src/HOL/Real/RealArith0.thy

 theory RealArith0 = RealBin
 files "real_arith0.ML":
+
+(*FIXME: move results further down to Ring_and_Field*)
 
 setup real_arith_setup
 
 subsection{*Facts that need the Arithmetic Decision Procedure*}
 

 
 lemma real_mult_div_cancel_disj:
      "(k*m) / (k*n) = (if k = (0::real) then 0 else m/n)"
   by (rule Ring_and_Field.mult_divide_cancel_eq_if) 
 
-
+subsection{* Simprules combining x+y and 0: ARE THEY NEEDED?*}
+
+lemma real_add_minus_iff [simp]: "(x + - a = (0::real)) = (x=a)"
+by arith
+
+lemma real_add_eq_0_iff [iff]: "(x+y = (0::real)) = (y = -x)"
+by auto
+
+lemma real_add_less_0_iff [iff]: "(x+y < (0::real)) = (y < -x)"
+by auto
+
+lemma real_0_less_add_iff [iff]: "((0::real) < x+y) = (-x < y)"
+by auto
+
+lemma real_add_le_0_iff [iff]: "(x+y \<le> (0::real)) = (y \<le> -x)"
+by auto
+
+lemma real_0_le_add_iff [iff]: "((0::real) \<le> x+y) = (-x \<le> y)"
+by auto
+
+
+(** Simprules combining x-y and 0; see also real_less_iff_diff_less_0, etc.,
+    in RealBin
+**)
+
+lemma real_0_less_diff_iff [iff]: "((0::real) < x-y) = (y < x)"
+by auto
+
+lemma real_0_le_diff_iff [iff]: "((0::real) \<le> x-y) = (y \<le> x)"
+by auto
+
+(*
+FIXME: we should have this, as for type int, but many proofs would break.
+It replaces x+-y by x-y.
+Addsimps [symmetric real_diff_def]
+*)
+
+
+(*FIXME: move most of these to Ring_and_Field*)
+
+subsection{*Simplification of Inequalities Involving Literal Divisors*}
+
+lemma pos_real_le_divide_eq: "0<z ==> ((x::real) \<le> y/z) = (x*z \<le> y)"
+apply (subgoal_tac " (x*z \<le> y) = (x*z \<le> (y/z) *z) ")
+ prefer 2 apply (simp add: real_divide_def real_mult_assoc) 
+apply (erule ssubst)
+apply (subst real_mult_le_cancel2, simp)
+done
+
+lemma neg_real_le_divide_eq: "z<0 ==> ((x::real) \<le> y/z) = (y \<le> x*z)"
+apply (subgoal_tac " (y \<le> x*z) = ((y/z) *z \<le> x*z) ")
+ prefer 2 apply (simp add: real_divide_def real_mult_assoc)
+apply (erule ssubst)
+apply (subst real_mult_le_cancel2, simp)
+done
+
+lemma real_le_divide_eq:
+  "((x::real) \<le> y/z) = (if 0<z then x*z \<le> y
+                        else if z<0 then y \<le> x*z
+                        else x\<le>0)"
+apply (case_tac "z=0", simp) 
+apply (simp add: pos_real_le_divide_eq neg_real_le_divide_eq) 
+done
+
+declare real_le_divide_eq [of _ _ "number_of w", standard, simp]
+
+lemma pos_real_divide_le_eq: "0<z ==> (y/z \<le> (x::real)) = (y \<le> x*z)"
+apply (subgoal_tac " (y \<le> x*z) = ((y/z) *z \<le> x*z) ")
+ prefer 2 apply (simp add: real_divide_def real_mult_assoc)
+apply (erule ssubst)
+apply (subst real_mult_le_cancel2, simp)
+done
+
+lemma neg_real_divide_le_eq: "z<0 ==> (y/z \<le> (x::real)) = (x*z \<le> y)"
+apply (subgoal_tac " (x*z \<le> y) = (x*z \<le> (y/z) *z) ")
+ prefer 2 apply (simp add: real_divide_def real_mult_assoc)
+apply (erule ssubst)
+apply (subst real_mult_le_cancel2, simp)
+done
+
+
+lemma real_divide_le_eq:
+  "(y/z \<le> (x::real)) = (if 0<z then y \<le> x*z
+                        else if z<0 then x*z \<le> y
+                        else 0 \<le> x)"
+apply (case_tac "z=0", simp) 
+apply (simp add: pos_real_divide_le_eq neg_real_divide_le_eq) 
+done
+
+declare real_divide_le_eq [of _ "number_of w", standard, simp]
+
+
+lemma pos_real_less_divide_eq: "0<z ==> ((x::real) < y/z) = (x*z < y)"
+apply (subgoal_tac " (x*z < y) = (x*z < (y/z) *z) ")
+ prefer 2 apply (simp add: real_divide_def real_mult_assoc)
+apply (erule ssubst)
+apply (subst real_mult_less_cancel2, simp)
+done
+
+lemma neg_real_less_divide_eq: "z<0 ==> ((x::real) < y/z) = (y < x*z)"
+apply (subgoal_tac " (y < x*z) = ((y/z) *z < x*z) ")
+ prefer 2 apply (simp add: real_divide_def real_mult_assoc)
+apply (erule ssubst)
+apply (subst real_mult_less_cancel2, simp)
+done
+
+lemma real_less_divide_eq:
+  "((x::real) < y/z) = (if 0<z then x*z < y
+                        else if z<0 then y < x*z
+                        else x<0)"
+apply (case_tac "z=0", simp) 
+apply (simp add: pos_real_less_divide_eq neg_real_less_divide_eq) 
+done
+
+declare real_less_divide_eq [of _ _ "number_of w", standard, simp]
+
+lemma pos_real_divide_less_eq: "0<z ==> (y/z < (x::real)) = (y < x*z)"
+apply (subgoal_tac " (y < x*z) = ((y/z) *z < x*z) ")
+ prefer 2 apply (simp add: real_divide_def real_mult_assoc)
+apply (erule ssubst)
+apply (subst real_mult_less_cancel2, simp)
+done
+
+lemma neg_real_divide_less_eq: "z<0 ==> (y/z < (x::real)) = (x*z < y)"
+apply (subgoal_tac " (x*z < y) = (x*z < (y/z) *z) ")
+ prefer 2 apply (simp add: real_divide_def real_mult_assoc)
+apply (erule ssubst)
+apply (subst real_mult_less_cancel2, simp)
+done
+
+lemma real_divide_less_eq:
+  "(y/z < (x::real)) = (if 0<z then y < x*z
+                        else if z<0 then x*z < y
+                        else 0 < x)"
+apply (case_tac "z=0", simp) 
+apply (simp add: pos_real_divide_less_eq neg_real_divide_less_eq) 
+done
+
+declare real_divide_less_eq [of _ "number_of w", standard, simp]
+
+lemma nonzero_real_eq_divide_eq: "z\<noteq>0 ==> ((x::real) = y/z) = (x*z = y)"
+apply (subgoal_tac " (x*z = y) = (x*z = (y/z) *z) ")
+ prefer 2 apply (simp add: real_divide_def real_mult_assoc)
+apply (erule ssubst)
+apply (subst real_mult_eq_cancel2, simp)
+done
+
+lemma real_eq_divide_eq:
+  "((x::real) = y/z) = (if z\<noteq>0 then x*z = y else x=0)"
+by (simp add: nonzero_real_eq_divide_eq) 
+
+declare real_eq_divide_eq [of _ _ "number_of w", standard, simp]
+
+lemma nonzero_real_divide_eq_eq: "z\<noteq>0 ==> (y/z = (x::real)) = (y = x*z)"
+apply (subgoal_tac " (y = x*z) = ((y/z) *z = x*z) ")
+ prefer 2 apply (simp add: real_divide_def real_mult_assoc)
+apply (erule ssubst)
+apply (subst real_mult_eq_cancel2, simp)
+done
+
+lemma real_divide_eq_eq:
+  "(y/z = (x::real)) = (if z\<noteq>0 then y = x*z else x=0)"
+by (simp add: nonzero_real_divide_eq_eq) 
+
+declare real_divide_eq_eq [of _ "number_of w", standard, simp]
+
+lemma real_divide_eq_cancel2: "(m/k = n/k) = (k = 0 | m = (n::real))"
+apply (case_tac "k=0", simp) 
+apply (simp add:divide_inverse) 
+done
+
+lemma real_divide_eq_cancel1: "(k/m = k/n) = (k = 0 | m = (n::real))" 
+by (force simp add: real_divide_eq_eq real_eq_divide_eq)
+
+lemma real_inverse_less_iff:
+     "[| 0 < r; 0 < x|] ==> (inverse x < inverse (r::real)) = (r < x)"
+by (rule Ring_and_Field.inverse_less_iff_less)
+
+lemma real_inverse_le_iff:
+     "[| 0 < r; 0 < x|] ==> (inverse x \<le> inverse r) = (r \<le> (x::real))"
+by (rule Ring_and_Field.inverse_le_iff_le)
+
+
+(** Division by 1, -1 **)
+
+lemma real_divide_1: "(x::real)/1 = x"
+by (simp add: real_divide_def)
+
+lemma real_divide_minus1 [simp]: "x/-1 = -(x::real)"
+by simp
+
+lemma real_minus1_divide [simp]: "-1/(x::real) = - (1/x)"
+by (simp add: real_divide_def real_minus_inverse)
+
+lemma real_lbound_gt_zero:
+     "[| (0::real) < d1; 0 < d2 |] ==> \<exists>e. 0 < e & e < d1 & e < d2"
+apply (rule_tac x = " (min d1 d2) /2" in exI)
+apply (simp add: min_def)
+done
+
+(*** Density of the Reals ***)
+
+lemma real_less_half_sum: "x < y ==> x < (x+y) / (2::real)"
+by auto
+
+lemma real_gt_half_sum: "x < y ==> (x+y)/(2::real) < y"
+by auto
+
+lemma real_dense: "x < y ==> \<exists>r::real. x < r & r < y"
+by (blast intro!: real_less_half_sum real_gt_half_sum)
 
 end