--- a/src/HOL/Hyperreal/HyperArith.thy Wed Dec 24 08:54:30 2003 +0100
+++ b/src/HOL/Hyperreal/HyperArith.thy Thu Dec 25 22:48:32 2003 +0100
@@ -4,6 +4,16 @@
setup hypreal_arith_setup
+text{*Used once in NSA*}
+lemma hypreal_add_minus_eq_minus: "x + y = (0::hypreal) ==> x = -y"
+apply arith
+done
+
+ML
+{*
+val hypreal_add_minus_eq_minus = thm "hypreal_add_minus_eq_minus";
+*}
+
subsubsection{*Division By @{term 1} and @{term "-1"}*}
lemma hypreal_divide_minus1 [simp]: "x/-1 = -(x::hypreal)"
--- a/src/HOL/Hyperreal/HyperBin.ML Wed Dec 24 08:54:30 2003 +0100
+++ b/src/HOL/Hyperreal/HyperBin.ML Thu Dec 25 22:48:32 2003 +0100
@@ -148,21 +148,6 @@
(**** Simprocs for numeric literals ****)
-(** Combining of literal coefficients in sums of products **)
-
-Goal "(x < y) = (x-y < (0::hypreal))";
-by (simp_tac (simpset() addsimps [hypreal_diff_less_eq]) 1);
-qed "hypreal_less_iff_diff_less_0";
-
-Goal "(x = y) = (x-y = (0::hypreal))";
-by (simp_tac (simpset() addsimps [hypreal_diff_eq_eq]) 1);
-qed "hypreal_eq_iff_diff_eq_0";
-
-Goal "(x <= y) = (x-y <= (0::hypreal))";
-by (simp_tac (simpset() addsimps [hypreal_diff_le_eq]) 1);
-qed "hypreal_le_iff_diff_le_0";
-
-
(** For combine_numerals **)
Goal "i*u + (j*u + k) = (i+j)*u + (k::hypreal)";
@@ -173,12 +158,12 @@
(** For cancel_numerals **)
val rel_iff_rel_0_rls =
- map (inst "y" "?u+?v")
- [hypreal_less_iff_diff_less_0, hypreal_eq_iff_diff_eq_0,
- hypreal_le_iff_diff_le_0] @
- map (inst "y" "n")
- [hypreal_less_iff_diff_less_0, hypreal_eq_iff_diff_eq_0,
- hypreal_le_iff_diff_le_0];
+ map (inst "b" "?u+?v")
+ [less_iff_diff_less_0, eq_iff_diff_eq_0,
+ le_iff_diff_le_0] @
+ map (inst "b" "n")
+ [less_iff_diff_less_0, eq_iff_diff_eq_0,
+ le_iff_diff_le_0];
Goal "!!i::hypreal. (i*u + m = j*u + n) = ((i-j)*u + m = n)";
by (asm_simp_tac
@@ -582,9 +567,9 @@
Addsimprocs [Hyperreal_Times_Assoc.conv];
(*Simplification of x-y < 0, etc.*)
-AddIffs [hypreal_less_iff_diff_less_0 RS sym];
-AddIffs [hypreal_eq_iff_diff_eq_0 RS sym];
-AddIffs [hypreal_le_iff_diff_le_0 RS sym];
+AddIffs [less_iff_diff_less_0 RS sym];
+AddIffs [eq_iff_diff_eq_0 RS sym];
+AddIffs [le_iff_diff_le_0 RS sym];
(** number_of related to hypreal_of_real **)
--- a/src/HOL/Hyperreal/HyperDef.thy Wed Dec 24 08:54:30 2003 +0100
+++ b/src/HOL/Hyperreal/HyperDef.thy Thu Dec 25 22:48:32 2003 +0100
@@ -84,12 +84,14 @@
hypreal_le_def:
"P <= (Q::hypreal) == ~(Q < P)"
-(*------------------------------------------------------------------------
- Proof that the set of naturals is not finite
- ------------------------------------------------------------------------*)
+ hrabs_def: "abs (r::hypreal) == (if 0 \<le> r then r else -r)"
+
+
+subsection{*The Set of Naturals is not Finite*}
(*** based on James' proof that the set of naturals is not finite ***)
-lemma finite_exhausts [rule_format (no_asm)]: "finite (A::nat set) --> (\<exists>n. \<forall>m. Suc (n + m) \<notin> A)"
+lemma finite_exhausts [rule_format]:
+ "finite (A::nat set) --> (\<exists>n. \<forall>m. Suc (n + m) \<notin> A)"
apply (rule impI)
apply (erule_tac F = A in finite_induct)
apply (blast, erule exE)
@@ -98,16 +100,18 @@
apply (auto simp add: add_ac)
done
-lemma finite_not_covers [rule_format (no_asm)]: "finite (A :: nat set) --> (\<exists>n. n \<notin>A)"
+lemma finite_not_covers [rule_format (no_asm)]:
+ "finite (A :: nat set) --> (\<exists>n. n \<notin>A)"
by (rule impI, drule finite_exhausts, blast)
lemma not_finite_nat: "~ finite(UNIV:: nat set)"
by (fast dest!: finite_exhausts)
-(*------------------------------------------------------------------------
- Existence of free ultrafilter over the naturals and proof of various
- properties of the FreeUltrafilterNat- an arbitrary free ultrafilter
- ------------------------------------------------------------------------*)
+
+subsection{*Existence of Free Ultrafilter over the Naturals*}
+
+text{*Also, proof of various properties of @{term FreeUltrafilterNat}:
+an arbitrary free ultrafilter*}
lemma FreeUltrafilterNat_Ex: "\<exists>U. U: FreeUltrafilter (UNIV::nat set)"
by (rule not_finite_nat [THEN FreeUltrafilter_Ex])
@@ -137,33 +141,39 @@
Filter_empty_not_mem)
done
-lemma FreeUltrafilterNat_Int: "[| X: FreeUltrafilterNat; Y: FreeUltrafilterNat |]
+lemma FreeUltrafilterNat_Int:
+ "[| X: FreeUltrafilterNat; Y: FreeUltrafilterNat |]
==> X Int Y \<in> FreeUltrafilterNat"
apply (cut_tac FreeUltrafilterNat_mem)
apply (blast dest: FreeUltrafilter_Ultrafilter Ultrafilter_Filter mem_FiltersetD1)
done
-lemma FreeUltrafilterNat_subset: "[| X: FreeUltrafilterNat; X <= Y |]
+lemma FreeUltrafilterNat_subset:
+ "[| X: FreeUltrafilterNat; X <= Y |]
==> Y \<in> FreeUltrafilterNat"
apply (cut_tac FreeUltrafilterNat_mem)
apply (blast dest: FreeUltrafilter_Ultrafilter Ultrafilter_Filter mem_FiltersetD2)
done
-lemma FreeUltrafilterNat_Compl: "X: FreeUltrafilterNat ==> -X \<notin> FreeUltrafilterNat"
+lemma FreeUltrafilterNat_Compl:
+ "X: FreeUltrafilterNat ==> -X \<notin> FreeUltrafilterNat"
apply safe
apply (drule FreeUltrafilterNat_Int, assumption, auto)
done
-lemma FreeUltrafilterNat_Compl_mem: "X\<notin> FreeUltrafilterNat ==> -X \<in> FreeUltrafilterNat"
+lemma FreeUltrafilterNat_Compl_mem:
+ "X\<notin> FreeUltrafilterNat ==> -X \<in> FreeUltrafilterNat"
apply (cut_tac FreeUltrafilterNat_mem [THEN FreeUltrafilter_iff [THEN iffD1]])
apply (safe, drule_tac x = X in bspec)
apply (auto simp add: UNIV_diff_Compl)
done
-lemma FreeUltrafilterNat_Compl_iff1: "(X \<notin> FreeUltrafilterNat) = (-X: FreeUltrafilterNat)"
+lemma FreeUltrafilterNat_Compl_iff1:
+ "(X \<notin> FreeUltrafilterNat) = (-X: FreeUltrafilterNat)"
by (blast dest: FreeUltrafilterNat_Compl FreeUltrafilterNat_Compl_mem)
-lemma FreeUltrafilterNat_Compl_iff2: "(X: FreeUltrafilterNat) = (-X \<notin> FreeUltrafilterNat)"
+lemma FreeUltrafilterNat_Compl_iff2:
+ "(X: FreeUltrafilterNat) = (-X \<notin> FreeUltrafilterNat)"
by (auto simp add: FreeUltrafilterNat_Compl_iff1 [symmetric])
lemma FreeUltrafilterNat_UNIV [simp]: "(UNIV::nat set) \<in> FreeUltrafilterNat"
@@ -172,7 +182,8 @@
lemma FreeUltrafilterNat_Nat_set [simp]: "UNIV \<in> FreeUltrafilterNat"
by auto
-lemma FreeUltrafilterNat_Nat_set_refl [intro]: "{n. P(n) = P(n)} \<in> FreeUltrafilterNat"
+lemma FreeUltrafilterNat_Nat_set_refl [intro]:
+ "{n. P(n) = P(n)} \<in> FreeUltrafilterNat"
by simp
lemma FreeUltrafilterNat_P: "{n::nat. P} \<in> FreeUltrafilterNat ==> P"
@@ -184,9 +195,8 @@
lemma FreeUltrafilterNat_all: "\<forall>n. P(n) ==> {n. P(n)} \<in> FreeUltrafilterNat"
by (auto intro: FreeUltrafilterNat_Nat_set)
-(*-------------------------------------------------------
- Define and use Ultrafilter tactics
- -------------------------------------------------------*)
+
+text{*Define and use Ultrafilter tactics*}
use "fuf.ML"
method_setup fuf = {*
@@ -204,21 +214,18 @@
"ultrafilter tactic"
-(*-------------------------------------------------------
- Now prove one further property of our free ultrafilter
- -------------------------------------------------------*)
-lemma FreeUltrafilterNat_Un: "X Un Y: FreeUltrafilterNat
+text{*One further property of our free ultrafilter*}
+lemma FreeUltrafilterNat_Un:
+ "X Un Y: FreeUltrafilterNat
==> X: FreeUltrafilterNat | Y: FreeUltrafilterNat"
apply auto
apply ultra
done
-(*-------------------------------------------------------
- Properties of hyprel
- -------------------------------------------------------*)
-(** Proving that hyprel is an equivalence relation **)
-(** Natural deduction for hyprel **)
+subsection{*Properties of @{term hyprel}*}
+
+text{*Proving that @{term hyprel} is an equivalence relation*}
lemma hyprel_iff: "((X,Y): hyprel) = ({n. X n = Y n}: FreeUltrafilterNat)"
by (unfold hyprel_def, fast)
@@ -281,9 +288,8 @@
by (cut_tac x = x in Rep_hypreal, auto)
-(*------------------------------------------------------------------------
- hypreal_of_real: the injection from real to hypreal
- ------------------------------------------------------------------------*)
+subsection{*@{term hypreal_of_real}:
+ the Injection from @{typ real} to @{typ hypreal}*}
lemma inj_hypreal_of_real: "inj(hypreal_of_real)"
apply (rule inj_onI)
@@ -302,7 +308,61 @@
apply (force simp add: Rep_hypreal_inverse)
done
-(**** hypreal_minus: additive inverse on hypreal ****)
+
+subsection{*Hyperreal Addition*}
+
+lemma hypreal_add_congruent2:
+ "congruent2 hyprel (%X Y. hyprel``{%n. X n + Y n})"
+apply (unfold congruent2_def, auto, ultra)
+done
+
+lemma hypreal_add:
+ "Abs_hypreal(hyprel``{%n. X n}) + Abs_hypreal(hyprel``{%n. Y n}) =
+ Abs_hypreal(hyprel``{%n. X n + Y n})"
+apply (unfold hypreal_add_def)
+apply (simp add: UN_equiv_class2 [OF equiv_hyprel hypreal_add_congruent2])
+done
+
+lemma hypreal_add_commute: "(z::hypreal) + w = w + z"
+apply (rule_tac z = z in eq_Abs_hypreal)
+apply (rule_tac z = w in eq_Abs_hypreal)
+apply (simp add: real_add_ac hypreal_add)
+done
+
+lemma hypreal_add_assoc: "((z1::hypreal) + z2) + z3 = z1 + (z2 + z3)"
+apply (rule_tac z = z1 in eq_Abs_hypreal)
+apply (rule_tac z = z2 in eq_Abs_hypreal)
+apply (rule_tac z = z3 in eq_Abs_hypreal)
+apply (simp add: hypreal_add real_add_assoc)
+done
+
+(*For AC rewriting*)
+lemma hypreal_add_left_commute: "(x::hypreal)+(y+z)=y+(x+z)"
+ apply (rule mk_left_commute [of "op +"])
+ apply (rule hypreal_add_assoc)
+ apply (rule hypreal_add_commute)
+ done
+
+(* hypreal addition is an AC operator *)
+lemmas hypreal_add_ac =
+ hypreal_add_assoc hypreal_add_commute hypreal_add_left_commute
+
+lemma hypreal_add_zero_left [simp]: "(0::hypreal) + z = z"
+apply (unfold hypreal_zero_def)
+apply (rule_tac z = z in eq_Abs_hypreal)
+apply (simp add: hypreal_add)
+done
+
+instance hypreal :: plus_ac0
+ by (intro_classes,
+ (assumption |
+ rule hypreal_add_commute hypreal_add_assoc hypreal_add_zero_left)+)
+
+lemma hypreal_add_zero_right [simp]: "z + (0::hypreal) = z"
+by (simp add: hypreal_add_zero_left hypreal_add_commute)
+
+
+subsection{*Additive inverse on @{typ hypreal}*}
lemma hypreal_minus_congruent:
"congruent hyprel (%X. hyprel``{%n. - (X n)})"
@@ -337,59 +397,12 @@
apply (auto simp add: hypreal_zero_def hypreal_minus)
done
-
-(**** hyperreal addition: hypreal_add ****)
-
-lemma hypreal_add_congruent2:
- "congruent2 hyprel (%X Y. hyprel``{%n. X n + Y n})"
-apply (unfold congruent2_def, auto, ultra)
-done
-
-lemma hypreal_add:
- "Abs_hypreal(hyprel``{%n. X n}) + Abs_hypreal(hyprel``{%n. Y n}) =
- Abs_hypreal(hyprel``{%n. X n + Y n})"
-apply (unfold hypreal_add_def)
-apply (simp add: UN_equiv_class2 [OF equiv_hyprel hypreal_add_congruent2])
-done
-
-lemma hypreal_diff: "Abs_hypreal(hyprel``{%n. X n}) - Abs_hypreal(hyprel``{%n. Y n}) =
+lemma hypreal_diff:
+ "Abs_hypreal(hyprel``{%n. X n}) - Abs_hypreal(hyprel``{%n. Y n}) =
Abs_hypreal(hyprel``{%n. X n - Y n})"
apply (simp add: hypreal_diff_def hypreal_minus hypreal_add)
done
-lemma hypreal_add_commute: "(z::hypreal) + w = w + z"
-apply (rule_tac z = z in eq_Abs_hypreal)
-apply (rule_tac z = w in eq_Abs_hypreal)
-apply (simp add: real_add_ac hypreal_add)
-done
-
-lemma hypreal_add_assoc: "((z1::hypreal) + z2) + z3 = z1 + (z2 + z3)"
-apply (rule_tac z = z1 in eq_Abs_hypreal)
-apply (rule_tac z = z2 in eq_Abs_hypreal)
-apply (rule_tac z = z3 in eq_Abs_hypreal)
-apply (simp add: hypreal_add real_add_assoc)
-done
-
-(*For AC rewriting*)
-lemma hypreal_add_left_commute: "(x::hypreal)+(y+z)=y+(x+z)"
- apply (rule mk_left_commute [of "op +"])
- apply (rule hypreal_add_assoc)
- apply (rule hypreal_add_commute)
- done
-
-(* hypreal addition is an AC operator *)
-lemmas hypreal_add_ac =
- hypreal_add_assoc hypreal_add_commute hypreal_add_left_commute
-
-lemma hypreal_add_zero_left [simp]: "(0::hypreal) + z = z"
-apply (unfold hypreal_zero_def)
-apply (rule_tac z = z in eq_Abs_hypreal)
-apply (simp add: hypreal_add)
-done
-
-lemma hypreal_add_zero_right [simp]: "z + (0::hypreal) = z"
-by (simp add: hypreal_add_zero_left hypreal_add_commute)
-
lemma hypreal_add_minus [simp]: "z + -z = (0::hypreal)"
apply (unfold hypreal_zero_def)
apply (rule_tac z = z in eq_Abs_hypreal)
@@ -399,42 +412,6 @@
lemma hypreal_add_minus_left [simp]: "-z + z = (0::hypreal)"
by (simp add: hypreal_add_commute hypreal_add_minus)
-lemma hypreal_minus_ex: "\<exists>y. (x::hypreal) + y = 0"
-by (fast intro: hypreal_add_minus)
-
-lemma hypreal_minus_ex1: "EX! y. (x::hypreal) + y = 0"
-apply (auto intro: hypreal_add_minus)
-apply (drule_tac f = "%x. ya+x" in arg_cong)
-apply (simp add: hypreal_add_assoc [symmetric])
-apply (simp add: hypreal_add_commute)
-done
-
-lemma hypreal_minus_left_ex1: "EX! y. y + (x::hypreal) = 0"
-apply (auto intro: hypreal_add_minus_left)
-apply (drule_tac f = "%x. x+ya" in arg_cong)
-apply (simp add: hypreal_add_assoc)
-apply (simp add: hypreal_add_commute)
-done
-
-lemma hypreal_add_minus_eq_minus: "x + y = (0::hypreal) ==> x = -y"
-apply (cut_tac z = y in hypreal_add_minus_left)
-apply (rule_tac x1 = y in hypreal_minus_left_ex1 [THEN ex1E], blast)
-done
-
-lemma hypreal_as_add_inverse_ex: "\<exists>y::hypreal. x = -y"
-apply (cut_tac x = x in hypreal_minus_ex)
-apply (erule exE, drule hypreal_add_minus_eq_minus, fast)
-done
-
-lemma hypreal_minus_add_distrib [simp]: "-(x + (y::hypreal)) = -x + -y"
-apply (rule_tac z = x in eq_Abs_hypreal)
-apply (rule_tac z = y in eq_Abs_hypreal)
-apply (auto simp add: hypreal_minus hypreal_add real_minus_add_distrib)
-done
-
-lemma hypreal_minus_distrib1: "-(y + -(x::hypreal)) = x + -y"
-by (simp add: hypreal_add_commute)
-
lemma hypreal_add_left_cancel: "((x::hypreal) + y = x + z) = (y = z)"
apply safe
apply (drule_tac f = "%t.-x + t" in arg_cong)
@@ -450,7 +427,8 @@
lemma hypreal_minus_add_cancelA [simp]: "(-z) + (z + w) = (w::hypreal)"
by (simp add: hypreal_add_assoc [symmetric])
-(**** hyperreal multiplication: hypreal_mult ****)
+
+subsection{*Hyperreal Multiplication*}
lemma hypreal_mult_congruent2:
"congruent2 hyprel (%X Y. hyprel``{%n. X n * Y n})"
@@ -530,30 +508,30 @@
lemma hypreal_minus_mult_commute: "(-x) * y = (x::hypreal) * -y"
by auto
-(*-----------------------------------------------------------------------------
- A few more theorems
- ----------------------------------------------------------------------------*)
-lemma hypreal_add_assoc_cong: "(z::hypreal) + v = z' + v' ==> z + (v + w) = z' + (v' + w)"
-by (simp add: hypreal_add_assoc [symmetric])
+subsection{*A few more theorems *}
-lemma hypreal_add_mult_distrib: "((z1::hypreal) + z2) * w = (z1 * w) + (z2 * w)"
+lemma hypreal_add_mult_distrib:
+ "((z1::hypreal) + z2) * w = (z1 * w) + (z2 * w)"
apply (rule_tac z = z1 in eq_Abs_hypreal)
apply (rule_tac z = z2 in eq_Abs_hypreal)
apply (rule_tac z = w in eq_Abs_hypreal)
apply (simp add: hypreal_mult hypreal_add real_add_mult_distrib)
done
-lemma hypreal_add_mult_distrib2: "(w::hypreal) * (z1 + z2) = (w * z1) + (w * z2)"
+lemma hypreal_add_mult_distrib2:
+ "(w::hypreal) * (z1 + z2) = (w * z1) + (w * z2)"
by (simp add: hypreal_mult_commute [of w] hypreal_add_mult_distrib)
-lemma hypreal_diff_mult_distrib: "((z1::hypreal) - z2) * w = (z1 * w) - (z2 * w)"
+lemma hypreal_diff_mult_distrib:
+ "((z1::hypreal) - z2) * w = (z1 * w) - (z2 * w)"
apply (unfold hypreal_diff_def)
apply (simp add: hypreal_add_mult_distrib)
done
-lemma hypreal_diff_mult_distrib2: "(w::hypreal) * (z1 - z2) = (w * z1) - (w * z2)"
+lemma hypreal_diff_mult_distrib2:
+ "(w::hypreal) * (z1 - z2) = (w * z1) - (w * z2)"
by (simp add: hypreal_mult_commute [of w] hypreal_diff_mult_distrib)
(*** one and zero are distinct ***)
@@ -563,7 +541,7 @@
done
-(**** multiplicative inverse on hypreal ****)
+subsection{*Multiplicative Inverse on @{typ hypreal} *}
lemma hypreal_inverse_congruent:
"congruent hyprel (%X. hyprel``{%n. if X n = 0 then 0 else inverse(X n)})"
@@ -586,19 +564,15 @@
lemma HYPREAL_DIVISION_BY_ZERO: "a / (0::hypreal) = 0"
by (simp add: hypreal_divide_def HYPREAL_INVERSE_ZERO)
-lemma hypreal_inverse_inverse [simp]: "inverse (inverse (z::hypreal)) = z"
-apply (case_tac "z=0", simp add: HYPREAL_INVERSE_ZERO)
-apply (rule_tac z = z in eq_Abs_hypreal)
-apply (simp add: hypreal_inverse hypreal_zero_def)
-done
-
-lemma hypreal_inverse_1 [simp]: "inverse((1::hypreal)) = (1::hypreal)"
-apply (unfold hypreal_one_def)
-apply (simp add: hypreal_inverse real_zero_not_eq_one [THEN not_sym])
-done
+instance hypreal :: division_by_zero
+proof
+ fix x :: hypreal
+ show "inverse 0 = (0::hypreal)" by (rule HYPREAL_INVERSE_ZERO)
+ show "x/0 = 0" by (rule HYPREAL_DIVISION_BY_ZERO)
+qed
-(*** existence of inverse ***)
+subsection{*Existence of Inverse*}
lemma hypreal_mult_inverse [simp]:
"x \<noteq> 0 ==> x*inverse(x) = (1::hypreal)"
@@ -609,99 +583,33 @@
apply (blast intro!: real_mult_inv_right FreeUltrafilterNat_subset)
done
-lemma hypreal_mult_inverse_left [simp]: "x \<noteq> 0 ==> inverse(x)*x = (1::hypreal)"
+lemma hypreal_mult_inverse_left [simp]:
+ "x \<noteq> 0 ==> inverse(x)*x = (1::hypreal)"
by (simp add: hypreal_mult_inverse hypreal_mult_commute)
-lemma hypreal_mult_left_cancel: "(c::hypreal) \<noteq> 0 ==> (c*a=c*b) = (a=b)"
-apply auto
-apply (drule_tac f = "%x. x*inverse c" in arg_cong)
-apply (simp add: hypreal_mult_inverse hypreal_mult_ac)
-done
-
-lemma hypreal_mult_right_cancel: "(c::hypreal) \<noteq> 0 ==> (a*c=b*c) = (a=b)"
-apply safe
-apply (drule_tac f = "%x. x*inverse c" in arg_cong)
-apply (simp add: hypreal_mult_inverse hypreal_mult_ac)
-done
+
+subsection{*Theorems for Ordering*}
+
+text{*TODO: define @{text "\<le>"} as the primitive concept and quickly
+establish membership in class @{text linorder}. Then proofs could be
+simplified, since properties of @{text "<"} would be generic.*}
-lemma hypreal_inverse_not_zero: "x \<noteq> 0 ==> inverse (x::hypreal) \<noteq> 0"
-apply (unfold hypreal_zero_def)
-apply (rule_tac z = x in eq_Abs_hypreal)
-apply (simp add: hypreal_inverse hypreal_mult)
-done
-
-
-lemma hypreal_mult_not_0: "[| x \<noteq> 0; y \<noteq> 0 |] ==> x * y \<noteq> (0::hypreal)"
-apply safe
-apply (drule_tac f = "%z. inverse x*z" in arg_cong)
-apply (simp add: hypreal_mult_assoc [symmetric])
-done
-
-lemma hypreal_mult_zero_disj: "x*y = (0::hypreal) ==> x = 0 | y = 0"
-by (auto intro: ccontr dest: hypreal_mult_not_0)
-
-lemma hypreal_minus_inverse: "inverse(-x) = -inverse(x::hypreal)"
-apply (case_tac "x=0", simp add: HYPREAL_INVERSE_ZERO)
-apply (rule hypreal_mult_right_cancel [of "-x", THEN iffD1], simp)
-apply (subst hypreal_mult_inverse_left, auto)
+text{*TODO: The following theorem should be used througout the proofs
+ as it probably makes many of them more straightforward.*}
+lemma hypreal_less:
+ "(Abs_hypreal(hyprel``{%n. X n}) < Abs_hypreal(hyprel``{%n. Y n})) =
+ ({n. X n < Y n} \<in> FreeUltrafilterNat)"
+apply (unfold hypreal_less_def)
+apply (auto intro!: lemma_hyprel_refl, ultra)
done
-lemma hypreal_inverse_distrib: "inverse(x*y) = inverse(x)*inverse(y::hypreal)"
-apply (case_tac "x=0", simp add: HYPREAL_INVERSE_ZERO)
-apply (case_tac "y=0", simp add: HYPREAL_INVERSE_ZERO)
-apply (frule_tac y = y in hypreal_mult_not_0, assumption)
-apply (rule_tac c1 = x in hypreal_mult_left_cancel [THEN iffD1])
-apply (auto simp add: hypreal_mult_assoc [symmetric])
-apply (rule_tac c1 = y in hypreal_mult_left_cancel [THEN iffD1])
-apply (auto simp add: hypreal_mult_left_commute)
-apply (simp add: hypreal_mult_assoc [symmetric])
-done
-
-(*------------------------------------------------------------------
- Theorems for ordering
- ------------------------------------------------------------------*)
-
(* prove introduction and elimination rules for hypreal_less *)
-lemma hypreal_less_iff:
- "(P < (Q::hypreal)) = (\<exists>X Y. X \<in> Rep_hypreal(P) &
- Y \<in> Rep_hypreal(Q) &
- {n. X n < Y n} \<in> FreeUltrafilterNat)"
-
-apply (unfold hypreal_less_def, fast)
-done
-
-lemma hypreal_lessI:
- "[| {n. X n < Y n} \<in> FreeUltrafilterNat;
- X \<in> Rep_hypreal(P);
- Y \<in> Rep_hypreal(Q) |] ==> P < (Q::hypreal)"
-apply (unfold hypreal_less_def, fast)
-done
-
-
-lemma hypreal_lessE:
- "!! R1. [| R1 < (R2::hypreal);
- !!X Y. {n. X n < Y n} \<in> FreeUltrafilterNat ==> P;
- !!X. X \<in> Rep_hypreal(R1) ==> P;
- !!Y. Y \<in> Rep_hypreal(R2) ==> P |]
- ==> P"
-
-apply (unfold hypreal_less_def, auto)
-done
-
-lemma hypreal_lessD:
- "R1 < (R2::hypreal) ==> (\<exists>X Y. {n. X n < Y n} \<in> FreeUltrafilterNat &
- X \<in> Rep_hypreal(R1) &
- Y \<in> Rep_hypreal(R2))"
-apply (unfold hypreal_less_def, fast)
-done
-
lemma hypreal_less_not_refl: "~ (R::hypreal) < R"
apply (rule_tac z = R in eq_Abs_hypreal)
apply (auto simp add: hypreal_less_def, ultra)
done
-(*** y < y ==> P ***)
lemmas hypreal_less_irrefl = hypreal_less_not_refl [THEN notE, standard]
declare hypreal_less_irrefl [elim!]
@@ -720,25 +628,10 @@
apply (simp add: hypreal_less_not_refl)
done
-(*-------------------------------------------------------
- TODO: The following theorem should have been proved
- first and then used througout the proofs as it probably
- makes many of them more straightforward.
- -------------------------------------------------------*)
-lemma hypreal_less:
- "(Abs_hypreal(hyprel``{%n. X n}) <
- Abs_hypreal(hyprel``{%n. Y n})) =
- ({n. X n < Y n} \<in> FreeUltrafilterNat)"
-apply (unfold hypreal_less_def)
-apply (auto intro!: lemma_hyprel_refl, ultra)
-done
-(*----------------------------------------------------------------------------
- Trichotomy: the hyperreals are linearly ordered
- ---------------------------------------------------------------------------*)
+subsection{*Trichotomy: the hyperreals are Linearly Ordered*}
lemma lemma_hyprel_0_mem: "\<exists>x. x: hyprel `` {%n. 0}"
-
apply (unfold hyprel_def)
apply (rule_tac x = "%n. 0" in exI, safe)
apply (auto intro!: FreeUltrafilterNat_Nat_set)
@@ -763,9 +656,7 @@
apply (insert hypreal_trichotomy [of x], blast)
done
-(*----------------------------------------------------------------------------
- More properties of <
- ----------------------------------------------------------------------------*)
+subsection{*More properties of Less Than*}
lemma hypreal_less_minus_iff: "((x::hypreal) < y) = (0 < y + -x)"
apply (rule_tac z = x in eq_Abs_hypreal)
@@ -789,24 +680,8 @@
apply (rule_tac x1 = "-x" in hypreal_add_right_cancel [THEN iffD1], auto)
done
-(* 07/00 *)
-lemma hypreal_diff_zero [simp]: "(0::hypreal) - x = -x"
-by (simp add: hypreal_diff_def)
-lemma hypreal_diff_zero_right [simp]: "x - (0::hypreal) = x"
-by (simp add: hypreal_diff_def)
-
-lemma hypreal_diff_self [simp]: "x - x = (0::hypreal)"
-by (simp add: hypreal_diff_def)
-
-lemma hypreal_eq_minus_iff3: "(x = y + z) = (x + -z = (y::hypreal))"
-by (auto simp add: hypreal_add_assoc)
-
-lemma hypreal_not_eq_minus_iff: "(x \<noteq> a) = (x + -a \<noteq> (0::hypreal))"
-by (auto dest: hypreal_eq_minus_iff [THEN iffD2])
-
-
-(*** linearity ***)
+subsection{*Linearity*}
lemma hypreal_linear: "(x::hypreal) < y | x = y | y < x"
apply (subst hypreal_eq_minus_iff2)
@@ -823,10 +698,8 @@
apply (cut_tac x = x and y = y in hypreal_linear, auto)
done
-(*------------------------------------------------------------------------------
- Properties of <=
- ------------------------------------------------------------------------------*)
-(*------ hypreal le iff reals le a.e ------*)
+
+subsection{*Properties of The @{text "\<le>"} Relation*}
lemma hypreal_le:
"(Abs_hypreal(hyprel``{%n. X n}) <=
@@ -837,8 +710,6 @@
apply (ultra+)
done
-(*---------------------------------------------------------*)
-(*---------------------------------------------------------*)
lemma hypreal_leI:
"~(w < z) ==> z <= (w::hypreal)"
apply (unfold hypreal_le_def, assumption)
@@ -894,17 +765,21 @@
apply (fast elim: hypreal_less_irrefl hypreal_less_asym)
done
-lemma not_less_not_eq_hypreal_less: "[| ~ y < x; y \<noteq> x |] ==> x < (y::hypreal)"
-apply (rule not_hypreal_leE)
-apply (fast dest: hypreal_le_imp_less_or_eq)
-done
-
(* Axiom 'order_less_le' of class 'order': *)
lemma hypreal_less_le: "((w::hypreal) < z) = (w <= z & w \<noteq> z)"
apply (simp add: hypreal_le_def hypreal_neq_iff)
apply (blast intro: hypreal_less_asym)
done
+instance hypreal :: order
+ by (intro_classes,
+ (assumption |
+ rule hypreal_le_refl hypreal_le_trans hypreal_le_anti_sym
+ hypreal_less_le)+)
+
+instance hypreal :: linorder
+ by (intro_classes, rule hypreal_le_linear)
+
lemma hypreal_minus_zero_less_iff [simp]: "(0 < -R) = (R < (0::hypreal))"
apply (rule_tac z = R in eq_Abs_hypreal)
apply (auto simp add: hypreal_zero_def hypreal_less hypreal_minus)
@@ -925,9 +800,141 @@
apply (simp add: hypreal_minus_zero_less_iff2)
done
-(*----------------------------------------------------------
- hypreal_of_real preserves field and order properties
- -----------------------------------------------------------*)
+
+lemma hypreal_self_eq_minus_self_zero: "x = -x ==> x = (0::hypreal)"
+apply (rule_tac z = x in eq_Abs_hypreal)
+apply (auto simp add: hypreal_minus hypreal_zero_def, ultra)
+done
+
+lemma hypreal_add_self_zero_cancel [simp]: "(x + x = 0) = (x = (0::hypreal))"
+apply (rule_tac z = x in eq_Abs_hypreal)
+apply (auto simp add: hypreal_add hypreal_zero_def)
+done
+
+lemma hypreal_add_self_zero_cancel2 [simp]:
+ "(x + x + y = y) = (x = (0::hypreal))"
+apply auto
+apply (drule hypreal_eq_minus_iff [THEN iffD1])
+apply (auto simp add: hypreal_add_assoc hypreal_self_eq_minus_self_zero)
+done
+
+lemma hypreal_minus_eq_swap: "(b = -a) = (-b = (a::hypreal))"
+by auto
+
+lemma hypreal_minus_eq_cancel [simp]: "(-b = -a) = (b = (a::hypreal))"
+by (simp add: hypreal_minus_eq_swap)
+
+lemma hypreal_add_less_mono1: "(A::hypreal) < B ==> A + C < B + C"
+apply (rule_tac z = A in eq_Abs_hypreal)
+apply (rule_tac z = B in eq_Abs_hypreal)
+apply (rule_tac z = C in eq_Abs_hypreal)
+apply (auto intro!: exI simp add: hypreal_less_def hypreal_add, ultra)
+done
+
+lemma hypreal_mult_order: "[| 0 < x; 0 < y |] ==> (0::hypreal) < x * y"
+apply (unfold hypreal_zero_def)
+apply (rule_tac z = x in eq_Abs_hypreal)
+apply (rule_tac z = y in eq_Abs_hypreal)
+apply (auto intro!: exI simp add: hypreal_less_def hypreal_mult, ultra)
+apply (auto intro: real_mult_order)
+done
+
+lemma hypreal_add_left_le_mono1: "(q1::hypreal) \<le> q2 ==> x + q1 \<le> x + q2"
+apply (drule order_le_imp_less_or_eq)
+apply (auto intro: order_less_imp_le hypreal_add_less_mono1 simp add: hypreal_add_commute)
+done
+
+lemma hypreal_mult_less_mono1: "[| (0::hypreal) < z; x < y |] ==> x*z < y*z"
+apply (rotate_tac 1)
+apply (drule hypreal_less_minus_iff [THEN iffD1])
+apply (rule hypreal_less_minus_iff [THEN iffD2])
+apply (drule hypreal_mult_order, assumption)
+apply (simp add: hypreal_add_mult_distrib2 hypreal_mult_commute)
+done
+
+lemma hypreal_mult_less_mono2: "[| (0::hypreal)<z; x<y |] ==> z*x<z*y"
+apply (simp (no_asm_simp) add: hypreal_mult_commute hypreal_mult_less_mono1)
+done
+
+subsection{*The Hyperreals Form an Ordered Field*}
+
+instance hypreal :: inverse ..
+
+instance hypreal :: ordered_field
+proof
+ fix x y z :: hypreal
+ show "(x + y) + z = x + (y + z)" by (rule hypreal_add_assoc)
+ show "x + y = y + x" by (rule hypreal_add_commute)
+ show "0 + x = x" by simp
+ show "- x + x = 0" by simp
+ show "x - y = x + (-y)" by (simp add: hypreal_diff_def)
+ show "(x * y) * z = x * (y * z)" by (rule hypreal_mult_assoc)
+ show "x * y = y * x" by (rule hypreal_mult_commute)
+ show "1 * x = x" by simp
+ show "(x + y) * z = x * z + y * z" by (simp add: hypreal_add_mult_distrib)
+ show "0 \<noteq> (1::hypreal)" by (rule hypreal_zero_not_eq_one)
+ show "x \<le> y ==> z + x \<le> z + y" by (rule hypreal_add_left_le_mono1)
+ show "x < y ==> 0 < z ==> z * x < z * y" by (simp add: hypreal_mult_less_mono2)
+ show "\<bar>x\<bar> = (if x < 0 then -x else x)"
+ by (auto dest: order_le_less_trans simp add: hrabs_def linorder_not_le)
+ show "x \<noteq> 0 ==> inverse x * x = 1" by simp
+ show "y \<noteq> 0 ==> x / y = x * inverse y" by (simp add: hypreal_divide_def)
+qed
+
+lemma hypreal_minus_add_distrib [simp]: "-(x + (y::hypreal)) = -x + -y"
+ by (rule Ring_and_Field.minus_add_distrib)
+
+(*Used ONCE: in NSA.ML*)
+lemma hypreal_minus_distrib1: "-(y + -(x::hypreal)) = x + -y"
+by (simp add: hypreal_add_commute)
+
+(*Used ONCE: in Lim.ML*)
+lemma hypreal_eq_minus_iff3: "(x = y + z) = (x + -z = (y::hypreal))"
+by (auto simp add: hypreal_add_assoc)
+
+lemma hypreal_not_eq_minus_iff: "(x \<noteq> a) = (x + -a \<noteq> (0::hypreal))"
+by (auto dest: hypreal_eq_minus_iff [THEN iffD2])
+
+lemma hypreal_mult_left_cancel: "(c::hypreal) \<noteq> 0 ==> (c*a=c*b) = (a=b)"
+apply auto
+done
+
+lemma hypreal_mult_right_cancel: "(c::hypreal) \<noteq> 0 ==> (a*c=b*c) = (a=b)"
+apply auto
+done
+
+lemma hypreal_inverse_not_zero: "x \<noteq> 0 ==> inverse (x::hypreal) \<noteq> 0"
+ by (rule Ring_and_Field.nonzero_imp_inverse_nonzero)
+
+lemma hypreal_mult_not_0: "[| x \<noteq> 0; y \<noteq> 0 |] ==> x * y \<noteq> (0::hypreal)"
+by simp
+
+lemma hypreal_minus_inverse: "inverse(-x) = -inverse(x::hypreal)"
+ by (rule Ring_and_Field.inverse_minus_eq)
+
+lemma hypreal_inverse_distrib: "inverse(x*y) = inverse(x)*inverse(y::hypreal)"
+ by (rule Ring_and_Field.inverse_mult_distrib)
+
+
+subsection{* Division lemmas *}
+
+lemma hypreal_divide_one: "x/(1::hypreal) = x"
+by (simp add: hypreal_divide_def)
+
+
+(** As with multiplication, pull minus signs OUT of the / operator **)
+
+lemma hypreal_add_divide_distrib: "(x+y)/(z::hypreal) = x/z + y/z"
+ by (rule Ring_and_Field.add_divide_distrib)
+
+lemma hypreal_inverse_add:
+ "[|(x::hypreal) \<noteq> 0; y \<noteq> 0 |]
+ ==> inverse(x) + inverse(y) = (x + y)*inverse(x*y)"
+by (simp add: Ring_and_Field.inverse_add mult_assoc)
+
+
+subsection{*@{term hypreal_of_real} Preserves Field and Order Properties*}
+
lemma hypreal_of_real_add [simp]:
"hypreal_of_real (z1 + z2) = hypreal_of_real z1 + hypreal_of_real z2"
apply (unfold hypreal_of_real_def)
@@ -953,10 +960,12 @@
apply (unfold hypreal_le_def real_le_def, auto)
done
-lemma hypreal_of_real_eq_iff [simp]: "(hypreal_of_real z1 = hypreal_of_real z2) = (z1 = z2)"
+lemma hypreal_of_real_eq_iff [simp]:
+ "(hypreal_of_real z1 = hypreal_of_real z2) = (z1 = z2)"
by (force intro: order_antisym hypreal_of_real_le_iff [THEN iffD1])
-lemma hypreal_of_real_minus [simp]: "hypreal_of_real (-r) = - hypreal_of_real r"
+lemma hypreal_of_real_minus [simp]:
+ "hypreal_of_real (-r) = - hypreal_of_real r"
apply (unfold hypreal_of_real_def)
apply (auto simp add: hypreal_minus)
done
@@ -970,146 +979,20 @@
lemma hypreal_of_real_zero_iff: "(hypreal_of_real r = 0) = (r = 0)"
by (auto intro: FreeUltrafilterNat_P simp add: hypreal_of_real_def hypreal_zero_def FreeUltrafilterNat_Nat_set)
-lemma hypreal_of_real_inverse [simp]: "hypreal_of_real (inverse r) = inverse (hypreal_of_real r)"
+lemma hypreal_of_real_inverse [simp]:
+ "hypreal_of_real (inverse r) = inverse (hypreal_of_real r)"
apply (case_tac "r=0")
apply (simp add: DIVISION_BY_ZERO INVERSE_ZERO HYPREAL_INVERSE_ZERO)
apply (rule_tac c1 = "hypreal_of_real r" in hypreal_mult_left_cancel [THEN iffD1])
apply (auto simp add: hypreal_of_real_zero_iff hypreal_of_real_mult [symmetric])
done
-lemma hypreal_of_real_divide [simp]: "hypreal_of_real (z1 / z2) = hypreal_of_real z1 / hypreal_of_real z2"
+lemma hypreal_of_real_divide [simp]:
+ "hypreal_of_real (z1 / z2) = hypreal_of_real z1 / hypreal_of_real z2"
by (simp add: hypreal_divide_def real_divide_def)
-(*** Division lemmas ***)
-
-lemma hypreal_zero_divide: "(0::hypreal)/x = 0"
-by (simp add: hypreal_divide_def)
-
-lemma hypreal_divide_one: "x/(1::hypreal) = x"
-by (simp add: hypreal_divide_def)
-declare hypreal_zero_divide [simp] hypreal_divide_one [simp]
-
-lemma hypreal_divide_divide1_eq [simp]: "(x::hypreal) / (y/z) = (x*z)/y"
-by (simp add: hypreal_divide_def hypreal_inverse_distrib hypreal_mult_ac)
-
-lemma hypreal_divide_divide2_eq [simp]: "((x::hypreal) / y) / z = x/(y*z)"
-by (simp add: hypreal_divide_def hypreal_inverse_distrib hypreal_mult_assoc)
-
-
-(** As with multiplication, pull minus signs OUT of the / operator **)
-
-lemma hypreal_minus_divide_eq [simp]: "(-x) / (y::hypreal) = - (x/y)"
-by (simp add: hypreal_divide_def)
-
-lemma hypreal_divide_minus_eq [simp]: "(x / -(y::hypreal)) = - (x/y)"
-by (simp add: hypreal_divide_def hypreal_minus_inverse)
-
-lemma hypreal_add_divide_distrib: "(x+y)/(z::hypreal) = x/z + y/z"
-by (simp add: hypreal_divide_def hypreal_add_mult_distrib)
-
-lemma hypreal_inverse_add: "[|(x::hypreal) \<noteq> 0; y \<noteq> 0 |]
- ==> inverse(x) + inverse(y) = (x + y)*inverse(x*y)"
-apply (simp add: hypreal_inverse_distrib hypreal_add_mult_distrib hypreal_mult_assoc [symmetric])
-apply (subst hypreal_mult_assoc)
-apply (rule hypreal_mult_left_commute [THEN subst])
-apply (simp add: hypreal_add_commute)
-done
-
-lemma hypreal_self_eq_minus_self_zero: "x = -x ==> x = (0::hypreal)"
-apply (rule_tac z = x in eq_Abs_hypreal)
-apply (auto simp add: hypreal_minus hypreal_zero_def, ultra)
-done
-
-lemma hypreal_add_self_zero_cancel [simp]: "(x + x = 0) = (x = (0::hypreal))"
-apply (rule_tac z = x in eq_Abs_hypreal)
-apply (auto simp add: hypreal_add hypreal_zero_def)
-done
-
-lemma hypreal_add_self_zero_cancel2 [simp]: "(x + x + y = y) = (x = (0::hypreal))"
-apply auto
-apply (drule hypreal_eq_minus_iff [THEN iffD1])
-apply (auto simp add: hypreal_add_assoc hypreal_self_eq_minus_self_zero)
-done
-
-lemma hypreal_add_self_zero_cancel2a [simp]: "(x + (x + y) = y) = (x = (0::hypreal))"
-by (simp add: hypreal_add_assoc [symmetric])
-
-lemma hypreal_minus_eq_swap: "(b = -a) = (-b = (a::hypreal))"
-by auto
-
-lemma hypreal_minus_eq_cancel [simp]: "(-b = -a) = (b = (a::hypreal))"
-by (simp add: hypreal_minus_eq_swap)
-
-lemma hypreal_less_eq_diff: "(x<y) = (x-y < (0::hypreal))"
-apply (unfold hypreal_diff_def)
-apply (rule hypreal_less_minus_iff2)
-done
-
-(*** Subtraction laws ***)
-
-lemma hypreal_add_diff_eq: "x + (y - z) = (x + y) - (z::hypreal)"
-by (simp add: hypreal_diff_def hypreal_add_ac)
-
-lemma hypreal_diff_add_eq: "(x - y) + z = (x + z) - (y::hypreal)"
-by (simp add: hypreal_diff_def hypreal_add_ac)
-
-lemma hypreal_diff_diff_eq: "(x - y) - z = x - (y + (z::hypreal))"
-by (simp add: hypreal_diff_def hypreal_add_ac)
-
-lemma hypreal_diff_diff_eq2: "x - (y - z) = (x + z) - (y::hypreal)"
-by (simp add: hypreal_diff_def hypreal_add_ac)
-
-lemma hypreal_diff_less_eq: "(x-y < z) = (x < z + (y::hypreal))"
-apply (subst hypreal_less_eq_diff)
-apply (rule_tac y1 = z in hypreal_less_eq_diff [THEN ssubst])
-apply (simp add: hypreal_diff_def hypreal_add_ac)
-done
-
-lemma hypreal_less_diff_eq: "(x < z-y) = (x + (y::hypreal) < z)"
-apply (subst hypreal_less_eq_diff)
-apply (rule_tac y1 = "z-y" in hypreal_less_eq_diff [THEN ssubst])
-apply (simp add: hypreal_diff_def hypreal_add_ac)
-done
-
-lemma hypreal_diff_le_eq: "(x-y <= z) = (x <= z + (y::hypreal))"
-apply (unfold hypreal_le_def)
-apply (simp add: hypreal_less_diff_eq)
-done
-
-lemma hypreal_le_diff_eq: "(x <= z-y) = (x + (y::hypreal) <= z)"
-apply (unfold hypreal_le_def)
-apply (simp add: hypreal_diff_less_eq)
-done
-
-lemma hypreal_diff_eq_eq: "(x-y = z) = (x = z + (y::hypreal))"
-apply (unfold hypreal_diff_def)
-apply (auto simp add: hypreal_add_assoc)
-done
-
-lemma hypreal_eq_diff_eq: "(x = z-y) = (x + (y::hypreal) = z)"
-apply (unfold hypreal_diff_def)
-apply (auto simp add: hypreal_add_assoc)
-done
-
-
-(** For the cancellation simproc.
- The idea is to cancel like terms on opposite sides by subtraction **)
-
-lemma hypreal_less_eqI: "(x::hypreal) - y = x' - y' ==> (x<y) = (x'<y')"
-apply (subst hypreal_less_eq_diff)
-apply (rule_tac y1 = y in hypreal_less_eq_diff [THEN ssubst], simp)
-done
-
-lemma hypreal_le_eqI: "(x::hypreal) - y = x' - y' ==> (y<=x) = (y'<=x')"
-apply (drule hypreal_less_eqI)
-apply (simp add: hypreal_le_def)
-done
-
-lemma hypreal_eq_eqI: "(x::hypreal) - y = x' - y' ==> (x=y) = (x'=y')"
-apply safe
-apply (simp_all add: hypreal_eq_diff_eq hypreal_diff_eq_eq)
-done
+subsection{*Misc Others*}
lemma hypreal_zero_num: "0 = Abs_hypreal (hyprel `` {%n. 0})"
by (simp add: hypreal_zero_def [THEN meta_eq_to_obj_eq, symmetric])
@@ -1122,8 +1005,19 @@
apply (auto simp add: hypreal_less hypreal_zero_num)
done
+
+lemma hypreal_hrabs:
+ "abs (Abs_hypreal (hyprel `` {X})) =
+ Abs_hypreal(hyprel `` {%n. abs (X n)})"
+apply (auto simp add: hrabs_def hypreal_zero_def hypreal_le hypreal_minus)
+apply (ultra, arith)+
+done
+
ML
{*
+val hrabs_def = thm "hrabs_def";
+val hypreal_hrabs = thm "hypreal_hrabs";
+
val hypreal_zero_def = thm "hypreal_zero_def";
val hypreal_one_def = thm "hypreal_one_def";
val hypreal_minus_def = thm "hypreal_minus_def";
@@ -1189,11 +1083,6 @@
val hypreal_add_zero_right = thm "hypreal_add_zero_right";
val hypreal_add_minus = thm "hypreal_add_minus";
val hypreal_add_minus_left = thm "hypreal_add_minus_left";
-val hypreal_minus_ex = thm "hypreal_minus_ex";
-val hypreal_minus_ex1 = thm "hypreal_minus_ex1";
-val hypreal_minus_left_ex1 = thm "hypreal_minus_left_ex1";
-val hypreal_add_minus_eq_minus = thm "hypreal_add_minus_eq_minus";
-val hypreal_as_add_inverse_ex = thm "hypreal_as_add_inverse_ex";
val hypreal_minus_add_distrib = thm "hypreal_minus_add_distrib";
val hypreal_minus_distrib1 = thm "hypreal_minus_distrib1";
val hypreal_add_left_cancel = thm "hypreal_add_left_cancel";
@@ -1214,7 +1103,6 @@
val hypreal_mult_minus_1 = thm "hypreal_mult_minus_1";
val hypreal_mult_minus_1_right = thm "hypreal_mult_minus_1_right";
val hypreal_minus_mult_commute = thm "hypreal_minus_mult_commute";
-val hypreal_add_assoc_cong = thm "hypreal_add_assoc_cong";
val hypreal_add_mult_distrib = thm "hypreal_add_mult_distrib";
val hypreal_add_mult_distrib2 = thm "hypreal_add_mult_distrib2";
val hypreal_diff_mult_distrib = thm "hypreal_diff_mult_distrib";
@@ -1224,35 +1112,24 @@
val hypreal_inverse = thm "hypreal_inverse";
val HYPREAL_INVERSE_ZERO = thm "HYPREAL_INVERSE_ZERO";
val HYPREAL_DIVISION_BY_ZERO = thm "HYPREAL_DIVISION_BY_ZERO";
-val hypreal_inverse_inverse = thm "hypreal_inverse_inverse";
-val hypreal_inverse_1 = thm "hypreal_inverse_1";
val hypreal_mult_inverse = thm "hypreal_mult_inverse";
val hypreal_mult_inverse_left = thm "hypreal_mult_inverse_left";
val hypreal_mult_left_cancel = thm "hypreal_mult_left_cancel";
val hypreal_mult_right_cancel = thm "hypreal_mult_right_cancel";
val hypreal_inverse_not_zero = thm "hypreal_inverse_not_zero";
val hypreal_mult_not_0 = thm "hypreal_mult_not_0";
-val hypreal_mult_zero_disj = thm "hypreal_mult_zero_disj";
val hypreal_minus_inverse = thm "hypreal_minus_inverse";
val hypreal_inverse_distrib = thm "hypreal_inverse_distrib";
-val hypreal_less_iff = thm "hypreal_less_iff";
-val hypreal_lessI = thm "hypreal_lessI";
-val hypreal_lessE = thm "hypreal_lessE";
-val hypreal_lessD = thm "hypreal_lessD";
val hypreal_less_not_refl = thm "hypreal_less_not_refl";
val hypreal_not_refl2 = thm "hypreal_not_refl2";
val hypreal_less_trans = thm "hypreal_less_trans";
val hypreal_less_asym = thm "hypreal_less_asym";
val hypreal_less = thm "hypreal_less";
val hypreal_trichotomy = thm "hypreal_trichotomy";
-val hypreal_trichotomyE = thm "hypreal_trichotomyE";
val hypreal_less_minus_iff = thm "hypreal_less_minus_iff";
val hypreal_less_minus_iff2 = thm "hypreal_less_minus_iff2";
val hypreal_eq_minus_iff = thm "hypreal_eq_minus_iff";
val hypreal_eq_minus_iff2 = thm "hypreal_eq_minus_iff2";
-val hypreal_diff_zero = thm "hypreal_diff_zero";
-val hypreal_diff_zero_right = thm "hypreal_diff_zero_right";
-val hypreal_diff_self = thm "hypreal_diff_self";
val hypreal_eq_minus_iff3 = thm "hypreal_eq_minus_iff3";
val hypreal_not_eq_minus_iff = thm "hypreal_not_eq_minus_iff";
val hypreal_linear = thm "hypreal_linear";
@@ -1270,7 +1147,6 @@
val hypreal_le_linear = thm "hypreal_le_linear";
val hypreal_le_trans = thm "hypreal_le_trans";
val hypreal_le_anti_sym = thm "hypreal_le_anti_sym";
-val not_less_not_eq_hypreal_less = thm "not_less_not_eq_hypreal_less";
val hypreal_less_le = thm "hypreal_less_le";
val hypreal_minus_zero_less_iff = thm "hypreal_minus_zero_less_iff";
val hypreal_minus_zero_less_iff2 = thm "hypreal_minus_zero_less_iff2";
@@ -1287,34 +1163,14 @@
val hypreal_of_real_zero_iff = thm "hypreal_of_real_zero_iff";
val hypreal_of_real_inverse = thm "hypreal_of_real_inverse";
val hypreal_of_real_divide = thm "hypreal_of_real_divide";
-val hypreal_zero_divide = thm "hypreal_zero_divide";
val hypreal_divide_one = thm "hypreal_divide_one";
-val hypreal_divide_divide1_eq = thm "hypreal_divide_divide1_eq";
-val hypreal_divide_divide2_eq = thm "hypreal_divide_divide2_eq";
-val hypreal_minus_divide_eq = thm "hypreal_minus_divide_eq";
-val hypreal_divide_minus_eq = thm "hypreal_divide_minus_eq";
val hypreal_add_divide_distrib = thm "hypreal_add_divide_distrib";
val hypreal_inverse_add = thm "hypreal_inverse_add";
val hypreal_self_eq_minus_self_zero = thm "hypreal_self_eq_minus_self_zero";
val hypreal_add_self_zero_cancel = thm "hypreal_add_self_zero_cancel";
val hypreal_add_self_zero_cancel2 = thm "hypreal_add_self_zero_cancel2";
-val hypreal_add_self_zero_cancel2a = thm "hypreal_add_self_zero_cancel2a";
val hypreal_minus_eq_swap = thm "hypreal_minus_eq_swap";
val hypreal_minus_eq_cancel = thm "hypreal_minus_eq_cancel";
-val hypreal_less_eq_diff = thm "hypreal_less_eq_diff";
-val hypreal_add_diff_eq = thm "hypreal_add_diff_eq";
-val hypreal_diff_add_eq = thm "hypreal_diff_add_eq";
-val hypreal_diff_diff_eq = thm "hypreal_diff_diff_eq";
-val hypreal_diff_diff_eq2 = thm "hypreal_diff_diff_eq2";
-val hypreal_diff_less_eq = thm "hypreal_diff_less_eq";
-val hypreal_less_diff_eq = thm "hypreal_less_diff_eq";
-val hypreal_diff_le_eq = thm "hypreal_diff_le_eq";
-val hypreal_le_diff_eq = thm "hypreal_le_diff_eq";
-val hypreal_diff_eq_eq = thm "hypreal_diff_eq_eq";
-val hypreal_eq_diff_eq = thm "hypreal_eq_diff_eq";
-val hypreal_less_eqI = thm "hypreal_less_eqI";
-val hypreal_le_eqI = thm "hypreal_le_eqI";
-val hypreal_eq_eqI = thm "hypreal_eq_eqI";
val hypreal_zero_num = thm "hypreal_zero_num";
val hypreal_one_num = thm "hypreal_one_num";
val hypreal_omega_gt_zero = thm "hypreal_omega_gt_zero";
--- a/src/HOL/Hyperreal/HyperOrd.thy Wed Dec 24 08:54:30 2003 +0100
+++ b/src/HOL/Hyperreal/HyperOrd.thy Thu Dec 25 22:48:32 2003 +0100
@@ -7,95 +7,6 @@
theory HyperOrd = HyperDef:
-instance hypreal :: division_by_zero
-proof
- fix x :: hypreal
- show "inverse 0 = (0::hypreal)" by (rule HYPREAL_INVERSE_ZERO)
- show "x/0 = 0" by (rule HYPREAL_DIVISION_BY_ZERO)
-qed
-
-
-defs (overloaded)
- hrabs_def: "abs (r::hypreal) == (if 0 \<le> r then r else -r)"
-
-
-lemma hypreal_hrabs:
- "abs (Abs_hypreal (hyprel `` {X})) =
- Abs_hypreal(hyprel `` {%n. abs (X n)})"
-apply (auto simp add: hrabs_def hypreal_zero_def hypreal_le hypreal_minus)
-apply (ultra, arith)+
-done
-
-instance hypreal :: order
- by (intro_classes,
- (assumption |
- rule hypreal_le_refl hypreal_le_trans hypreal_le_anti_sym
- hypreal_less_le)+)
-
-instance hypreal :: linorder
- by (intro_classes, rule hypreal_le_linear)
-
-instance hypreal :: plus_ac0
- by (intro_classes,
- (assumption |
- rule hypreal_add_commute hypreal_add_assoc hypreal_add_zero_left)+)
-
-lemma hypreal_add_less_mono1: "(A::hypreal) < B ==> A + C < B + C"
-apply (rule_tac z = A in eq_Abs_hypreal)
-apply (rule_tac z = B in eq_Abs_hypreal)
-apply (rule_tac z = C in eq_Abs_hypreal)
-apply (auto intro!: exI simp add: hypreal_less_def hypreal_add, ultra)
-done
-
-lemma hypreal_mult_order: "[| 0 < x; 0 < y |] ==> (0::hypreal) < x * y"
-apply (unfold hypreal_zero_def)
-apply (rule_tac z = x in eq_Abs_hypreal)
-apply (rule_tac z = y in eq_Abs_hypreal)
-apply (auto intro!: exI simp add: hypreal_less_def hypreal_mult, ultra)
-apply (auto intro: real_mult_order)
-done
-
-lemma hypreal_add_left_le_mono1: "(q1::hypreal) \<le> q2 ==> x + q1 \<le> x + q2"
-apply (drule order_le_imp_less_or_eq)
-apply (auto intro: order_less_imp_le hypreal_add_less_mono1 simp add: hypreal_add_commute)
-done
-
-lemma hypreal_mult_less_mono1: "[| (0::hypreal) < z; x < y |] ==> x*z < y*z"
-apply (rotate_tac 1)
-apply (drule hypreal_less_minus_iff [THEN iffD1])
-apply (rule hypreal_less_minus_iff [THEN iffD2])
-apply (drule hypreal_mult_order, assumption)
-apply (simp add: hypreal_add_mult_distrib2 hypreal_mult_commute)
-done
-
-lemma hypreal_mult_less_mono2: "[| (0::hypreal)<z; x<y |] ==> z*x<z*y"
-apply (simp (no_asm_simp) add: hypreal_mult_commute hypreal_mult_less_mono1)
-done
-
-subsection{*The Hyperreals Form an Ordered Field*}
-
-instance hypreal :: inverse ..
-
-instance hypreal :: ordered_field
-proof
- fix x y z :: hypreal
- show "(x + y) + z = x + (y + z)" by (rule hypreal_add_assoc)
- show "x + y = y + x" by (rule hypreal_add_commute)
- show "0 + x = x" by simp
- show "- x + x = 0" by simp
- show "x - y = x + (-y)" by (simp add: hypreal_diff_def)
- show "(x * y) * z = x * (y * z)" by (rule hypreal_mult_assoc)
- show "x * y = y * x" by (rule hypreal_mult_commute)
- show "1 * x = x" by simp
- show "(x + y) * z = x * z + y * z" by (simp add: hypreal_add_mult_distrib)
- show "0 \<noteq> (1::hypreal)" by (rule hypreal_zero_not_eq_one)
- show "x \<le> y ==> z + x \<le> z + y" by (rule hypreal_add_left_le_mono1)
- show "x < y ==> 0 < z ==> z * x < z * y" by (simp add: hypreal_mult_less_mono2)
- show "\<bar>x\<bar> = (if x < 0 then -x else x)"
- by (auto dest: order_le_less_trans simp add: hrabs_def linorder_not_le)
- show "x \<noteq> 0 ==> inverse x * x = 1" by simp
- show "y \<noteq> 0 ==> x / y = x * inverse y" by (simp add: hypreal_divide_def)
-qed
(*** Monotonicity results ***)
@@ -277,9 +188,6 @@
ML
{*
-val hrabs_def = thm "hrabs_def";
-val hypreal_hrabs = thm "hypreal_hrabs";
-
val hypreal_add_less_mono1 = thm"hypreal_add_less_mono1";
val hypreal_add_less_mono2 = thm"hypreal_add_less_mono2";
val hypreal_mult_order = thm"hypreal_mult_order";
--- a/src/HOL/Hyperreal/Integration.ML Wed Dec 24 08:54:30 2003 +0100
+++ b/src/HOL/Hyperreal/Integration.ML Thu Dec 25 22:48:32 2003 +0100
@@ -183,7 +183,7 @@
by (dres_inst_tac [("x","psize D - Suc n")] spec 2);
by (thin_tac "ALL n. psize D <= n --> D n = b" 2);
by (Asm_full_simp_tac 2);
-by (Blast_tac 1);
+by (arith_tac 1);
qed "partition_ub";
Goal "[| partition(a,b) D; n < psize D |] ==> D(n) < b";
--- a/src/HOL/Hyperreal/NSA.ML Wed Dec 24 08:54:30 2003 +0100
+++ b/src/HOL/Hyperreal/NSA.ML Thu Dec 25 22:48:32 2003 +0100
@@ -350,7 +350,7 @@
by (forw_inst_tac [("x1","r"),("z","abs x")]
(hypreal_inverse_gt_0 RS order_less_trans) 1);
by (assume_tac 1);
-by (dtac ((hypreal_inverse_inverse RS sym) RS subst) 1);
+by (dtac ((inverse_inverse_eq RS sym) RS subst) 1);
by (rtac (hypreal_inverse_less_iff RS iffD1) 1);
by (auto_tac (claset(), simpset() addsimps [SReal_inverse]));
qed "HInfinite_inverse_Infinitesimal";
@@ -2244,7 +2244,7 @@
Goal "(u = inverse (real(Suc n))) = (real(Suc n) = inverse u)";
by (auto_tac (claset(),
- simpset() addsimps [real_inverse_inverse, real_of_nat_Suc_gt_zero,
+ simpset() addsimps [inverse_inverse_eq, real_of_nat_Suc_gt_zero,
real_not_refl2 RS not_sym]));
qed "real_of_nat_inverse_eq_iff";
--- a/src/HOL/Hyperreal/SEQ.ML Wed Dec 24 08:54:30 2003 +0100
+++ b/src/HOL/Hyperreal/SEQ.ML Thu Dec 25 22:48:32 2003 +0100
@@ -1130,9 +1130,9 @@
by (ftac order_less_trans 1 THEN assume_tac 1);
by (forw_inst_tac [("x","f n")] real_inverse_gt_0 1);
by (asm_simp_tac (simpset() addsimps [abs_eqI2]) 1);
-by (res_inst_tac [("t","r")] (real_inverse_inverse RS subst) 1);
+by (res_inst_tac [("t","r")] (inverse_inverse_eq RS subst) 1);
by (auto_tac (claset() addIs [inverse_less_iff_less RS iffD2],
- simpset() delsimps [thm"Ring_and_Field.inverse_inverse_eq"]));
+ simpset() delsimps [inverse_inverse_eq]));
qed "LIMSEQ_inverse_zero";
Goal "ALL y. EX N. ALL n. N <= n --> y < f(n) \
--- a/src/HOL/Integ/int_arith1.ML Wed Dec 24 08:54:30 2003 +0100
+++ b/src/HOL/Integ/int_arith1.ML Thu Dec 25 22:48:32 2003 +0100
@@ -5,6 +5,43 @@
Simprocs and decision procedure for linear arithmetic.
*)
+(** Misc ML bindings **)
+
+val left_inverse = thm "left_inverse";
+val right_inverse = thm "right_inverse";
+val inverse_less_iff_less = thm"Ring_and_Field.inverse_less_iff_less";
+val inverse_eq_divide = thm"Ring_and_Field.inverse_eq_divide";
+val inverse_minus_eq = thm "inverse_minus_eq";
+val inverse_mult_distrib = thm "inverse_mult_distrib";
+val inverse_add = thm "inverse_add";
+val inverse_inverse_eq = thm "inverse_inverse_eq";
+
+val add_right_mono = thm"Ring_and_Field.add_right_mono";
+val times_divide_eq_left = thm "times_divide_eq_left";
+val times_divide_eq_right = thm "times_divide_eq_right";
+val minus_minus = thm "minus_minus";
+val minus_mult_left = thm "minus_mult_left";
+val minus_mult_right = thm "minus_mult_right";
+
+val pos_real_less_divide_eq = thm"pos_less_divide_eq";
+val pos_real_divide_less_eq = thm"pos_divide_less_eq";
+val pos_real_le_divide_eq = thm"pos_le_divide_eq";
+val pos_real_divide_le_eq = thm"pos_divide_le_eq";
+
+val mult_less_cancel_left = thm"Ring_and_Field.mult_less_cancel_left";
+val mult_le_cancel_left = thm"Ring_and_Field.mult_le_cancel_left";
+val mult_less_cancel_right = thm"Ring_and_Field.mult_less_cancel_right";
+val mult_le_cancel_right = thm"Ring_and_Field.mult_le_cancel_right";
+val mult_cancel_left = thm"Ring_and_Field.mult_cancel_left";
+val mult_cancel_right = thm"Ring_and_Field.mult_cancel_right";
+
+val field_mult_cancel_left = thm "field_mult_cancel_left";
+val field_mult_cancel_right = thm "field_mult_cancel_right";
+
+val mult_divide_cancel_left = thm"Ring_and_Field.mult_divide_cancel_left";
+val mult_divide_cancel_right = thm "Ring_and_Field.mult_divide_cancel_right";
+val mult_divide_cancel_eq_if = thm"Ring_and_Field.mult_divide_cancel_eq_if";
+
val NCons_Pls = thm"NCons_Pls";
val NCons_Min = thm"NCons_Min";
val NCons_BIT = thm"NCons_BIT";
--- a/src/HOL/Real/HahnBanach/Subspace.thy Wed Dec 24 08:54:30 2003 +0100
+++ b/src/HOL/Real/HahnBanach/Subspace.thy Thu Dec 25 22:48:32 2003 +0100
@@ -329,13 +329,13 @@
proof (rule add_minus_eq)
show "u1 \<in> E" ..
show "u2 \<in> E" ..
- from u u' and direct show "u1 - u2 = 0" by blast
+ from u u' and direct show "u1 - u2 = 0" by force
qed
show "v1 = v2"
proof (rule add_minus_eq [symmetric])
show "v1 \<in> E" ..
show "v2 \<in> E" ..
- from v v' and direct show "v2 - v1 = 0" by blast
+ from v v' and direct show "v2 - v1 = 0" by force
qed
qed
--- a/src/HOL/Real/RealArith.thy Wed Dec 24 08:54:30 2003 +0100
+++ b/src/HOL/Real/RealArith.thy Thu Dec 25 22:48:32 2003 +0100
@@ -31,9 +31,7 @@
by auto
-(** Simprules combining x-y and 0; see also real_less_iff_diff_less_0, etc.,
- in RealBin
-**)
+(** Simprules combining x-y and 0 (needed??) **)
lemma real_0_less_diff_iff [iff]: "((0::real) < x-y) = (y < x)"
by auto
--- a/src/HOL/Real/RealBin.ML Wed Dec 24 08:54:30 2003 +0100
+++ b/src/HOL/Real/RealBin.ML Thu Dec 25 22:48:32 2003 +0100
@@ -192,17 +192,6 @@
(**** Simprocs for numeric literals ****)
-(** Combining of literal coefficients in sums of products **)
-
-Goal "(x = y) = (x-y = (0::real))";
-by (simp_tac (simpset() addsimps compare_rls) 1);
-qed "real_eq_iff_diff_eq_0";
-
-Goal "(x <= y) = (x-y <= (0::real))";
-by (simp_tac (simpset() addsimps compare_rls) 1);
-qed "real_le_iff_diff_le_0";
-
-
(** For combine_numerals **)
Goal "i*u + (j*u + k) = (i+j)*u + (k::real)";
@@ -212,12 +201,10 @@
(** For cancel_numerals **)
-val rel_iff_rel_0_rls = map (inst "y" "?u+?v")
- [real_less_eq_diff, real_eq_iff_diff_eq_0,
- real_le_iff_diff_le_0] @
- map (inst "y" "n")
- [real_less_eq_diff, real_eq_iff_diff_eq_0,
- real_le_iff_diff_le_0];
+val rel_iff_rel_0_rls = map (inst "b" "?u+?v")
+ [less_iff_diff_less_0, eq_iff_diff_eq_0, le_iff_diff_le_0] @
+ map (inst "b" "n")
+ [less_iff_diff_less_0, eq_iff_diff_eq_0, 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]@
@@ -603,9 +590,9 @@
Addsimprocs [Real_Times_Assoc.conv];
(*Simplification of x-y < 0, etc.*)
-AddIffs [real_less_eq_diff RS sym];
-AddIffs [real_eq_iff_diff_eq_0 RS sym];
-AddIffs [real_le_iff_diff_le_0 RS sym];
+AddIffs [less_iff_diff_less_0 RS sym];
+AddIffs [eq_iff_diff_eq_0 RS sym];
+AddIffs [le_iff_diff_le_0 RS sym];
(** <= monotonicity results: needed for arithmetic **)
--- a/src/HOL/Real/RealDef.thy Wed Dec 24 08:54:30 2003 +0100
+++ b/src/HOL/Real/RealDef.thy Thu Dec 25 22:48:32 2003 +0100
@@ -93,17 +93,17 @@
real_of_nat_def: "real n == real_of_posnat n + (- 1)"
real_add_def:
- "P+Q == Abs_REAL(UN p1:Rep_REAL(P). UN p2:Rep_REAL(Q).
+ "P+Q == Abs_REAL(\<Union>p1\<in>Rep_REAL(P). \<Union>p2\<in>Rep_REAL(Q).
(%(x1,y1). (%(x2,y2). realrel``{(x1+x2, y1+y2)}) p2) p1)"
real_mult_def:
- "P*Q == Abs_REAL(UN p1:Rep_REAL(P). UN p2:Rep_REAL(Q).
+ "P*Q == Abs_REAL(\<Union>p1\<in>Rep_REAL(P). \<Union>p2\<in>Rep_REAL(Q).
(%(x1,y1). (%(x2,y2). realrel``{(x1*x2+y1*y2,x1*y2+x2*y1)})
p2) p1)"
real_less_def:
"P<Q == \<exists>x1 y1 x2 y2. x1 + y2 < x2 + y1 &
- (x1,y1):Rep_REAL(P) & (x2,y2):Rep_REAL(Q)"
+ (x1,y1)\<in>Rep_REAL(P) & (x2,y2)\<in>Rep_REAL(Q)"
real_le_def:
"P \<le> (Q::real) == ~(Q < P)"
@@ -112,7 +112,7 @@
Nats :: "'a set" ("\<nat>")
-(*** Proving that realrel is an equivalence relation ***)
+subsection{*Proving that realrel is an equivalence relation*}
lemma preal_trans_lemma:
"[| (x1::preal) + y2 = x2 + y1; x2 + y3 = x3 + y2 |]
@@ -225,7 +225,7 @@
declare real_minus_zero_iff [simp]
-(*** Congruence property for addition ***)
+subsection{*Congruence property for addition*}
lemma real_add_congruent2_lemma:
"[|a + ba = aa + b; ab + bc = ac + bb|]
@@ -298,9 +298,10 @@
declare real_add_minus_left [simp]
-(*** Congruence property for multiplication ***)
+subsection{*Congruence property for multiplication*}
-lemma real_mult_congruent2_lemma: "!!(x1::preal). [| x1 + y2 = x2 + y1 |] ==>
+lemma real_mult_congruent2_lemma:
+ "!!(x1::preal). [| x1 + y2 = x2 + y1 |] ==>
x * x1 + y * y1 + (x * y2 + x2 * y) =
x * x2 + y * y2 + (x * y1 + x1 * y)"
apply (simp add: preal_add_left_commute preal_add_assoc [symmetric] preal_add_mult_distrib2 [symmetric])
@@ -407,7 +408,8 @@
(** Lemmas **)
-lemma real_add_assoc_cong: "(z::real) + v = z' + v' ==> z + (v + w) = z' + (v' + w)"
+lemma real_add_assoc_cong:
+ "(z::real) + v = z' + v' ==> z + (v + w) = z' + (v' + w)"
by (simp add: real_add_assoc [symmetric])
lemma real_add_mult_distrib: "((z1::real) + z2) * w = (z1 * w) + (z2 * w)"
@@ -428,13 +430,13 @@
lemma real_diff_mult_distrib2: "(w::real) * (z1 - z2) = (w * z1) - (w * z2)"
by (simp add: real_mult_commute [of w] real_diff_mult_distrib)
-(*** one and zero are distinct ***)
+text{*one and zero are distinct*}
lemma real_zero_not_eq_one: "0 ~= (1::real)"
apply (unfold real_zero_def real_one_def)
apply (auto simp add: preal_self_less_add_left [THEN preal_not_refl2])
done
-(*** existence of inverse ***)
+subsection{*existence of inverse*}
(** lemma -- alternative definition of 0 **)
lemma real_zero_iff: "0 = Abs_REAL (realrel `` {(x, x)})"
apply (unfold real_zero_def)
@@ -449,7 +451,9 @@
apply (auto dest!: preal_less_add_left_Ex simp add: real_zero_iff [symmetric])
apply (rule_tac x = "Abs_REAL (realrel `` { (preal_of_prat (prat_of_pnat 1), pinv (D) + preal_of_prat (prat_of_pnat 1))}) " in exI)
apply (rule_tac [2] x = "Abs_REAL (realrel `` { (pinv (D) + preal_of_prat (prat_of_pnat 1), preal_of_prat (prat_of_pnat 1))}) " in exI)
-apply (auto simp add: real_mult pnat_one_def preal_mult_1_right preal_add_mult_distrib2 preal_add_mult_distrib preal_mult_1 preal_mult_inv_right preal_add_ac preal_mult_ac)
+apply (auto simp add: real_mult pnat_one_def preal_mult_1_right
+ preal_add_mult_distrib2 preal_add_mult_distrib preal_mult_1
+ preal_mult_inv_right preal_add_ac preal_mult_ac)
done
lemma real_mult_inv_left_ex: "x ~= 0 ==> \<exists>y. y*x = (1::real)"
@@ -471,21 +475,21 @@
declare real_mult_inv_right [simp]
-(*---------------------------------------------------------
- Theorems for ordering
- --------------------------------------------------------*)
-(* prove introduction and elimination rules for real_less *)
+subsection{*Theorems for Ordering*}
+
+(* real_less is a strict order: irreflexive *)
-(* real_less is a strong order i.e. nonreflexive and transitive *)
-
-(*** lemmas ***)
-lemma preal_lemma_eq_rev_sum: "!!(x::preal). [| x = y; x1 = y1 |] ==> x + y1 = x1 + y"
+text{*lemmas*}
+lemma preal_lemma_eq_rev_sum:
+ "!!(x::preal). [| x = y; x1 = y1 |] ==> x + y1 = x1 + y"
by (simp add: preal_add_commute)
-lemma preal_add_left_commute_cancel: "!!(b::preal). x + (b + y) = x1 + (b + y1) ==> x + y = x1 + y1"
+lemma preal_add_left_commute_cancel:
+ "!!(b::preal). x + (b + y) = x1 + (b + y1) ==> x + y = x1 + y1"
by (simp add: preal_add_ac)
-lemma preal_lemma_for_not_refl: "!!(x::preal). [| x + y2a = x2a + y;
+lemma preal_lemma_for_not_refl:
+ "!!(x::preal). [| x + y2a = x2a + y;
x + y2b = x2b + y |]
==> x2a + y2b = x2b + y2a"
apply (drule preal_lemma_eq_rev_sum, assumption)
@@ -569,10 +573,11 @@
apply (simp add: preal_add_assoc [symmetric] preal_self_less_add_left)
done
-lemma real_of_preal_iff: "(\<exists>m. Abs_REAL (realrel `` {(x,y)}) = real_of_preal m) = (y < x)"
+lemma real_of_preal_iff:
+ "(\<exists>m. Abs_REAL (realrel `` {(x,y)}) = real_of_preal m) = (y < x)"
by (blast intro!: real_of_preal_ExI real_of_preal_ExD)
-(*** Gleason prop 9-4.4 p 127 ***)
+text{*Gleason prop 9-4.4 p 127*}
lemma real_of_preal_trichotomy:
"\<exists>m. (x::real) = real_of_preal m | x = 0 | x = -(real_of_preal m)"
apply (unfold real_of_preal_def real_zero_def)
@@ -583,7 +588,8 @@
apply (auto simp add: preal_add_commute)
done
-lemma real_of_preal_trichotomyE: "!!P. [| !!m. x = real_of_preal m ==> P;
+lemma real_of_preal_trichotomyE:
+ "!!P. [| !!m. x = real_of_preal m ==> P;
x = 0 ==> P;
!!m. x = -(real_of_preal m) ==> P |] ==> P"
apply (cut_tac x = x in real_of_preal_trichotomy, auto)
@@ -606,7 +612,8 @@
apply (simp add: preal_self_less_add_left del: preal_add_less_iff2)
done
-lemma real_of_preal_less_iff1: "(real_of_preal m1 < real_of_preal m2) = (m1 < m2)"
+lemma real_of_preal_less_iff1:
+ "(real_of_preal m1 < real_of_preal m2) = (m1 < m2)"
by (blast intro: real_of_preal_lessI real_of_preal_lessD)
declare real_of_preal_less_iff1 [simp]
@@ -677,7 +684,8 @@
apply (blast dest: real_less_trans elim: real_less_irrefl)
done
-lemma real_of_preal_minus_less_rev1: "- real_of_preal m1 < - real_of_preal m2
+lemma real_of_preal_minus_less_rev1:
+ "- real_of_preal m1 < - real_of_preal m2
==> real_of_preal m2 < real_of_preal m1"
apply (auto simp add: real_of_preal_def real_less_def real_minus)
apply (rule exI)+
@@ -688,7 +696,8 @@
apply (auto simp add: preal_add_ac)
done
-lemma real_of_preal_minus_less_rev2: "real_of_preal m1 < real_of_preal m2
+lemma real_of_preal_minus_less_rev2:
+ "real_of_preal m1 < real_of_preal m2
==> - real_of_preal m2 < - real_of_preal m1"
apply (auto simp add: real_of_preal_def real_less_def real_minus)
apply (rule exI)+
@@ -699,7 +708,8 @@
apply (auto simp add: preal_add_ac)
done
-lemma real_of_preal_minus_less_rev_iff: "(- real_of_preal m1 < - real_of_preal m2) =
+lemma real_of_preal_minus_less_rev_iff:
+ "(- real_of_preal m1 < - real_of_preal m2) =
(real_of_preal m2 < real_of_preal m1)"
apply (blast intro!: real_of_preal_minus_less_rev1 real_of_preal_minus_less_rev2)
done
@@ -721,7 +731,8 @@
by (cut_tac real_linear, blast)
-lemma real_linear_less2: "!!(R1::real). [| R1 < R2 ==> P; R1 = R2 ==> P;
+lemma real_linear_less2:
+ "!!(R1::real). [| R1 < R2 ==> P; R1 = R2 ==> P;
R2 < R1 ==> P |] ==> P"
apply (cut_tac x = R1 and y = R2 in real_linear, auto)
done
--- a/src/HOL/Real/RealOrd.thy Wed Dec 24 08:54:30 2003 +0100
+++ b/src/HOL/Real/RealOrd.thy Thu Dec 25 22:48:32 2003 +0100
@@ -56,7 +56,7 @@
done
(* Axiom 'order_less_le' of class 'order': *)
-lemma real_less_le: "((w::real) < z) = (w \<le> z & w ~= z)"
+lemma real_less_le: "((w::real) < z) = (w \<le> z & w \<noteq> z)"
apply (simp add: real_le_def real_neq_iff)
apply (blast elim!: real_less_asym)
done
@@ -84,11 +84,13 @@
apply (blast elim!: real_less_irrefl real_of_preal_not_minus_gt_zero [THEN notE])
done
-lemma real_gt_preal_preal_Ex: "real_of_preal z < x ==> \<exists>y. x = real_of_preal y"
+lemma real_gt_preal_preal_Ex:
+ "real_of_preal z < x ==> \<exists>y. x = real_of_preal y"
by (blast dest!: real_of_preal_zero_less [THEN real_less_trans]
intro: real_gt_zero_preal_Ex [THEN iffD1])
-lemma real_ge_preal_preal_Ex: "real_of_preal z \<le> x ==> \<exists>y. x = real_of_preal y"
+lemma real_ge_preal_preal_Ex:
+ "real_of_preal z \<le> x ==> \<exists>y. x = real_of_preal y"
by (blast dest: order_le_imp_less_or_eq real_gt_preal_preal_Ex)
lemma real_less_all_preal: "y \<le> 0 ==> \<forall>x. y < real_of_preal x"
@@ -99,7 +101,8 @@
lemma real_less_all_real2: "~ 0 < y ==> \<forall>x. y < real_of_preal x"
by (blast intro!: real_less_all_preal real_leI)
-lemma real_of_preal_le_iff: "(real_of_preal m1 \<le> real_of_preal m2) = (m1 \<le> m2)"
+lemma real_of_preal_le_iff:
+ "(real_of_preal m1 \<le> real_of_preal m2) = (m1 \<le> m2)"
apply (auto intro!: preal_leI simp add: linorder_not_less [symmetric])
done
@@ -241,7 +244,7 @@
apply (auto simp add: real_zero_not_eq_one)
done
-lemma DIVISION_BY_ZERO [simp]: "a / (0::real) = 0"
+lemma DIVISION_BY_ZERO: "a / (0::real) = 0"
by (simp add: real_divide_def INVERSE_ZERO)
instance real :: division_by_zero
@@ -251,27 +254,24 @@
show "x/0 = 0" by (rule DIVISION_BY_ZERO)
qed
-lemma real_mult_left_cancel: "(c::real) ~= 0 ==> (c*a=c*b) = (a=b)"
+lemma real_mult_left_cancel: "(c::real) \<noteq> 0 ==> (c*a=c*b) = (a=b)"
by auto
-lemma real_mult_right_cancel: "(c::real) ~= 0 ==> (a*c=b*c) = (a=b)"
+lemma real_mult_right_cancel: "(c::real) \<noteq> 0 ==> (a*c=b*c) = (a=b)"
by auto
-lemma real_mult_left_cancel_ccontr: "c*a ~= c*b ==> a ~= b"
+lemma real_mult_left_cancel_ccontr: "c*a \<noteq> c*b ==> a \<noteq> b"
by auto
-lemma real_mult_right_cancel_ccontr: "a*c ~= b*c ==> a ~= b"
+lemma real_mult_right_cancel_ccontr: "a*c \<noteq> b*c ==> a \<noteq> b"
by auto
-lemma real_inverse_not_zero: "x ~= 0 ==> inverse(x::real) ~= 0"
+lemma real_inverse_not_zero: "x \<noteq> 0 ==> inverse(x::real) \<noteq> 0"
by (rule Ring_and_Field.nonzero_imp_inverse_nonzero)
-lemma real_mult_not_zero: "[| x ~= 0; y ~= 0 |] ==> x * y ~= (0::real)"
+lemma real_mult_not_zero: "[| x \<noteq> 0; y \<noteq> 0 |] ==> x * y \<noteq> (0::real)"
by simp
-lemma real_inverse_inverse: "inverse(inverse (x::real)) = x"
- by (rule Ring_and_Field.inverse_inverse_eq)
-
lemma real_inverse_1: "inverse((1::real)) = (1::real)"
by (rule Ring_and_Field.inverse_1)
@@ -301,7 +301,8 @@
apply (erule add_left_mono)
done
-lemma real_add_le_less_mono: "!!z z'::real. [| w'\<le>w; z'<z |] ==> w' + z' < w + z"
+lemma real_add_le_less_mono:
+ "!!z z'::real. [| w'\<le>w; z'<z |] ==> w' + z' < w + z"
apply (erule add_right_mono [THEN order_le_less_trans])
apply (erule add_strict_left_mono)
done
@@ -371,7 +372,8 @@
lemma real_mult_is_0 [iff]: "(x*y = 0) = (x = 0 | y = (0::real))"
by (rule Ring_and_Field.mult_eq_0_iff)
-lemma real_inverse_add: "[| x \<noteq> 0; y \<noteq> 0 |]
+lemma real_inverse_add:
+ "[| x \<noteq> 0; y \<noteq> 0 |]
==> inverse x + inverse y = (x + y) * inverse (x * (y::real))"
by (simp add: Ring_and_Field.inverse_add mult_assoc)
@@ -436,13 +438,15 @@
apply (simp (no_asm_use) add: real_of_posnat_one [symmetric] real_of_posnat_def real_of_preal_add [symmetric] preal_of_prat_add [symmetric] prat_of_pnat_add [symmetric] pnat_of_nat_add)
done
-lemma real_of_posnat_add_one: "real_of_posnat (n + 1) = real_of_posnat n + (1::real)"
+lemma real_of_posnat_add_one:
+ "real_of_posnat (n + 1) = real_of_posnat n + (1::real)"
apply (rule_tac x1 = " (1::real) " in real_add_right_cancel [THEN iffD1])
apply (rule real_of_posnat_add [THEN subst])
apply (simp (no_asm_use) add: real_of_posnat_two real_add_assoc)
done
-lemma real_of_posnat_Suc: "real_of_posnat (Suc n) = real_of_posnat n + (1::real)"
+lemma real_of_posnat_Suc:
+ "real_of_posnat (Suc n) = real_of_posnat n + (1::real)"
by (subst real_of_posnat_add_one [symmetric], simp)
lemma inj_real_of_posnat: "inj(real_of_posnat)"
@@ -535,7 +539,7 @@
done
lemma real_inverse_unique: "x*y = (1::real) ==> y = inverse x"
-apply (case_tac "x ~= 0")
+apply (case_tac "x \<noteq> 0")
apply (rule_tac c1 = x in real_mult_left_cancel [THEN iffD1], auto)
done
@@ -564,7 +568,8 @@
done
declare real_of_nat_ge_zero_cancel_iff [simp]
-lemma real_of_nat_num_if: "real n = (if n=0 then 0 else 1 + real ((n::nat) - 1))"
+lemma real_of_nat_num_if:
+ "real n = (if n=0 then 0 else 1 + real ((n::nat) - 1))"
apply (case_tac "n", simp)
apply (simp add: real_of_nat_Suc add_commute)
done
@@ -721,7 +726,6 @@
val real_mult_right_cancel_ccontr = thm"real_mult_right_cancel_ccontr";
val real_inverse_not_zero = thm"real_inverse_not_zero";
val real_mult_not_zero = thm"real_mult_not_zero";
-val real_inverse_inverse = thm"real_inverse_inverse";
val real_inverse_1 = thm"real_inverse_1";
val real_minus_inverse = thm"real_minus_inverse";
val real_inverse_distrib = thm"real_inverse_distrib";
--- a/src/HOL/Real/real_arith0.ML Wed Dec 24 08:54:30 2003 +0100
+++ b/src/HOL/Real/real_arith0.ML Thu Dec 25 22:48:32 2003 +0100
@@ -6,43 +6,6 @@
Instantiation of the generic linear arithmetic package for type real.
*)
-(** Misc ML bindings **)
-(*FIXME: move to Integ or earlier*)
-
-val left_inverse = thm "left_inverse";
-val right_inverse = thm "right_inverse";
-val inverse_less_iff_less = thm"Ring_and_Field.inverse_less_iff_less";
-val inverse_eq_divide = thm"Ring_and_Field.inverse_eq_divide";
-val inverse_minus_eq = thm "inverse_minus_eq";
-val inverse_mult_distrib = thm "inverse_mult_distrib";
-val inverse_add = thm "inverse_add";
-
-val add_right_mono = thm"Ring_and_Field.add_right_mono";
-val times_divide_eq_left = thm "times_divide_eq_left";
-val times_divide_eq_right = thm "times_divide_eq_right";
-val minus_minus = thm "minus_minus";
-val minus_mult_left = thm "minus_mult_left";
-val minus_mult_right = thm "minus_mult_right";
-
-val pos_real_less_divide_eq = thm"pos_less_divide_eq";
-val pos_real_divide_less_eq = thm"pos_divide_less_eq";
-val pos_real_le_divide_eq = thm"pos_le_divide_eq";
-val pos_real_divide_le_eq = thm"pos_divide_le_eq";
-
-val mult_less_cancel_left = thm"Ring_and_Field.mult_less_cancel_left";
-val mult_le_cancel_left = thm"Ring_and_Field.mult_le_cancel_left";
-val mult_less_cancel_right = thm"Ring_and_Field.mult_less_cancel_right";
-val mult_le_cancel_right = thm"Ring_and_Field.mult_le_cancel_right";
-val mult_cancel_left = thm"Ring_and_Field.mult_cancel_left";
-val mult_cancel_right = thm"Ring_and_Field.mult_cancel_right";
-
-val field_mult_cancel_left = thm "field_mult_cancel_left";
-val field_mult_cancel_right = thm "field_mult_cancel_right";
-
-val mult_divide_cancel_left = thm"Ring_and_Field.mult_divide_cancel_left";
-val mult_divide_cancel_right = thm "Ring_and_Field.mult_divide_cancel_right";
-val mult_divide_cancel_eq_if = thm"Ring_and_Field.mult_divide_cancel_eq_if";
-
local