--- a/src/HOL/Tools/int_factor_simprocs.ML Fri May 08 08:01:09 2009 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,391 +0,0 @@
-(* Title: HOL/int_factor_simprocs.ML
- Author: Lawrence C Paulson, Cambridge University Computer Laboratory
- Copyright 2000 University of Cambridge
-
-Factor cancellation simprocs for the integers (and for fields).
-
-This file can't be combined with int_arith1 because it requires IntDiv.thy.
-*)
-
-
-(*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.
-*)
-
-val rel_number_of = [@{thm eq_number_of_eq}, @{thm less_number_of}, @{thm le_number_of}];
-
-local
- open Int_Numeral_Simprocs
-in
-
-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 rel_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)];
-
-(* referenced by rat_arith.ML *)
-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)]
-
-end;
-
-Addsimprocs 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)";
-*)
-
-
-(** Declarations for ExtractCommonTerm **)
-
-local
- open Int_Numeral_Simprocs
-in
-
-(*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 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";
-*)