# HG changeset patch
# User paulson
# Date 960999560 -7200
# Node ID 202fdf72cf23ee1ec2aacbdfe8e5a51e00e14359
# Parent  64464b5f686722d3e1e04c30630c00cb91ca4a85
installing the cancel_numerals and combine_numerals simprocs

diff -r 64464b5f6867 -r 202fdf72cf23 src/HOL/Real/RealBin.ML
--- a/src/HOL/Real/RealBin.ML	Wed Jun 14 18:00:46 2000 +0200
+++ b/src/HOL/Real/RealBin.ML	Wed Jun 14 18:19:20 2000 +0200
@@ -121,7 +121,8 @@
     HOL_ss addsimps [zero_eq_numeral_0, one_eq_numeral_1, 
 		     minus_numeral_one];
 
-fun rename_numerals thy th = simplify real_numeral_ss (change_theory thy th);
+fun rename_numerals thy th = 
+    asm_full_simplify real_numeral_ss (change_theory thy th);
 
 
 (*Now insert some identities previously stated for 0 and 1r*)
@@ -138,14 +139,14 @@
 
 AddIffs (map (rename_numerals thy) [real_mult_is_0, real_0_is_mult]);
 
-bind_thm ("real_0_less_times_iff", 
-	  rename_numerals thy real_zero_less_times_iff);
-bind_thm ("real_0_le_times_iff", 
-	  rename_numerals thy real_zero_le_times_iff);
-bind_thm ("real_times_less_0_iff", 
-	  rename_numerals thy real_times_less_zero_iff);
-bind_thm ("real_times_le_0_iff", 
-	  rename_numerals thy real_times_le_zero_iff);
+bind_thm ("real_0_less_mult_iff", 
+	  rename_numerals thy real_zero_less_mult_iff);
+bind_thm ("real_0_le_mult_iff", 
+	  rename_numerals thy real_zero_le_mult_iff);
+bind_thm ("real_mult_less_0_iff", 
+	  rename_numerals thy real_mult_less_zero_iff);
+bind_thm ("real_mult_le_0_iff", 
+	  rename_numerals thy real_mult_le_zero_iff);
 
 
 (*Perhaps add some theorems that aren't in the default simpset, as
@@ -207,10 +208,6 @@
 
 Addsimps [zero_eq_numeral_0,one_eq_numeral_1];
 
-(* added by jdf *)
-fun full_rename_numerals thy th = 
-    asm_full_simplify real_numeral_ss (change_theory thy th);
-
 
 (** Simplification of arithmetic when nested to the right **)
 
@@ -236,3 +233,461 @@
 Addsimps [real_add_number_of_left, real_mult_number_of_left,
 	  real_add_number_of_diff1, real_add_number_of_diff2]; 
 
+
+(*"neg" is used in rewrite rules for binary comparisons*)
+Goal "real_of_nat (number_of v :: nat) = \
+\        (if neg (number_of v) then #0 \
+\         else (number_of v :: real))";
+by (simp_tac
+    (simpset_of Int.thy addsimps [nat_number_of_def, real_of_nat_real_of_int,
+				  real_of_nat_neg_int, real_number_of,
+				  zero_eq_numeral_0]) 1);
+qed "real_of_nat_number_of";
+Addsimps [real_of_nat_number_of];
+
+
+(**** Simprocs for numeric literals ****)
+
+(** Combining of literal coefficients in sums of products **)
+
+Goal "(x < y) = (x-y < (#0::real))";
+by Auto_tac; 
+qed "real_less_iff_diff_less_0";
+
+Goal "(x = y) = (x-y = (#0::real))";
+by Auto_tac; 
+qed "real_eq_iff_diff_eq_0";
+
+Goal "(x <= y) = (x-y <= (#0::real))";
+by Auto_tac; 
+qed "real_le_iff_diff_le_0";
+
+
+(** For combine_numerals **)
+
+Goal "i*u + (j*u + k) = (i+j)*u + (k::real)";
+by (asm_simp_tac (simpset() addsimps [real_add_mult_distrib]) 1);
+qed "left_real_add_mult_distrib";
+
+
+(** For cancel_numerals **)
+
+val rel_iff_rel_0_rls = map (inst "y" "?u+?v")
+                          [real_less_iff_diff_less_0, real_eq_iff_diff_eq_0, 
+			   real_le_iff_diff_le_0] @
+		        map (inst "y" "n")
+                          [real_less_iff_diff_less_0, real_eq_iff_diff_eq_0, 
+			   real_le_iff_diff_le_0];
+
+Goal "!!i::real. (i*u + m = j*u + n) = ((i-j)*u + m = n)";
+by (asm_simp_tac (simpset() addsimps [real_diff_def, real_add_mult_distrib]@
+		                     real_add_ac@rel_iff_rel_0_rls) 1);
+qed "real_eq_add_iff1";
+
+Goal "!!i::real. (i*u + m = j*u + n) = (m = (j-i)*u + n)";
+by (asm_simp_tac (simpset() addsimps [real_diff_def, real_add_mult_distrib]@
+                                     real_add_ac@rel_iff_rel_0_rls) 1);
+qed "real_eq_add_iff2";
+
+Goal "!!i::real. (i*u + m < j*u + n) = ((i-j)*u + m < n)";
+by (asm_simp_tac (simpset() addsimps [real_diff_def, real_add_mult_distrib]@
+                                     real_add_ac@rel_iff_rel_0_rls) 1);
+qed "real_less_add_iff1";
+
+Goal "!!i::real. (i*u + m < j*u + n) = (m < (j-i)*u + n)";
+by (asm_simp_tac (simpset() addsimps [real_diff_def, real_add_mult_distrib]@
+                                     real_add_ac@rel_iff_rel_0_rls) 1);
+qed "real_less_add_iff2";
+
+Goal "!!i::real. (i*u + m <= j*u + n) = ((i-j)*u + m <= n)";
+by (asm_simp_tac (simpset() addsimps [real_diff_def, real_add_mult_distrib]@
+                                     real_add_ac@rel_iff_rel_0_rls) 1);
+qed "real_le_add_iff1";
+
+Goal "!!i::real. (i*u + m <= j*u + n) = (m <= (j-i)*u + n)";
+by (asm_simp_tac (simpset() addsimps [real_diff_def, real_add_mult_distrib]
+                                     @real_add_ac@rel_iff_rel_0_rls) 1);
+qed "real_le_add_iff2";
+
+
+structure Real_Numeral_Simprocs =
+struct
+
+(*Utilities*)
+
+fun mk_numeral n = HOLogic.number_of_const HOLogic.realT $ 
+                   NumeralSyntax.mk_bin n;
+
+(*Decodes a binary real constant*)
+fun dest_numeral (Const("Numeral.number_of", _) $ w) = 
+     (NumeralSyntax.dest_bin w
+      handle Match => raise TERM("Real_Numeral_Simprocs.dest_numeral:1", [w]))
+  | dest_numeral t = raise TERM("Real_Numeral_Simprocs.dest_numeral:2", [t]);
+
+fun find_first_numeral past (t::terms) =
+	((dest_numeral t, rev past @ terms)
+	 handle TERM _ => find_first_numeral (t::past) terms)
+  | find_first_numeral past [] = raise TERM("find_first_numeral", []);
+
+val zero = mk_numeral 0;
+val mk_plus = HOLogic.mk_binop "op +";
+
+val uminus_const = Const ("uminus", HOLogic.realT --> HOLogic.realT);
+
+(*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 []        = zero
+  | long_mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
+
+val dest_plus = HOLogic.dest_bin "op +" HOLogic.realT;
+
+(*decompose additions AND subtractions as a sum*)
+fun dest_summing (pos, Const ("op +", _) $ t $ u, ts) =
+        dest_summing (pos, t, dest_summing (pos, u, ts))
+  | dest_summing (pos, Const ("op -", _) $ t $ u, ts) =
+        dest_summing (pos, t, dest_summing (not pos, u, ts))
+  | dest_summing (pos, t, ts) =
+	if pos then t::ts else uminus_const$t :: ts;
+
+fun dest_sum t = dest_summing (true, t, []);
+
+val mk_diff = HOLogic.mk_binop "op -";
+val dest_diff = HOLogic.dest_bin "op -" HOLogic.realT;
+
+val one = mk_numeral 1;
+val mk_times = HOLogic.mk_binop "op *";
+
+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 "op *" HOLogic.realT;
+
+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, ts) = mk_times (mk_numeral k, ts);
+
+(*Express t as a product of (possibly) a numeral with other sorted terms*)
+fun dest_coeff sign (Const ("uminus", _) $ t) = dest_coeff (~sign) t
+  | dest_coeff sign t =
+    let val ts = sort Term.term_ord (dest_prod t)
+	val (n, ts') = find_first_numeral [] ts
+                          handle TERM _ => (1, ts)
+    in (sign*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 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;
+
+
+(*Simplify #1*n and n*#1 to n*)
+val add_0s = map (rename_numerals thy) 
+                 [real_add_zero_left, real_add_zero_right];
+val mult_1s = map (rename_numerals thy) 
+                  [real_mult_1, real_mult_1_right, 
+		   real_mult_minus_1, real_mult_minus_1_right];
+
+(*To perform binary arithmetic*)
+val bin_simps = 
+    [add_real_number_of, real_add_number_of_left, minus_real_number_of, 
+     diff_real_number_of] @ 
+    bin_arith_simps @ bin_rel_simps;
+
+(*To evaluate binary negations of coefficients*)
+val real_minus_simps = NCons_simps @
+                   [minus_real_number_of, 
+		    bin_minus_1, bin_minus_0, bin_minus_Pls, bin_minus_Min,
+		    bin_pred_1, bin_pred_0, bin_pred_Pls, bin_pred_Min];
+
+(*To let us treat subtraction as addition*)
+val diff_simps = [real_diff_def, real_minus_add_distrib, real_minus_minus];
+
+(*Apply the given rewrite (if present) just once*)
+fun trans_tac None      = all_tac
+  | trans_tac (Some th) = ALLGOALS (rtac (th RS trans));
+
+fun prove_conv name tacs sg (t, u) =
+  if t aconv u then None
+  else
+  let val ct = cterm_of sg (HOLogic.mk_Trueprop (HOLogic.mk_eq (t, u)))
+  in Some
+     (prove_goalw_cterm [] ct (K tacs)
+      handle ERROR => error 
+	  ("The error(s) above occurred while trying to prove " ^
+	   string_of_cterm ct ^ "\nInternal failure of simproc " ^ name))
+  end;
+
+(*Final simplification: cancel + and *  *)
+val simplify_meta_eq = 
+    Int_Numeral_Simprocs.simplify_meta_eq
+         [real_add_zero_left, real_add_zero_right,
+ 	  real_mult_0, real_mult_0_right, real_mult_1, real_mult_1_right];
+
+fun prep_simproc (name, pats, proc) = Simplifier.mk_simproc name pats proc;
+fun prep_pat s = Thm.read_cterm (Theory.sign_of RealInt.thy) (s, HOLogic.termT);
+val prep_pats = map prep_pat;
+
+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         = trans_tac
+  val norm_tac = ALLGOALS (simp_tac (HOL_ss addsimps add_0s@mult_1s@diff_simps@
+                                                real_minus_simps@real_add_ac))
+                 THEN ALLGOALS
+                    (simp_tac (HOL_ss addsimps [real_minus_mult_eq2]@
+                                         bin_simps@real_add_ac@real_mult_ac))
+  val numeral_simp_tac	= ALLGOALS (simp_tac (HOL_ss addsimps add_0s@bin_simps))
+  val simplify_meta_eq  = simplify_meta_eq
+  end;
+
+
+structure EqCancelNumerals = CancelNumeralsFun
+ (open CancelNumeralsCommon
+  val prove_conv = prove_conv "realeq_cancel_numerals"
+  val mk_bal   = HOLogic.mk_eq
+  val dest_bal = HOLogic.dest_bin "op =" HOLogic.realT
+  val bal_add1 = real_eq_add_iff1 RS trans
+  val bal_add2 = real_eq_add_iff2 RS trans
+);
+
+structure LessCancelNumerals = CancelNumeralsFun
+ (open CancelNumeralsCommon
+  val prove_conv = prove_conv "realless_cancel_numerals"
+  val mk_bal   = HOLogic.mk_binrel "op <"
+  val dest_bal = HOLogic.dest_bin "op <" HOLogic.realT
+  val bal_add1 = real_less_add_iff1 RS trans
+  val bal_add2 = real_less_add_iff2 RS trans
+);
+
+structure LeCancelNumerals = CancelNumeralsFun
+ (open CancelNumeralsCommon
+  val prove_conv = prove_conv "realle_cancel_numerals"
+  val mk_bal   = HOLogic.mk_binrel "op <="
+  val dest_bal = HOLogic.dest_bin "op <=" HOLogic.realT
+  val bal_add1 = real_le_add_iff1 RS trans
+  val bal_add2 = real_le_add_iff2 RS trans
+);
+
+val cancel_numerals = 
+  map prep_simproc
+   [("realeq_cancel_numerals",
+     prep_pats ["(l::real) + m = n", "(l::real) = m + n", 
+		"(l::real) - m = n", "(l::real) = m - n", 
+		"(l::real) * m = n", "(l::real) = m * n"], 
+     EqCancelNumerals.proc),
+    ("realless_cancel_numerals", 
+     prep_pats ["(l::real) + m < n", "(l::real) < m + n", 
+		"(l::real) - m < n", "(l::real) < m - n", 
+		"(l::real) * m < n", "(l::real) < m * n"], 
+     LessCancelNumerals.proc),
+    ("realle_cancel_numerals", 
+     prep_pats ["(l::real) + m <= n", "(l::real) <= m + n", 
+		"(l::real) - m <= n", "(l::real) <= m - n", 
+		"(l::real) * m <= n", "(l::real) <= m * n"], 
+     LeCancelNumerals.proc)];
+
+
+structure CombineNumeralsData =
+  struct
+  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	= left_real_add_mult_distrib RS trans
+  val prove_conv	= prove_conv "real_combine_numerals"
+  val trans_tac          = trans_tac
+  val norm_tac = ALLGOALS
+                   (simp_tac (HOL_ss addsimps add_0s@mult_1s@diff_simps@
+                                              real_minus_simps@real_add_ac))
+                 THEN ALLGOALS
+                    (simp_tac (HOL_ss addsimps [real_minus_mult_eq2]@
+                                               bin_simps@real_add_ac@real_mult_ac))
+  val numeral_simp_tac	= ALLGOALS 
+                    (simp_tac (HOL_ss addsimps add_0s@bin_simps))
+  val simplify_meta_eq  = simplify_meta_eq
+  end;
+
+structure CombineNumerals = CombineNumeralsFun(CombineNumeralsData);
+  
+val combine_numerals = 
+    prep_simproc ("real_combine_numerals",
+		  prep_pats ["(i::real) + j", "(i::real) - j"],
+		  CombineNumerals.proc);
+
+end;
+
+
+Addsimprocs Real_Numeral_Simprocs.cancel_numerals;
+Addsimprocs [Real_Numeral_Simprocs.combine_numerals];
+
+(*The Abel_Cancel simprocs are now obsolete*)
+Delsimprocs [Real_Cancel.sum_conv, Real_Cancel.rel_conv];
+
+(*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::real)";
+
+test "#2*u = (u::real)";
+test "(i + j + #12 + (k::real)) - #15 = y";
+test "(i + j + #12 + (k::real)) - #5 = y";
+
+test "y - b < (b::real)";
+test "y - (#3*b + c) < (b::real) - #2*c";
+
+test "(#2*x - (u*v) + y) - v*#3*u = (w::real)";
+test "(#2*x*u*v + (u*v)*#4 + y) - v*u*#4 = (w::real)";
+test "(#2*x*u*v + (u*v)*#4 + y) - v*u = (w::real)";
+test "u*v - (x*u*v + (u*v)*#4 + y) = (w::real)";
+
+test "(i + j + #12 + (k::real)) = u + #15 + y";
+test "(i + j*#2 + #12 + (k::real)) = 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::real)";
+
+test "a + -(b+c) + b = (d::real)";
+test "a + -(b+c) - b = (d::real)";
+
+(*negative numerals*)
+test "(i + j + #-2 + (k::real)) - (u + #5 + y) = zz";
+test "(i + j + #-3 + (k::real)) < u + #5 + y";
+test "(i + j + #3 + (k::real)) < u + #-6 + y";
+test "(i + j + #-12 + (k::real)) - #15 = y";
+test "(i + j + #12 + (k::real)) - #-15 = y";
+test "(i + j + #-12 + (k::real)) - #-15 = y";
+*)
+
+
+(** Constant folding for real plus and times **)
+
+(*We do not need
+    structure Real_Plus_Assoc = Assoc_Fold (Real_Plus_Assoc_Data);
+  because combine_numerals does the same thing*)
+
+structure Real_Times_Assoc_Data : ASSOC_FOLD_DATA =
+struct
+  val ss		= HOL_ss
+  val eq_reflection	= eq_reflection
+  val thy    = RealBin.thy
+  val T	     = HOLogic.realT
+  val plus   = Const ("op *", [HOLogic.realT,HOLogic.realT] ---> HOLogic.realT)
+  val add_ac = real_mult_ac
+end;
+
+structure Real_Times_Assoc = Assoc_Fold (Real_Times_Assoc_Data);
+
+Addsimprocs [Real_Times_Assoc.conv];
+
+
+(*** decision procedure for linear arithmetic ***)
+
+(*---------------------------------------------------------------------------*)
+(* Linear arithmetic                                                         *)
+(*---------------------------------------------------------------------------*)
+
+(*
+Instantiation of the generic linear arithmetic package for real.
+*)
+
+(* Update parameters of arithmetic prover *)
+let
+
+(* reduce contradictory <= to False *)
+val add_rules =  
+    real_diff_def ::
+    map (rename_numerals thy) 
+        [real_add_zero_left, real_add_zero_right, 
+	 real_add_minus, real_add_minus_left, 
+	 real_mult_0, real_mult_0_right, 
+	 real_mult_1, real_mult_1_right, 
+	 real_mult_minus_1, real_mult_minus_1_right];
+
+val simprocs = [Real_Times_Assoc.conv, Real_Numeral_Simprocs.combine_numerals]@
+               Real_Numeral_Simprocs.cancel_numerals;
+
+val add_mono_thms =
+  map (fn s => prove_goal RealBin.thy s
+                 (fn prems => [cut_facts_tac prems 1,
+                      asm_simp_tac (simpset() addsimps [real_add_le_mono]) 1]))
+    ["(i <= j) & (k <= l) ==> i + k <= j + (l::real)",
+     "(i  = j) & (k <= l) ==> i + k <= j + (l::real)",
+     "(i <= j) & (k  = l) ==> i + k <= j + (l::real)",
+     "(i  = j) & (k  = l) ==> i + k  = j + (l::real)"
+    ];
+
+in
+LA_Data_Ref.add_mono_thms := !LA_Data_Ref.add_mono_thms @ add_mono_thms;
+(*We don't change LA_Data_Ref.lessD and LA_Data_Ref.discrete
+ because the real ordering is dense!*)
+LA_Data_Ref.ss_ref := !LA_Data_Ref.ss_ref addsimps add_rules
+                      addsimprocs simprocs
+                      addcongs [if_weak_cong]
+end;
+
+let
+val real_arith_simproc_pats =
+  map (fn s => Thm.read_cterm (Theory.sign_of RealDef.thy) (s, HOLogic.boolT))
+      ["(m::real) < n","(m::real) <= n", "(m::real) = n"];
+
+val fast_real_arith_simproc = mk_simproc
+  "fast_real_arith" real_arith_simproc_pats Fast_Arith.lin_arith_prover;
+in
+Addsimprocs [fast_real_arith_simproc]
+end;
+
+(* Some test data [omitting examples thet assume the ordering to be discrete!]
+Goal "!!a::real. [| a <= b; c <= d; x+y<z |] ==> a+c <= b+d";
+by (fast_arith_tac 1);
+Goal "!!a::real. [| a <= b; b+b <= c |] ==> a+a <= c";
+by (fast_arith_tac 1);
+Goal "!!a::real. [| a+b <= i+j; a<=b; i<=j |] ==> a+a <= j+j";
+by (fast_arith_tac 1);
+Goal "!!a::real. a+b+c <= i+j+k & a<=b & b<=c & i<=j & j<=k --> a+a+a <= k+k+k";
+by (arith_tac 1);
+Goal "!!a::real. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \
+\     ==> a <= l";
+by (fast_arith_tac 1);
+Goal "!!a::real. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \
+\     ==> a+a+a+a <= l+l+l+l";
+by (fast_arith_tac 1);
+Goal "!!a::real. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \
+\     ==> a+a+a+a+a <= l+l+l+l+i";
+by (fast_arith_tac 1);
+Goal "!!a::real. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \
+\     ==> a+a+a+a+a+a <= l+l+l+l+i+l";
+by (fast_arith_tac 1);
+Goal "!!a::real. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \
+\     ==> #6*a <= #5*l+i";
+by (fast_arith_tac 1);
+*)
+
+(*---------------------------------------------------------------------------*)
+(* End of linear arithmetic                                                  *)
+(*---------------------------------------------------------------------------*)
+
+(*useful??*)
+Goal "(z = z + w) = (w = (#0::real))";
+by Auto_tac;
+qed "real_add_left_cancel0";
+Addsimps [real_add_left_cancel0];