src/HOL/Tools/int_factor_simprocs.ML
 changeset 28952 15a4b2cf8c34 parent 27651 16a26996c30e child 29038 90f42c138585
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+++ b/src/HOL/Tools/int_factor_simprocs.ML	Wed Dec 03 15:58:44 2008 +0100
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+(*  Title:      HOL/int_factor_simprocs.ML
+    ID:         \$Id\$
+    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_eq_neg}, @{thm le_number_of_eq}];
+
+local
+  open Int_Numeral_Simprocs
+in
+
+structure CancelNumeralFactorCommon =
+  struct
+  val mk_coeff          = mk_coeff
+  val dest_coeff        = dest_coeff 1
+  val trans_tac         = fn _ => 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 = Int_Numeral_Simprocs.simplify_meta_eq
+      @{thm mult_zero_right}, @{thm mult_Bit1}, @{thm mult_1_right}];
+  end
+
+(*Version for integer division*)
+structure IntDivCancelNumeralFactor = CancelNumeralFactorFun
+ (open CancelNumeralFactorCommon
+  val prove_conv = Int_Numeral_Base_Simprocs.prove_conv
+  val mk_bal   = HOLogic.mk_binop @{const_name Divides.div}
+  val dest_bal = HOLogic.dest_bin @{const_name Divides.div} HOLogic.intT
+  val cancel = @{thm zdiv_zmult_zmult1} RS trans
+  val neg_exchanges = false
+)
+
+(*Version for fields*)
+structure DivideCancelNumeralFactor = CancelNumeralFactorFun
+ (open CancelNumeralFactorCommon
+  val prove_conv = Int_Numeral_Base_Simprocs.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 = Int_Numeral_Base_Simprocs.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 = Int_Numeral_Base_Simprocs.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 = Int_Numeral_Base_Simprocs.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 Int_Numeral_Base_Simprocs.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::int) * m) div n", "(l::int) div (m * n)"],
+     K IntDivCancelNumeralFactor.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 Int_Numeral_Base_Simprocs.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;
+
+
+(*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 = Int_Numeral_Simprocs.simplify_meta_eq
+  [@{thm mult_1_left}, @{thm mult_1_right}, @{thm zdiv_1}, @{thm numeral_1_eq_1}];
+
+fun cancel_simplify_meta_eq cancel_th ss th =
+    simplify_one ss (([th, cancel_th]) MRS trans);
+
+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         = fn _ => 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))
+  end;
+
+(*mult_cancel_left requires a ring with no zero divisors.*)
+structure EqCancelFactor = ExtractCommonTermFun
+ (open CancelFactorCommon
+  val prove_conv = Int_Numeral_Base_Simprocs.prove_conv
+  val mk_bal   = HOLogic.mk_eq
+  val dest_bal = HOLogic.dest_bin "op =" Term.dummyT
+  val simplify_meta_eq  = cancel_simplify_meta_eq @{thm mult_cancel_left}
+);
+
+(*zdiv_zmult_zmult1_if is for integer division (div).*)
+structure IntDivCancelFactor = ExtractCommonTermFun
+ (open CancelFactorCommon
+  val prove_conv = Int_Numeral_Base_Simprocs.prove_conv
+  val mk_bal   = HOLogic.mk_binop @{const_name Divides.div}
+  val dest_bal = HOLogic.dest_bin @{const_name Divides.div} HOLogic.intT
+  val simplify_meta_eq  = cancel_simplify_meta_eq @{thm zdiv_zmult_zmult1_if}
+);
+
+structure IntModCancelFactor = ExtractCommonTermFun
+ (open CancelFactorCommon
+  val prove_conv = Int_Numeral_Base_Simprocs.prove_conv
+  val mk_bal   = HOLogic.mk_binop @{const_name Divides.mod}
+  val dest_bal = HOLogic.dest_bin @{const_name Divides.mod} HOLogic.intT
+  val simplify_meta_eq  = cancel_simplify_meta_eq @{thm zmod_zmult_zmult1}
+);
+
+structure IntDvdCancelFactor = ExtractCommonTermFun
+ (open CancelFactorCommon
+  val prove_conv = Int_Numeral_Base_Simprocs.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.intT
+  val simplify_meta_eq  = cancel_simplify_meta_eq @{thm zdvd_zmult_cancel_disj}
+);
+
+(*Version for all fields, including unordered ones (type complex).*)
+structure DivideCancelFactor = ExtractCommonTermFun
+ (open CancelFactorCommon
+  val prove_conv = Int_Numeral_Base_Simprocs.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 simplify_meta_eq  = cancel_simplify_meta_eq @{thm mult_divide_mult_cancel_left_if}
+);
+
+val cancel_factors =
+  map Int_Numeral_Base_Simprocs.prep_simproc
+   [("ring_eq_cancel_factor",
+     ["(l::'a::{idom}) * m = n",
+      "(l::'a::{idom}) = m * n"],
+    K EqCancelFactor.proc),
+    ("int_div_cancel_factor",
+     ["((l::int) * m) div n", "(l::int) div (m * n)"],
+     K IntDivCancelFactor.proc),
+    ("int_mod_cancel_factor",
+     ["((l::int) * m) mod n", "(l::int) mod (m * n)"],
+     K IntModCancelFactor.proc),
+    ("int_dvd_cancel_factor",
+     ["((l::int) * m) dvd n", "(l::int) dvd (m * n)"],
+     K IntDvdCancelFactor.proc),
+    ("divide_cancel_factor",
+     ["((l::'a::{division_by_zero,field}) * m) / n",
+      "(l::'a::{division_by_zero,field}) / (m * n)"],
+     K DivideCancelFactor.proc)];
+
+end;
+
+
+
+(*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)";
+