--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Real/real_arith.ML Tue Jul 25 00:06:46 2000 +0200
@@ -0,0 +1,109 @@
+(* Title: HOL/Real/real_arith.ML
+ ID: $Id$
+ Author: Tobias Nipkow, TU Muenchen
+ Copyright 1999 TU Muenchen
+
+Instantiation of the generic linear arithmetic package for type real.
+*)
+
+local
+
+(* reduce contradictory <= to False *)
+val simps = [order_less_irrefl, zero_eq_numeral_0, one_eq_numeral_1,
+ add_real_number_of, minus_real_number_of, diff_real_number_of,
+ mult_real_number_of, eq_real_number_of, less_real_number_of,
+ le_real_number_of_eq_not_less, real_diff_def,
+ real_minus_add_distrib, real_minus_minus];
+
+val add_rules =
+ map rename_numerals
+ [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 mono_ss = simpset() addsimps
+ [real_add_le_mono,real_add_less_mono,
+ real_add_less_le_mono,real_add_le_less_mono];
+
+val add_mono_thms_real =
+ map (fn s => prove_goal (the_context ()) s
+ (fn prems => [cut_facts_tac prems 1, asm_simp_tac mono_ss 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)",
+ "(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)",
+ "(i < j) & (k < l) ==> i + k < j + (l::real)"];
+
+val real_arith_simproc_pats =
+ map (fn s => Thm.read_cterm (Theory.sign_of (the_context ())) (s, HOLogic.boolT))
+ ["(m::real) < n","(m::real) <= n", "(m::real) = n"];
+
+in
+
+val fast_real_arith_simproc = mk_simproc
+ "fast_real_arith" real_arith_simproc_pats Fast_Arith.lin_arith_prover;
+
+val real_arith_setup =
+ [Fast_Arith.map_data (fn {add_mono_thms, lessD, simpset} =>
+ {add_mono_thms = add_mono_thms @ add_mono_thms_real,
+ lessD = lessD, (*We don't change LA_Data_Ref.lessD because the real ordering is dense!*)
+ simpset = simpset addsimps simps@add_rules
+ addsimprocs simprocs
+ addcongs [if_weak_cong]}),
+ arith_discrete ("RealDef.real",false),
+ Simplifier.change_simpset_of (op addsimprocs) [fast_real_arith_simproc]];
+
+end;
+
+
+(* Some test data [omitting examples that 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);
+qed "";
+
+Goal "!!a::real. [| a <= b; b+b <= c |] ==> a+a <= c";
+by (fast_arith_tac 1);
+qed "";
+
+Goal "!!a::real. [| a+b <= i+j; a<=b; i<=j |] ==> a+a <= j+j";
+by (fast_arith_tac 1);
+qed "";
+
+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);
+qed "";
+
+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);
+qed "";
+
+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);
+qed "";
+
+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);
+qed "";
+
+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);
+qed "";
+
+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);
+qed "";
+*)