val prove_conv = Arith_Data.prove_conv_nohyps
+ val trans_tac = K Arith_Data.trans_tac
+
+ val norm_ss1a = norm_ss1 addsimps inverse_1s @ divide_simps
+ fun norm_tac ss =
+ ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss1a))
+ THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss2))
+ THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss3))
+
+ val numeral_simp_ss = HOL_ss addsimps add_0s @ simps @ [@{thm add_frac_eq}]
+ fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
+ val simplify_meta_eq = Arith_Data.simplify_meta_eq (add_0s @ mult_1s @ divide_1s)
+ end;
+
+structure FieldCombineNumerals = CombineNumeralsFun(FieldCombineNumeralsData);
+
+val combine_numerals =
+ Arith_Data.prep_simproc
+ ("int_combine_numerals",
+ ["(i::'a::number_ring) + j", "(i::'a::number_ring) - j"],
+ K CombineNumerals.proc);
+
+val field_combine_numerals =
+ Arith_Data.prep_simproc
+ ("field_combine_numerals",
+ ["(i::'a::{number_ring,field,division_by_zero}) + j",
+ "(i::'a::{number_ring,field,division_by_zero}) - j"],
+ K FieldCombineNumerals.proc);
+
+(** Constant folding for multiplication in semirings **)
+
+(*We do not need folding for addition: combine_numerals does the same thing*)
+
+structure Semiring_Times_Assoc_Data : ASSOC_FOLD_DATA =
+struct
+ val assoc_ss = HOL_ss addsimps @{thms mult_ac}
+ val eq_reflection = eq_reflection
+ fun is_numeral (Const(@{const_name Int.number_of}, _) $ _) = true
+ | is_numeral _ = false;
+end;
+
+structure Semiring_Times_Assoc = Assoc_Fold (Semiring_Times_Assoc_Data);
+
+val assoc_fold_simproc =
+ Arith_Data.prep_simproc
+ ("semiring_assoc_fold", ["(a::'a::comm_semiring_1_cancel) * b"],
+ K Semiring_Times_Assoc.proc);
+
+structure CancelNumeralFactorCommon =
+ struct
+ val mk_coeff = mk_coeff
+ val dest_coeff = dest_coeff 1
+ val trans_tac = K Arith_Data.trans_tac
+
+ val norm_ss1 = HOL_ss addsimps minus_from_mult_simps @ mult_1s
+ val norm_ss2 = HOL_ss addsimps simps @ mult_minus_simps
+ val norm_ss3 = HOL_ss addsimps @{thms mult_ac}
+ fun norm_tac ss =
+ ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss1))
+ THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss2))
+ THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss3))
+
+ val numeral_simp_ss = HOL_ss addsimps
+ [@{thm eq_number_of_eq}, @{thm less_number_of}, @{thm le_number_of}] @ simps
+ fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
+ val simplify_meta_eq = Arith_Data.simplify_meta_eq
+ [@{thm add_0}, @{thm add_0_right}, @{thm mult_zero_left},
+ @{thm mult_zero_right}, @{thm mult_Bit1}, @{thm mult_1_right}];
+ end
+
+(*Version for semiring_div*)
+structure DivCancelNumeralFactor = CancelNumeralFactorFun
+ (open CancelNumeralFactorCommon
+ val prove_conv = Arith_Data.prove_conv
+ val mk_bal = HOLogic.mk_binop @{const_name Divides.div}
+ val dest_bal = HOLogic.dest_bin @{const_name Divides.div} Term.dummyT
+ val cancel = @{thm div_mult_mult1} RS trans
+ val neg_exchanges = false
+)
+
+(*Version for fields*)
+structure DivideCancelNumeralFactor = CancelNumeralFactorFun
+ (open CancelNumeralFactorCommon
+ val prove_conv = Arith_Data.prove_conv
+ val mk_bal = HOLogic.mk_binop @{const_name HOL.divide}
+ val dest_bal = HOLogic.dest_bin @{const_name HOL.divide} Term.dummyT
+ val cancel = @{thm mult_divide_mult_cancel_left} RS trans
+ val neg_exchanges = false
+)
+
+structure EqCancelNumeralFactor = CancelNumeralFactorFun
+ (open CancelNumeralFactorCommon
+ val prove_conv = Arith_Data.prove_conv
+ val mk_bal = HOLogic.mk_eq
+ val dest_bal = HOLogic.dest_bin "op =" Term.dummyT
+ val cancel = @{thm mult_cancel_left} RS trans
+ val neg_exchanges = false
+)
+
+structure LessCancelNumeralFactor = CancelNumeralFactorFun
+ (open CancelNumeralFactorCommon
+ val prove_conv = Arith_Data.prove_conv
+ val mk_bal = HOLogic.mk_binrel @{const_name HOL.less}
+ val dest_bal = HOLogic.dest_bin @{const_name HOL.less} Term.dummyT
+ val cancel = @{thm mult_less_cancel_left} RS trans
+ val neg_exchanges = true
+)
+
+structure LeCancelNumeralFactor = CancelNumeralFactorFun
+ (open CancelNumeralFactorCommon
+ val prove_conv = Arith_Data.prove_conv
+ val mk_bal = HOLogic.mk_binrel @{const_name HOL.less_eq}
+ val dest_bal = HOLogic.dest_bin @{const_name HOL.less_eq} Term.dummyT
+ val cancel = @{thm mult_le_cancel_left} RS trans
+ val neg_exchanges = true
+)
+
+val cancel_numeral_factors =
+ map Arith_Data.prep_simproc
+ [("ring_eq_cancel_numeral_factor",
+ ["(l::'a::{idom,number_ring}) * m = n",
+ "(l::'a::{idom,number_ring}) = m * n"],
+ K EqCancelNumeralFactor.proc),
+ ("ring_less_cancel_numeral_factor",
+ ["(l::'a::{ordered_idom,number_ring}) * m < n",
+ "(l::'a::{ordered_idom,number_ring}) < m * n"],
+ K LessCancelNumeralFactor.proc),
+ ("ring_le_cancel_numeral_factor",
+ ["(l::'a::{ordered_idom,number_ring}) * m <= n",
+ "(l::'a::{ordered_idom,number_ring}) <= m * n"],
+ K LeCancelNumeralFactor.proc),
+ ("int_div_cancel_numeral_factors",
+ ["((l::'a::{semiring_div,number_ring}) * m) div n",
+ "(l::'a::{semiring_div,number_ring}) div (m * n)"],
+ K DivCancelNumeralFactor.proc),
+ ("divide_cancel_numeral_factor",
+ ["((l::'a::{division_by_zero,field,number_ring}) * m) / n",
+ "(l::'a::{division_by_zero,field,number_ring}) / (m * n)",
+ "((number_of v)::'a::{division_by_zero,field,number_ring}) / (number_of w)"],
+ K DivideCancelNumeralFactor.proc)];
+
+val field_cancel_numeral_factors =
+ map Arith_Data.prep_simproc
+ [("field_eq_cancel_numeral_factor",
+ ["(l::'a::{field,number_ring}) * m = n",
+ "(l::'a::{field,number_ring}) = m * n"],
+ K EqCancelNumeralFactor.proc),
+ ("field_cancel_numeral_factor",
+ ["((l::'a::{division_by_zero,field,number_ring}) * m) / n",
+ "(l::'a::{division_by_zero,field,number_ring}) / (m * n)",
+ "((number_of v)::'a::{division_by_zero,field,number_ring}) / (number_of w)"],
+ K DivideCancelNumeralFactor.proc)]
+
+
+(** Declarations for ExtractCommonTerm **)
+
+(*Find first term that matches u*)
+fun find_first_t past u [] = raise TERM ("find_first_t", [])
+ | find_first_t past u (t::terms) =
+ if u aconv t then (rev past @ terms)
+ else find_first_t (t::past) u terms
+ handle TERM _ => find_first_t (t::past) u terms;
+
+(** Final simplification for the CancelFactor simprocs **)
+val simplify_one = Arith_Data.simplify_meta_eq
+ [@{thm mult_1_left}, @{thm mult_1_right}, @{thm div_by_1}, @{thm numeral_1_eq_1}];
+
+fun cancel_simplify_meta_eq ss cancel_th th =
+ simplify_one ss (([th, cancel_th]) MRS trans);
+
+local
+ val Tp_Eq = Thm.reflexive (Thm.cterm_of @{theory HOL} HOLogic.Trueprop)
+ fun Eq_True_elim Eq =
+ Thm.equal_elim (Thm.combination Tp_Eq (Thm.symmetric Eq)) @{thm TrueI}
+in
+fun sign_conv pos_th neg_th ss t =
+ let val T = fastype_of t;
+ val zero = Const(@{const_name HOL.zero}, T);
+ val less = Const(@{const_name HOL.less}, [T,T] ---> HOLogic.boolT);
+ val pos = less $ zero $ t and neg = less $ t $ zero
+ fun prove p =
+ Option.map Eq_True_elim (Lin_Arith.lin_arith_simproc ss p)
+ handle THM _ => NONE
+ in case prove pos of
+ SOME th => SOME(th RS pos_th)
+ | NONE => (case prove neg of
+ SOME th => SOME(th RS neg_th)
+ | NONE => NONE)
+ end;
+end
+
+structure CancelFactorCommon =
+ struct
+ val mk_sum = long_mk_prod
+ val dest_sum = dest_prod
+ val mk_coeff = mk_coeff
+ val dest_coeff = dest_coeff
+ val find_first = find_first_t []
+ val trans_tac = K Arith_Data.trans_tac
+ val norm_ss = HOL_ss addsimps mult_1s @ @{thms mult_ac}
+ fun norm_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss))
+ val simplify_meta_eq = cancel_simplify_meta_eq
+ end;
+
+(*mult_cancel_left requires a ring with no zero divisors.*)
+structure EqCancelFactor = ExtractCommonTermFun
+ (open CancelFactorCommon
+ val prove_conv = Arith_Data.prove_conv
+ val mk_bal = HOLogic.mk_eq
+ val dest_bal = HOLogic.dest_bin "op =" Term.dummyT
+ val simp_conv = K (K (SOME @{thm mult_cancel_left}))
+);
+
+(*for ordered rings*)
+structure LeCancelFactor = ExtractCommonTermFun
+ (open CancelFactorCommon
+ val prove_conv = Arith_Data.prove_conv
+ val mk_bal = HOLogic.mk_binrel @{const_name HOL.less_eq}
+ val dest_bal = HOLogic.dest_bin @{const_name HOL.less_eq} Term.dummyT
+ val simp_conv = sign_conv
+ @{thm mult_le_cancel_left_pos} @{thm mult_le_cancel_left_neg}
+);
+
+(*for ordered rings*)
+structure LessCancelFactor = ExtractCommonTermFun
+ (open CancelFactorCommon
+ val prove_conv = Arith_Data.prove_conv
+ val mk_bal = HOLogic.mk_binrel @{const_name HOL.less}
+ val dest_bal = HOLogic.dest_bin @{const_name HOL.less} Term.dummyT
+ val simp_conv = sign_conv
+ @{thm mult_less_cancel_left_pos} @{thm mult_less_cancel_left_neg}
+);
+
+(*for semirings with division*)
+structure DivCancelFactor = ExtractCommonTermFun
+ (open CancelFactorCommon
+ val prove_conv = Arith_Data.prove_conv
+ val mk_bal = HOLogic.mk_binop @{const_name Divides.div}
+ val dest_bal = HOLogic.dest_bin @{const_name Divides.div} Term.dummyT
+ val simp_conv = K (K (SOME @{thm div_mult_mult1_if}))
+);
+
+structure ModCancelFactor = ExtractCommonTermFun
+ (open CancelFactorCommon
+ val prove_conv = Arith_Data.prove_conv
+ val mk_bal = HOLogic.mk_binop @{const_name Divides.mod}
+ val dest_bal = HOLogic.dest_bin @{const_name Divides.mod} Term.dummyT
+ val simp_conv = K (K (SOME @{thm mod_mult_mult1}))
+);
+
+(*for idoms*)
+structure DvdCancelFactor = ExtractCommonTermFun
+ (open CancelFactorCommon
+ val prove_conv = Arith_Data.prove_conv
+ val mk_bal = HOLogic.mk_binrel @{const_name Ring_and_Field.dvd}
+ val dest_bal = HOLogic.dest_bin @{const_name Ring_and_Field.dvd} Term.dummyT
+ val simp_conv = K (K (SOME @{thm dvd_mult_cancel_left}))
+);
+
+(*Version for all fields, including unordered ones (type complex).*)
+structure DivideCancelFactor = ExtractCommonTermFun
+ (open CancelFactorCommon
+ val prove_conv = Arith_Data.prove_conv
+ val mk_bal = HOLogic.mk_binop @{const_name HOL.divide}
+ val dest_bal = HOLogic.dest_bin @{const_name HOL.divide} Term.dummyT
+ val simp_conv = K (K (SOME @{thm mult_divide_mult_cancel_left_if}))
+);
+
+val cancel_factors =
+ map Arith_Data.prep_simproc
+ [("ring_eq_cancel_factor",
+ ["(l::'a::idom) * m = n",
+ "(l::'a::idom) = m * n"],
+ K EqCancelFactor.proc),
+ ("ordered_ring_le_cancel_factor",
+ ["(l::'a::ordered_ring) * m <= n",
+ "(l::'a::ordered_ring) <= m * n"],
+ K LeCancelFactor.proc),
+ ("ordered_ring_less_cancel_factor",
+ ["(l::'a::ordered_ring) * m < n",
+ "(l::'a::ordered_ring) < m * n"],
+ K LessCancelFactor.proc),
+ ("int_div_cancel_factor",
+ ["((l::'a::semiring_div) * m) div n", "(l::'a::semiring_div) div (m * n)"],
+ K DivCancelFactor.proc),
+ ("int_mod_cancel_factor",
+ ["((l::'a::semiring_div) * m) mod n", "(l::'a::semiring_div) mod (m * n)"],
+ K ModCancelFactor.proc),
+ ("dvd_cancel_factor",
+ ["((l::'a::idom) * m) dvd n", "(l::'a::idom) dvd (m * n)"],
+ K DvdCancelFactor.proc),
+ ("divide_cancel_factor",
+ ["((l::'a::{division_by_zero,field}) * m) / n",
+ "(l::'a::{division_by_zero,field}) / (m * n)"],
+ K DivideCancelFactor.proc)];
+
+end;
+
+Addsimprocs Numeral_Simprocs.cancel_numerals;
+Addsimprocs [Numeral_Simprocs.combine_numerals];
+Addsimprocs [Numeral_Simprocs.field_combine_numerals];
+Addsimprocs [Numeral_Simprocs.assoc_fold_simproc];
+
+(*examples:
+print_depth 22;
+set timing;
+set trace_simp;
+fun test s = (Goal s, by (Simp_tac 1));
+
+test "l + 2 + 2 + 2 + (l + 2) + (oo + 2) = (uu::int)";
+
+test "2*u = (u::int)";
+test "(i + j + 12 + (k::int)) - 15 = y";
+test "(i + j + 12 + (k::int)) - 5 = y";
+
+test "y - b < (b::int)";
+test "y - (3*b + c) < (b::int) - 2*c";
+
+test "(2*x - (u*v) + y) - v*3*u = (w::int)";
+test "(2*x*u*v + (u*v)*4 + y) - v*u*4 = (w::int)";
+test "(2*x*u*v + (u*v)*4 + y) - v*u = (w::int)";
+test "u*v - (x*u*v + (u*v)*4 + y) = (w::int)";
+
+test "(i + j + 12 + (k::int)) = u + 15 + y";
+test "(i + j*2 + 12 + (k::int)) = j + 5 + y";
+
+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::int)";
+
+test "a + -(b+c) + b = (d::int)";
+test "a + -(b+c) - b = (d::int)";
+
+(*negative numerals*)
+test "(i + j + -2 + (k::int)) - (u + 5 + y) = zz";
+test "(i + j + -3 + (k::int)) < u + 5 + y";
+test "(i + j + 3 + (k::int)) < u + -6 + y";
+test "(i + j + -12 + (k::int)) - 15 = y";
+test "(i + j + 12 + (k::int)) - -15 = y";
+test "(i + j + -12 + (k::int)) - -15 = y";
+*)
+
+Addsimprocs Numeral_Simprocs.cancel_numeral_factors;
+
+(*examples:
+print_depth 22;
+set timing;
+set trace_simp;
+fun test s = (Goal s; by (Simp_tac 1));
+
+test "9*x = 12 * (y::int)";
+test "(9*x) div (12 * (y::int)) = z";
+test "9*x < 12 * (y::int)";
+test "9*x <= 12 * (y::int)";
+
+test "-99*x = 132 * (y::int)";
+test "(-99*x) div (132 * (y::int)) = z";
+test "-99*x < 132 * (y::int)";
+test "-99*x <= 132 * (y::int)";
+
+test "999*x = -396 * (y::int)";
+test "(999*x) div (-396 * (y::int)) = z";
+test "999*x < -396 * (y::int)";
+test "999*x <= -396 * (y::int)";
+
+test "-99*x = -81 * (y::int)";
+test "(-99*x) div (-81 * (y::int)) = z";
+test "-99*x <= -81 * (y::int)";
+test "-99*x < -81 * (y::int)";
+
+test "-2 * x = -1 * (y::int)";
+test "-2 * x = -(y::int)";
+test "(-2 * x) div (-1 * (y::int)) = z";
+test "-2 * x < -(y::int)";
+test "-2 * x <= -1 * (y::int)";
+test "-x < -23 * (y::int)";
+test "-x <= -23 * (y::int)";
+*)
+
+(*And the same examples for fields such as rat or real:
+test "0 <= (y::rat) * -2";
+test "9*x = 12 * (y::rat)";
+test "(9*x) / (12 * (y::rat)) = z";
+test "9*x < 12 * (y::rat)";
+test "9*x <= 12 * (y::rat)";
+
+test "-99*x = 132 * (y::rat)";
+test "(-99*x) / (132 * (y::rat)) = z";
+test "-99*x < 132 * (y::rat)";
+test "-99*x <= 132 * (y::rat)";
+
+test "999*x = -396 * (y::rat)";
+test "(999*x) / (-396 * (y::rat)) = z";
+test "999*x < -396 * (y::rat)";
+test "999*x <= -396 * (y::rat)";
+
+test "(- ((2::rat) * x) <= 2 * y)";
+test "-99*x = -81 * (y::rat)";
+test "(-99*x) / (-81 * (y::rat)) = z";
+test "-99*x <= -81 * (y::rat)";
+test "-99*x < -81 * (y::rat)";
+
+test "-2 * x = -1 * (y::rat)";
+test "-2 * x = -(y::rat)";
+test "(-2 * x) / (-1 * (y::rat)) = z";
+test "-2 * x < -(y::rat)";
+test "-2 * x <= -1 * (y::rat)";
+test "-x < -23 * (y::rat)";
+test "-x <= -23 * (y::rat)";
+*)
+
+Addsimprocs Numeral_Simprocs.cancel_factors;
+
+
+(*examples:
+print_depth 22;
+set timing;
+set trace_simp;
+fun test s = (Goal s; by (Asm_simp_tac 1));
+
+test "x*k = k*(y::int)";
+test "k = k*(y::int)";
+test "a*(b*c) = (b::int)";
+test "a*(b*c) = d*(b::int)*(x*a)";
+
+test "(x*k) div (k*(y::int)) = (uu::int)";
+test "(k) div (k*(y::int)) = (uu::int)";
+test "(a*(b*c)) div ((b::int)) = (uu::int)";
+test "(a*(b*c)) div (d*(b::int)*(x*a)) = (uu::int)";
+*)
+
+(*And the same examples for fields such as rat or real:
+print_depth 22;
+set timing;
+set trace_simp;
+fun test s = (Goal s; by (Asm_simp_tac 1));
+
+test "x*k = k*(y::rat)";
+test "k = k*(y::rat)";
+test "a*(b*c) = (b::rat)";
+test "a*(b*c) = d*(b::rat)*(x*a)";
+
+
+test "(x*k) / (k*(y::rat)) = (uu::rat)";
+test "(k) / (k*(y::rat)) = (uu::rat)";
+test "(a*(b*c)) / ((b::rat)) = (uu::rat)";
+test "(a*(b*c)) / (d*(b::rat)*(x*a)) = (uu::rat)";
+
+(*FIXME: what do we do about this?*)
+test "a*(b*c)/(y*z) = d*(b::rat)*(x*a)/z";
+*)