--- 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];