--- a/src/HOL/Tools/nat_simprocs.ML Fri May 08 09:48:07 2009 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,574 +0,0 @@
-(* Title: HOL/Tools/nat_simprocs.ML
- Author: Lawrence C Paulson, Cambridge University Computer Laboratory
-
-Simprocs for nat numerals.
-*)
-
-structure Nat_Numeral_Simprocs =
-struct
-
-(*Maps n to #n for n = 0, 1, 2*)
-val numeral_syms = [@{thm nat_numeral_0_eq_0} RS sym, @{thm nat_numeral_1_eq_1} RS sym, @{thm numeral_2_eq_2} RS sym];
-val numeral_sym_ss = HOL_ss addsimps numeral_syms;
-
-fun rename_numerals th =
- simplify numeral_sym_ss (Thm.transfer (the_context ()) th);
-
-(*Utilities*)
-
-fun mk_number n = HOLogic.number_of_const HOLogic.natT $ HOLogic.mk_numeral n;
-fun dest_number t = Int.max (0, snd (HOLogic.dest_number t));
-
-fun find_first_numeral past (t::terms) =
- ((dest_number t, t, rev past @ terms)
- handle TERM _ => find_first_numeral (t::past) terms)
- | find_first_numeral past [] = raise TERM("find_first_numeral", []);
-
-val zero = mk_number 0;
-val mk_plus = HOLogic.mk_binop @{const_name HOL.plus};
-
-(*Thus mk_sum[t] yields t+0; longer sums don't have a trailing zero*)
-fun mk_sum [] = zero
- | mk_sum [t,u] = mk_plus (t, u)
- | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
-
-(*this version ALWAYS includes a trailing zero*)
-fun long_mk_sum [] = HOLogic.zero
- | long_mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
-
-val dest_plus = HOLogic.dest_bin @{const_name HOL.plus} HOLogic.natT;
-
-
-(** Other simproc items **)
-
-val bin_simps =
- [@{thm nat_numeral_0_eq_0} RS sym, @{thm nat_numeral_1_eq_1} RS sym,
- @{thm add_nat_number_of}, @{thm nat_number_of_add_left},
- @{thm diff_nat_number_of}, @{thm le_number_of_eq_not_less},
- @{thm mult_nat_number_of}, @{thm nat_number_of_mult_left},
- @{thm less_nat_number_of},
- @{thm Let_number_of}, @{thm nat_number_of}] @
- @{thms arith_simps} @ @{thms rel_simps} @ @{thms neg_simps};
-
-
-(*** CancelNumerals simprocs ***)
-
-val one = mk_number 1;
-val mk_times = HOLogic.mk_binop @{const_name HOL.times};
-
-fun mk_prod [] = one
- | mk_prod [t] = t
- | mk_prod (t :: ts) = if t = one then mk_prod ts
- else mk_times (t, mk_prod ts);
-
-val dest_times = HOLogic.dest_bin @{const_name HOL.times} HOLogic.natT;
-
-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 k, t);
-
-(*Express t as a product of (possibly) a numeral with other factors, sorted*)
-fun dest_coeff t =
- let val ts = sort TermOrd.term_ord (dest_prod t)
- val (n, _, ts') = find_first_numeral [] ts
- handle TERM _ => (1, one, ts)
- in (n, mk_prod 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 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;
-
-
-(*Split up a sum into the list of its constituent terms, on the way removing any
- Sucs and counting them.*)
-fun dest_Suc_sum (Const ("Suc", _) $ t, (k,ts)) = dest_Suc_sum (t, (k+1,ts))
- | dest_Suc_sum (t, (k,ts)) =
- let val (t1,t2) = dest_plus t
- in dest_Suc_sum (t1, dest_Suc_sum (t2, (k,ts))) end
- handle TERM _ => (k, t::ts);
-
-(*Code for testing whether numerals are already used in the goal*)
-fun is_numeral (Const(@{const_name Int.number_of}, _) $ w) = true
- | is_numeral _ = false;
-
-fun prod_has_numeral t = exists is_numeral (dest_prod t);
-
-(*The Sucs found in the term are converted to a binary numeral. If relaxed is false,
- an exception is raised unless the original expression contains at least one
- numeral in a coefficient position. This prevents nat_combine_numerals from
- introducing numerals to goals.*)
-fun dest_Sucs_sum relaxed t =
- let val (k,ts) = dest_Suc_sum (t,(0,[]))
- in
- if relaxed orelse exists prod_has_numeral ts then
- if k=0 then ts
- else mk_number k :: ts
- else raise TERM("Nat_Numeral_Simprocs.dest_Sucs_sum", [t])
- end;
-
-
-(*Simplify 1*n and n*1 to n*)
-val add_0s = map rename_numerals [@{thm add_0}, @{thm add_0_right}];
-val mult_1s = map rename_numerals [@{thm nat_mult_1}, @{thm nat_mult_1_right}];
-
-(*Final simplification: cancel + and *; replace Numeral0 by 0 and Numeral1 by 1*)
-
-(*And these help the simproc return False when appropriate, which helps
- the arith prover.*)
-val contra_rules = [@{thm add_Suc}, @{thm add_Suc_right}, @{thm Zero_not_Suc},
- @{thm Suc_not_Zero}, @{thm le_0_eq}];
-
-val simplify_meta_eq =
- Arith_Data.simplify_meta_eq
- ([@{thm nat_numeral_0_eq_0}, @{thm numeral_1_eq_Suc_0}, @{thm add_0}, @{thm add_0_right},
- @{thm mult_0}, @{thm mult_0_right}, @{thm mult_1}, @{thm mult_1_right}] @ contra_rules);
-
-
-(*** Applying CancelNumeralsFun ***)
-
-structure CancelNumeralsCommon =
- struct
- val mk_sum = (fn T:typ => mk_sum)
- val dest_sum = dest_Sucs_sum true
- val mk_coeff = mk_coeff
- val dest_coeff = dest_coeff
- val find_first_coeff = find_first_coeff []
- val trans_tac = K Arith_Data.trans_tac
-
- val norm_ss1 = Int_Numeral_Simprocs.num_ss addsimps numeral_syms @ add_0s @ mult_1s @
- [@{thm Suc_eq_add_numeral_1_left}] @ @{thms add_ac}
- val norm_ss2 = Int_Numeral_Simprocs.num_ss addsimps bin_simps @ @{thms add_ac} @ @{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))
-
- val numeral_simp_ss = HOL_ss addsimps add_0s @ bin_simps;
- fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss));
- val simplify_meta_eq = simplify_meta_eq
- 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 =" HOLogic.natT
- val bal_add1 = @{thm nat_eq_add_iff1} RS trans
- val bal_add2 = @{thm nat_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} HOLogic.natT
- val bal_add1 = @{thm nat_less_add_iff1} RS trans
- val bal_add2 = @{thm nat_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} HOLogic.natT
- val bal_add1 = @{thm nat_le_add_iff1} RS trans
- val bal_add2 = @{thm nat_le_add_iff2} RS trans
-);
-
-structure DiffCancelNumerals = CancelNumeralsFun
- (open CancelNumeralsCommon
- val prove_conv = Arith_Data.prove_conv
- val mk_bal = HOLogic.mk_binop @{const_name HOL.minus}
- val dest_bal = HOLogic.dest_bin @{const_name HOL.minus} HOLogic.natT
- val bal_add1 = @{thm nat_diff_add_eq1} RS trans
- val bal_add2 = @{thm nat_diff_add_eq2} RS trans
-);
-
-
-val cancel_numerals =
- map Arith_Data.prep_simproc
- [("nateq_cancel_numerals",
- ["(l::nat) + m = n", "(l::nat) = m + n",
- "(l::nat) * m = n", "(l::nat) = m * n",
- "Suc m = n", "m = Suc n"],
- K EqCancelNumerals.proc),
- ("natless_cancel_numerals",
- ["(l::nat) + m < n", "(l::nat) < m + n",
- "(l::nat) * m < n", "(l::nat) < m * n",
- "Suc m < n", "m < Suc n"],
- K LessCancelNumerals.proc),
- ("natle_cancel_numerals",
- ["(l::nat) + m <= n", "(l::nat) <= m + n",
- "(l::nat) * m <= n", "(l::nat) <= m * n",
- "Suc m <= n", "m <= Suc n"],
- K LeCancelNumerals.proc),
- ("natdiff_cancel_numerals",
- ["((l::nat) + m) - n", "(l::nat) - (m + n)",
- "(l::nat) * m - n", "(l::nat) - m * n",
- "Suc m - n", "m - Suc n"],
- K DiffCancelNumerals.proc)];
-
-
-(*** Applying CombineNumeralsFun ***)
-
-structure CombineNumeralsData =
- struct
- type coeff = int
- val iszero = (fn x => x = 0)
- val add = op +
- val mk_sum = (fn T:typ => long_mk_sum) (*to work for 2*x + 3*x *)
- val dest_sum = dest_Sucs_sum false
- val mk_coeff = mk_coeff
- val dest_coeff = dest_coeff
- val left_distrib = @{thm left_add_mult_distrib} RS trans
- val prove_conv = Arith_Data.prove_conv_nohyps
- val trans_tac = K Arith_Data.trans_tac
-
- val norm_ss1 = Int_Numeral_Simprocs.num_ss addsimps numeral_syms @ add_0s @ mult_1s @ [@{thm Suc_eq_add_numeral_1}] @ @{thms add_ac}
- val norm_ss2 = Int_Numeral_Simprocs.num_ss addsimps bin_simps @ @{thms add_ac} @ @{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))
-
- val numeral_simp_ss = HOL_ss addsimps add_0s @ bin_simps;
- fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
- val simplify_meta_eq = simplify_meta_eq
- end;
-
-structure CombineNumerals = CombineNumeralsFun(CombineNumeralsData);
-
-val combine_numerals =
- Arith_Data.prep_simproc ("nat_combine_numerals", ["(i::nat) + j", "Suc (i + j)"], K CombineNumerals.proc);
-
-
-(*** Applying CancelNumeralFactorFun ***)
-
-structure CancelNumeralFactorCommon =
- struct
- val mk_coeff = mk_coeff
- val dest_coeff = dest_coeff
- val trans_tac = K Arith_Data.trans_tac
-
- val norm_ss1 = Int_Numeral_Simprocs.num_ss addsimps
- numeral_syms @ add_0s @ mult_1s @ [@{thm Suc_eq_add_numeral_1_left}] @ @{thms add_ac}
- val norm_ss2 = Int_Numeral_Simprocs.num_ss addsimps bin_simps @ @{thms add_ac} @ @{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))
-
- val numeral_simp_ss = HOL_ss addsimps bin_simps
- fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
- val simplify_meta_eq = simplify_meta_eq
- end
-
-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} HOLogic.natT
- val cancel = @{thm nat_mult_div_cancel1} RS trans
- val neg_exchanges = false
-)
-
-structure DvdCancelNumeralFactor = CancelNumeralFactorFun
- (open CancelNumeralFactorCommon
- 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} HOLogic.natT
- val cancel = @{thm nat_mult_dvd_cancel1} 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 =" HOLogic.natT
- val cancel = @{thm nat_mult_eq_cancel1} 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} HOLogic.natT
- val cancel = @{thm nat_mult_less_cancel1} 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} HOLogic.natT
- val cancel = @{thm nat_mult_le_cancel1} RS trans
- val neg_exchanges = true
-)
-
-val cancel_numeral_factors =
- map Arith_Data.prep_simproc
- [("nateq_cancel_numeral_factors",
- ["(l::nat) * m = n", "(l::nat) = m * n"],
- K EqCancelNumeralFactor.proc),
- ("natless_cancel_numeral_factors",
- ["(l::nat) * m < n", "(l::nat) < m * n"],
- K LessCancelNumeralFactor.proc),
- ("natle_cancel_numeral_factors",
- ["(l::nat) * m <= n", "(l::nat) <= m * n"],
- K LeCancelNumeralFactor.proc),
- ("natdiv_cancel_numeral_factors",
- ["((l::nat) * m) div n", "(l::nat) div (m * n)"],
- K DivCancelNumeralFactor.proc),
- ("natdvd_cancel_numeral_factors",
- ["((l::nat) * m) dvd n", "(l::nat) dvd (m * n)"],
- K DvdCancelNumeralFactor.proc)];
-
-
-
-(*** Applying ExtractCommonTermFun ***)
-
-(*this version ALWAYS includes a trailing one*)
-fun long_mk_prod [] = one
- | long_mk_prod (t :: ts) = mk_times (t, mk_prod ts);
-
-(*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_1}, @{thm numeral_1_eq_Suc_0}];
-
-fun cancel_simplify_meta_eq ss cancel_th th =
- simplify_one ss (([th, cancel_th]) MRS trans);
-
-structure CancelFactorCommon =
- struct
- val mk_sum = (fn T:typ => 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;
-
-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 =" HOLogic.natT
- val simp_conv = K(K (SOME @{thm nat_mult_eq_cancel_disj}))
-);
-
-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} HOLogic.natT
- val simp_conv = K(K (SOME @{thm nat_mult_less_cancel_disj}))
-);
-
-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} HOLogic.natT
- val simp_conv = K(K (SOME @{thm nat_mult_le_cancel_disj}))
-);
-
-structure DivideCancelFactor = 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} HOLogic.natT
- val simp_conv = K(K (SOME @{thm nat_mult_div_cancel_disj}))
-);
-
-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} HOLogic.natT
- val simp_conv = K(K (SOME @{thm nat_mult_dvd_cancel_disj}))
-);
-
-val cancel_factor =
- map Arith_Data.prep_simproc
- [("nat_eq_cancel_factor",
- ["(l::nat) * m = n", "(l::nat) = m * n"],
- K EqCancelFactor.proc),
- ("nat_less_cancel_factor",
- ["(l::nat) * m < n", "(l::nat) < m * n"],
- K LessCancelFactor.proc),
- ("nat_le_cancel_factor",
- ["(l::nat) * m <= n", "(l::nat) <= m * n"],
- K LeCancelFactor.proc),
- ("nat_divide_cancel_factor",
- ["((l::nat) * m) div n", "(l::nat) div (m * n)"],
- K DivideCancelFactor.proc),
- ("nat_dvd_cancel_factor",
- ["((l::nat) * m) dvd n", "(l::nat) dvd (m * n)"],
- K DvdCancelFactor.proc)];
-
-end;
-
-
-Addsimprocs Nat_Numeral_Simprocs.cancel_numerals;
-Addsimprocs [Nat_Numeral_Simprocs.combine_numerals];
-Addsimprocs Nat_Numeral_Simprocs.cancel_numeral_factors;
-Addsimprocs Nat_Numeral_Simprocs.cancel_factor;
-
-
-(*examples:
-print_depth 22;
-set timing;
-set trace_simp;
-fun test s = (Goal s; by (Simp_tac 1));
-
-(*cancel_numerals*)
-test "l +( 2) + (2) + 2 + (l + 2) + (oo + 2) = (uu::nat)";
-test "(2*length xs < 2*length xs + j)";
-test "(2*length xs < length xs * 2 + j)";
-test "2*u = (u::nat)";
-test "2*u = Suc (u)";
-test "(i + j + 12 + (k::nat)) - 15 = y";
-test "(i + j + 12 + (k::nat)) - 5 = y";
-test "Suc u - 2 = y";
-test "Suc (Suc (Suc u)) - 2 = y";
-test "(i + j + 2 + (k::nat)) - 1 = y";
-test "(i + j + 1 + (k::nat)) - 2 = y";
-
-test "(2*x + (u*v) + y) - v*3*u = (w::nat)";
-test "(2*x*u*v + 5 + (u*v)*4 + y) - v*u*4 = (w::nat)";
-test "(2*x*u*v + (u*v)*4 + y) - v*u = (w::nat)";
-test "Suc (Suc (2*x*u*v + u*4 + y)) - u = w";
-test "Suc ((u*v)*4) - v*3*u = w";
-test "Suc (Suc ((u*v)*3)) - v*3*u = w";
-
-test "(i + j + 12 + (k::nat)) = u + 15 + y";
-test "(i + j + 32 + (k::nat)) - (u + 15 + y) = zz";
-test "(i + j + 12 + (k::nat)) = u + 5 + y";
-(*Suc*)
-test "(i + j + 12 + k) = Suc (u + y)";
-test "Suc (Suc (Suc (Suc (Suc (u + y))))) <= ((i + j) + 41 + k)";
-test "(i + j + 5 + k) < Suc (Suc (Suc (Suc (Suc (u + y)))))";
-test "Suc (Suc (Suc (Suc (Suc (u + y))))) - 5 = v";
-test "(i + j + 5 + k) = Suc (Suc (Suc (Suc (Suc (Suc (Suc (u + y)))))))";
-test "2*y + 3*z + 2*u = Suc (u)";
-test "2*y + 3*z + 6*w + 2*y + 3*z + 2*u = Suc (u)";
-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::nat)";
-test "6 + 2*y + 3*z + 4*u = Suc (vv + 2*u + z)";
-test "(2*n*m) < (3*(m*n)) + (u::nat)";
-
-test "(Suc (Suc (Suc (Suc (Suc (Suc (case length (f c) of 0 => 0 | Suc k => k)))))) <= Suc 0)";
-
-test "Suc (Suc (Suc (Suc (Suc (Suc (length l1 + length l2)))))) <= length l1";
-
-test "( (Suc (Suc (Suc (Suc (Suc (length (compT P E A ST mxr e) + length l3)))))) <= length (compT P E A ST mxr e))";
-
-test "( (Suc (Suc (Suc (Suc (Suc (length (compT P E A ST mxr e) + length (compT P E (A Un \<A> e) ST mxr c))))))) <= length (compT P E A ST mxr e))";
-
-
-(*negative numerals: FAIL*)
-test "(i + j + -23 + (k::nat)) < u + 15 + y";
-test "(i + j + 3 + (k::nat)) < u + -15 + y";
-test "(i + j + -12 + (k::nat)) - 15 = y";
-test "(i + j + 12 + (k::nat)) - -15 = y";
-test "(i + j + -12 + (k::nat)) - -15 = y";
-
-(*combine_numerals*)
-test "k + 3*k = (u::nat)";
-test "Suc (i + 3) = u";
-test "Suc (i + j + 3 + k) = u";
-test "k + j + 3*k + j = (u::nat)";
-test "Suc (j*i + i + k + 5 + 3*k + i*j*4) = (u::nat)";
-test "(2*n*m) + (3*(m*n)) = (u::nat)";
-(*negative numerals: FAIL*)
-test "Suc (i + j + -3 + k) = u";
-
-(*cancel_numeral_factors*)
-test "9*x = 12 * (y::nat)";
-test "(9*x) div (12 * (y::nat)) = z";
-test "9*x < 12 * (y::nat)";
-test "9*x <= 12 * (y::nat)";
-
-(*cancel_factor*)
-test "x*k = k*(y::nat)";
-test "k = k*(y::nat)";
-test "a*(b*c) = (b::nat)";
-test "a*(b*c) = d*(b::nat)*(x*a)";
-
-test "x*k < k*(y::nat)";
-test "k < k*(y::nat)";
-test "a*(b*c) < (b::nat)";
-test "a*(b*c) < d*(b::nat)*(x*a)";
-
-test "x*k <= k*(y::nat)";
-test "k <= k*(y::nat)";
-test "a*(b*c) <= (b::nat)";
-test "a*(b*c) <= d*(b::nat)*(x*a)";
-
-test "(x*k) div (k*(y::nat)) = (uu::nat)";
-test "(k) div (k*(y::nat)) = (uu::nat)";
-test "(a*(b*c)) div ((b::nat)) = (uu::nat)";
-test "(a*(b*c)) div (d*(b::nat)*(x*a)) = (uu::nat)";
-*)
-
-
-(*** Prepare linear arithmetic for nat numerals ***)
-
-local
-
-(* reduce contradictory <= to False *)
-val add_rules = @{thms ring_distribs} @
- [@{thm Let_number_of}, @{thm Let_0}, @{thm Let_1}, @{thm nat_0}, @{thm nat_1},
- @{thm add_nat_number_of}, @{thm diff_nat_number_of}, @{thm mult_nat_number_of},
- @{thm eq_nat_number_of}, @{thm less_nat_number_of}, @{thm le_number_of_eq_not_less},
- @{thm le_Suc_number_of}, @{thm le_number_of_Suc},
- @{thm less_Suc_number_of}, @{thm less_number_of_Suc},
- @{thm Suc_eq_number_of}, @{thm eq_number_of_Suc},
- @{thm mult_Suc}, @{thm mult_Suc_right},
- @{thm add_Suc}, @{thm add_Suc_right},
- @{thm eq_number_of_0}, @{thm eq_0_number_of}, @{thm less_0_number_of},
- @{thm of_int_number_of_eq}, @{thm of_nat_number_of_eq}, @{thm nat_number_of}, @{thm if_True}, @{thm if_False}];
-
-(* Products are multiplied out during proof (re)construction via
-ring_distribs. Ideally they should remain atomic. But that is
-currently not possible because 1 is replaced by Suc 0, and then some
-simprocs start to mess around with products like (n+1)*m. The rule
-1 == Suc 0 is necessary for early parts of HOL where numerals and
-simprocs are not yet available. But then it is difficult to remove
-that rule later on, because it may find its way back in when theories
-(and thus lin-arith simpsets) are merged. Otherwise one could turn the
-rule around (Suc n = n+1) and see if that helps products being left
-alone. *)
-
-val simprocs = Nat_Numeral_Simprocs.combine_numerals
- :: Nat_Numeral_Simprocs.cancel_numerals;
-
-in
-
-val nat_simprocs_setup =
- Lin_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset} =>
- {add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms,
- inj_thms = inj_thms, lessD = lessD, neqE = neqE,
- simpset = simpset addsimps add_rules
- addsimprocs simprocs});
-
-end;