--- a/src/HOL/Complex/CStar.ML Wed Jan 28 17:01:01 2004 +0100
+++ b/src/HOL/Complex/CStar.ML Thu Jan 29 16:51:17 2004 +0100
@@ -95,7 +95,7 @@
Goal "*sc* (A - B) = *sc* A - *sc* B";
by (auto_tac (claset(),simpset() addsimps
- [set_diff_iff2,STARC_Int,STARC_Compl]));
+ [Diff_eq,STARC_Int,STARC_Compl]));
qed "STARC_diff";
Goalw [starsetC_n_def]
--- a/src/HOL/Complex/NSComplex.thy Wed Jan 28 17:01:01 2004 +0100
+++ b/src/HOL/Complex/NSComplex.thy Thu Jan 29 16:51:17 2004 +0100
@@ -6,17 +6,15 @@
theory NSComplex = NSInduct:
-(* Move to one of the hyperreal theories *)
+(* ???MOVE to one of the hyperreal theories: HRealAbs??? *)
lemma hypreal_of_nat: "hypreal_of_nat m = Abs_hypreal(hyprel `` {%n. real m})"
apply (induct_tac "m")
apply (auto simp add: hypreal_zero_def hypreal_of_nat_Suc hypreal_zero_num hypreal_one_num hypreal_add real_of_nat_Suc)
done
-(* not proved already? strange! *)
lemma hypreal_of_nat_le_iff:
"(hypreal_of_nat n \<le> hypreal_of_nat m) = (n \<le> m)"
-apply (unfold hypreal_le_def)
-apply auto
+apply (auto simp add: linorder_not_less [symmetric])
done
declare hypreal_of_nat_le_iff [simp]
@@ -1365,7 +1363,7 @@
lemma cos_harg_i_mult_zero [simp]:
"y \<noteq> 0 ==> ( *f* cos) (harg(iii * hcomplex_of_hypreal y)) = 0"
-apply (cut_tac x = "y" and y = "0" in hypreal_linear)
+apply (cut_tac x = "y" and y = "0" in linorder_less_linear)
apply (auto simp add: cos_harg_i_mult_zero_pos cos_harg_i_mult_zero_neg)
done
--- a/src/HOL/Hyperreal/HLog.ML Wed Jan 28 17:01:01 2004 +0100
+++ b/src/HOL/Hyperreal/HLog.ML Thu Jan 29 16:51:17 2004 +0100
@@ -114,7 +114,7 @@
Addsimps [powhr_less_cancel_iff];
Goal "1 < x ==> (x powhr a <= x powhr b) = (a <= b)";
-by (auto_tac (claset(),simpset() addsimps [hypreal_le_def]));
+by (auto_tac (claset(),simpset() addsimps [linorder_not_less RS sym]));
qed "powhr_le_cancel_iff";
Addsimps [powhr_le_cancel_iff];
@@ -246,7 +246,7 @@
Addsimps [hlog_less_cancel_iff];
Goal "[| 1 < a; 0 < x; 0 < y |] ==> (hlog a x <= hlog a y) = (x <= y)";
-by (auto_tac (claset(),simpset() addsimps [hypreal_le_def]));
+by (auto_tac (claset(),simpset() addsimps [linorder_not_less RS sym]));
qed "hlog_le_cancel_iff";
Addsimps [hlog_le_cancel_iff];
--- a/src/HOL/Hyperreal/HRealAbs.thy Wed Jan 28 17:01:01 2004 +0100
+++ b/src/HOL/Hyperreal/HRealAbs.thy Thu Jan 29 16:51:17 2004 +0100
@@ -31,7 +31,7 @@
by (simp add: order_less_imp_le hrabs_eqI1)
lemma hrabs_minus_eqI2: "x<(0::hypreal) ==> abs x = -x"
-by (simp add: hypreal_le_def hrabs_def)
+by (simp add: linorder_not_less [symmetric] hrabs_def)
lemma hrabs_minus_eqI1: "x<=(0::hypreal) ==> abs x = -x"
by (auto dest: order_antisym simp add: hrabs_def)
--- a/src/HOL/Hyperreal/HTranscendental.ML Wed Jan 28 17:01:01 2004 +0100
+++ b/src/HOL/Hyperreal/HTranscendental.ML Thu Jan 29 16:51:17 2004 +0100
@@ -63,7 +63,7 @@
Goal "[|0<=x; 0<=y |] ==> \
\ ( *f* sqrt)(x*y) = ( *f* sqrt)(x) * ( *f* sqrt)(y)";
by (auto_tac (claset() addIs [hypreal_sqrt_mult_distrib],
- simpset() addsimps [hypreal_le_eq_less_or_eq]));
+ simpset() addsimps [order_le_less]));
qed "hypreal_sqrt_mult_distrib2";
Goal "0 < x ==> (( *f* sqrt)(x) @= 0) = (x @= 0)";
@@ -102,7 +102,7 @@
Goal "0 <= x ==> 0 <= ( *f* sqrt)(x)";
by (auto_tac (claset() addIs [hypreal_sqrt_gt_zero],
- simpset() addsimps [hypreal_le_eq_less_or_eq ]));
+ simpset() addsimps [order_le_less ]));
qed "hypreal_sqrt_ge_zero";
Goal "( *f* sqrt)(x ^ 2) = abs(x)";
@@ -148,14 +148,14 @@
Addsimps [hypreal_sqrt_sum_squares_ge1];
Goal "[| 0 <= x; x : HFinite |] ==> ( *f* sqrt) x : HFinite";
-by (auto_tac (claset(),simpset() addsimps [hypreal_le_eq_less_or_eq]));
+by (auto_tac (claset(),simpset() addsimps [order_le_less]));
by (rtac (HFinite_square_iff RS iffD1) 1);
by (dtac hypreal_sqrt_gt_zero_pow2 1);
by (auto_tac (claset(),simpset() addsimps [numeral_2_eq_2]));
qed "HFinite_hypreal_sqrt";
Goal "[| 0 <= x; ( *f* sqrt) x : HFinite |] ==> x : HFinite";
-by (auto_tac (claset(),simpset() addsimps [hypreal_le_eq_less_or_eq]));
+by (auto_tac (claset(),simpset() addsimps [order_le_less]));
by (dtac (HFinite_square_iff RS iffD2) 1);
by (dtac hypreal_sqrt_gt_zero_pow2 1);
by (auto_tac (claset(),simpset() addsimps [numeral_2_eq_2] delsimps [HFinite_square_iff]));
@@ -175,14 +175,14 @@
Addsimps [HFinite_sqrt_sum_squares];
Goal "[| 0 <= x; x : Infinitesimal |] ==> ( *f* sqrt) x : Infinitesimal";
-by (auto_tac (claset(),simpset() addsimps [hypreal_le_eq_less_or_eq]));
+by (auto_tac (claset(),simpset() addsimps [order_le_less]));
by (rtac (Infinitesimal_square_iff RS iffD2) 1);
by (dtac hypreal_sqrt_gt_zero_pow2 1);
by (auto_tac (claset(),simpset() addsimps [numeral_2_eq_2]));
qed "Infinitesimal_hypreal_sqrt";
Goal "[| 0 <= x; ( *f* sqrt) x : Infinitesimal |] ==> x : Infinitesimal";
-by (auto_tac (claset(),simpset() addsimps [hypreal_le_eq_less_or_eq]));
+by (auto_tac (claset(),simpset() addsimps [order_le_less]));
by (dtac (Infinitesimal_square_iff RS iffD1) 1);
by (dtac hypreal_sqrt_gt_zero_pow2 1);
by (auto_tac (claset(),simpset() addsimps [numeral_2_eq_2]
@@ -203,14 +203,14 @@
Addsimps [Infinitesimal_sqrt_sum_squares];
Goal "[| 0 <= x; x : HInfinite |] ==> ( *f* sqrt) x : HInfinite";
-by (auto_tac (claset(),simpset() addsimps [hypreal_le_eq_less_or_eq]));
+by (auto_tac (claset(),simpset() addsimps [order_le_less]));
by (rtac (HInfinite_square_iff RS iffD1) 1);
by (dtac hypreal_sqrt_gt_zero_pow2 1);
by (auto_tac (claset(),simpset() addsimps [numeral_2_eq_2]));
qed "HInfinite_hypreal_sqrt";
Goal "[| 0 <= x; ( *f* sqrt) x : HInfinite |] ==> x : HInfinite";
-by (auto_tac (claset(),simpset() addsimps [hypreal_le_eq_less_or_eq]));
+by (auto_tac (claset(),simpset() addsimps [order_le_less]));
by (dtac (HInfinite_square_iff RS iffD2) 1);
by (dtac hypreal_sqrt_gt_zero_pow2 1);
by (auto_tac (claset(),simpset() addsimps [numeral_2_eq_2]
--- a/src/HOL/Hyperreal/HyperArith.thy Wed Jan 28 17:01:01 2004 +0100
+++ b/src/HOL/Hyperreal/HyperArith.thy Thu Jan 29 16:51:17 2004 +0100
@@ -143,6 +143,9 @@
lemma hypreal_add_minus_eq_minus: "x + y = (0::hypreal) ==> x = -y"
by arith
+lemma hypreal_le_add_order: "[| 0 \<le> x; 0 \<le> y |] ==> (0::hypreal) \<le> x + y"
+by arith
+
subsubsection{*Division By @{term 1} and @{term "-1"}*}
@@ -197,6 +200,7 @@
ML
{*
val hypreal_add_minus_eq_minus = thm "hypreal_add_minus_eq_minus";
+val hypreal_le_add_order = thm"hypreal_le_add_order";
*}
end
--- a/src/HOL/Hyperreal/HyperDef.thy Wed Jan 28 17:01:01 2004 +0100
+++ b/src/HOL/Hyperreal/HyperDef.thy Thu Jan 29 16:51:17 2004 +0100
@@ -67,7 +67,7 @@
epsilon :: hypreal ("\<epsilon>")
-defs
+defs (overloaded)
hypreal_add_def:
"P + Q == Abs_hypreal(\<Union>X \<in> Rep_hypreal(P). \<Union>Y \<in> Rep_hypreal(Q).
@@ -77,12 +77,12 @@
"P * Q == Abs_hypreal(\<Union>X \<in> Rep_hypreal(P). \<Union>Y \<in> Rep_hypreal(Q).
hyprel``{%n::nat. X n * Y n})"
- hypreal_less_def:
- "P < (Q::hypreal) == \<exists>X Y. X \<in> Rep_hypreal(P) &
+ hypreal_le_def:
+ "P \<le> (Q::hypreal) == \<exists>X Y. X \<in> Rep_hypreal(P) &
Y \<in> Rep_hypreal(Q) &
- {n::nat. X n < Y n} \<in> FreeUltrafilterNat"
- hypreal_le_def:
- "P \<le> (Q::hypreal) == ~(Q < P)"
+ {n::nat. X n \<le> Y n} \<in> FreeUltrafilterNat"
+
+ hypreal_less_def: "(x < (y::hypreal)) == (x \<le> y & x \<noteq> y)"
hrabs_def: "abs (r::hypreal) == (if 0 \<le> r then r else -r)"
@@ -494,205 +494,88 @@
qed
-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.*}
-
-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_less_not_refl: "~ (R::hypreal) < R"
-apply (rule_tac z = R in eq_Abs_hypreal)
-apply (auto simp add: hypreal_less_def, ultra)
-done
-
-lemmas hypreal_less_irrefl = hypreal_less_not_refl [THEN notE, standard]
-declare hypreal_less_irrefl [elim!]
-
-lemma hypreal_not_refl2: "!!(x::hypreal). x < y ==> x \<noteq> y"
-by (auto simp add: hypreal_less_not_refl)
-
-lemma hypreal_less_trans: "!!(R1::hypreal). [| R1 < R2; R2 < R3 |] ==> R1 < R3"
-apply (rule_tac z = R1 in eq_Abs_hypreal)
-apply (rule_tac z = R2 in eq_Abs_hypreal)
-apply (rule_tac z = R3 in eq_Abs_hypreal)
-apply (auto intro!: exI simp add: hypreal_less_def, ultra)
-done
-
-lemma hypreal_less_asym: "!! (R1::hypreal). [| R1 < R2; R2 < R1 |] ==> P"
-apply (drule hypreal_less_trans, assumption)
-apply (simp add: hypreal_less_not_refl)
-done
-
-
-subsection{*Trichotomy: the hyperreals are Linearly Ordered*}
-
-lemma lemma_hyprel_0_mem: "\<exists>x. x \<in> hyprel `` {%n. 0}"
-apply (unfold hyprel_def)
-apply (rule_tac x = "%n. 0" in exI, safe)
-apply (auto intro!: FreeUltrafilterNat_Nat_set)
-done
-
-lemma hypreal_trichotomy: "0 < x | x = 0 | x < (0::hypreal)"
-apply (unfold hypreal_zero_def)
-apply (rule_tac z = x in eq_Abs_hypreal)
-apply (auto simp add: hypreal_less_def)
-apply (cut_tac lemma_hyprel_0_mem, erule exE)
-apply (drule_tac x = xa in spec)
-apply (drule_tac x = x in spec)
-apply (cut_tac x = x in lemma_hyprel_refl, auto)
-apply (drule_tac x = x in spec)
-apply (drule_tac x = xa in spec, auto, ultra)
-done
-
-lemma hypreal_trichotomyE:
- "[| (0::hypreal) < x ==> P;
- x = 0 ==> P;
- x < 0 ==> P |] ==> P"
-apply (insert hypreal_trichotomy [of x], blast)
-done
-
-lemma hypreal_less_minus_iff: "((x::hypreal) < y) = (0 < y + -x)"
-apply (rule_tac z = x in eq_Abs_hypreal)
-apply (rule_tac z = y in eq_Abs_hypreal)
-apply (auto simp add: hypreal_add hypreal_zero_def hypreal_minus hypreal_less)
-done
-
-lemma hypreal_less_minus_iff2: "((x::hypreal) < y) = (x + -y < 0)"
-apply (rule_tac z = x in eq_Abs_hypreal)
-apply (rule_tac z = y in eq_Abs_hypreal)
-apply (auto simp add: hypreal_add hypreal_zero_def hypreal_minus hypreal_less)
-done
-
-lemma hypreal_eq_minus_iff2: "((x::hypreal) = y) = (0 = y + - x)"
-apply auto
-apply (rule Ring_and_Field.add_right_cancel [of _ "-x", THEN iffD1], auto)
-done
-
-lemma hypreal_linear: "(x::hypreal) < y | x = y | y < x"
-apply (subst hypreal_eq_minus_iff2)
-apply (rule_tac x1 = x in hypreal_less_minus_iff [THEN ssubst])
-apply (rule_tac x1 = y in hypreal_less_minus_iff2 [THEN ssubst])
-apply (rule hypreal_trichotomyE, auto)
-done
-
-lemma hypreal_neq_iff: "((w::hypreal) \<noteq> z) = (w<z | z<w)"
-by (cut_tac hypreal_linear, blast)
-
-
subsection{*Properties of The @{text "\<le>"} Relation*}
lemma hypreal_le:
"(Abs_hypreal(hyprel``{%n. X n}) \<le> Abs_hypreal(hyprel``{%n. Y n})) =
({n. X n \<le> Y n} \<in> FreeUltrafilterNat)"
-apply (auto simp add: hypreal_less hypreal_le_def linorder_not_less[symmetric])
-apply (ultra+)
-done
-
-lemma hypreal_le_imp_less_or_eq: "!!(x::hypreal). x \<le> y ==> x < y | x = y"
apply (unfold hypreal_le_def)
-apply (cut_tac hypreal_linear)
-apply (fast elim: hypreal_less_irrefl hypreal_less_asym)
+apply (auto intro!: lemma_hyprel_refl)
+apply (ultra)
done
-lemma hypreal_less_or_eq_imp_le: "z<w | z=w ==> z \<le>(w::hypreal)"
-apply (unfold hypreal_le_def)
-apply (cut_tac hypreal_linear)
-apply (fast elim: hypreal_less_irrefl hypreal_less_asym)
-done
-
-lemma hypreal_le_eq_less_or_eq: "(x \<le> (y::hypreal)) = (x < y | x=y)"
-by (blast intro!: hypreal_less_or_eq_imp_le dest: hypreal_le_imp_less_or_eq)
-
-lemmas hypreal_le_less = hypreal_le_eq_less_or_eq
-
lemma hypreal_le_refl: "w \<le> (w::hypreal)"
-by (simp add: hypreal_le_eq_less_or_eq)
-
-(* Axiom 'linorder_linear' of class 'linorder': *)
-lemma hypreal_le_linear: "(z::hypreal) \<le> w | w \<le> z"
-apply (simp add: hypreal_le_less)
-apply (cut_tac hypreal_linear, blast)
+apply (rule eq_Abs_hypreal [of w])
+apply (simp add: hypreal_le)
done
lemma hypreal_le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::hypreal)"
-apply (drule hypreal_le_imp_less_or_eq)
-apply (drule hypreal_le_imp_less_or_eq)
-apply (rule hypreal_less_or_eq_imp_le)
-apply (blast intro: hypreal_less_trans)
+apply (rule eq_Abs_hypreal [of i])
+apply (rule eq_Abs_hypreal [of j])
+apply (rule eq_Abs_hypreal [of k])
+apply (simp add: hypreal_le)
+apply ultra
done
lemma hypreal_le_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::hypreal)"
-apply (drule hypreal_le_imp_less_or_eq)
-apply (drule hypreal_le_imp_less_or_eq)
-apply (fast elim: hypreal_less_irrefl hypreal_less_asym)
+apply (rule eq_Abs_hypreal [of z])
+apply (rule eq_Abs_hypreal [of w])
+apply (simp add: hypreal_le)
+apply ultra
done
(* Axiom 'order_less_le' of class 'order': *)
lemma hypreal_less_le: "((w::hypreal) < z) = (w \<le> z & w \<noteq> z)"
-apply (simp add: hypreal_le_def hypreal_neq_iff)
-apply (blast intro: hypreal_less_asym)
+apply (simp add: hypreal_less_def)
done
instance hypreal :: order
- by (intro_classes,
- (assumption |
- rule hypreal_le_refl hypreal_le_trans hypreal_le_anti_sym
- hypreal_less_le)+)
+proof qed
+ (assumption |
+ rule hypreal_le_refl hypreal_le_trans hypreal_le_anti_sym hypreal_less_le)+
+
+
+(* Axiom 'linorder_linear' of class 'linorder': *)
+lemma hypreal_le_linear: "(z::hypreal) \<le> w | w \<le> z"
+apply (rule eq_Abs_hypreal [of z])
+apply (rule eq_Abs_hypreal [of w])
+apply (auto simp add: hypreal_le)
+apply ultra
+done
instance hypreal :: linorder
by (intro_classes, rule hypreal_le_linear)
-
-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_not_refl2: "!!(x::hypreal). x < y ==> x \<noteq> y"
+by (auto simp add: order_less_irrefl)
-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: right_distrib hypreal_mult_commute)
+lemma hypreal_add_left_mono: "x \<le> y ==> z + x \<le> z + (y::hypreal)"
+apply (rule eq_Abs_hypreal [of x])
+apply (rule eq_Abs_hypreal [of y])
+apply (rule eq_Abs_hypreal [of z])
+apply (auto simp add: hypreal_le hypreal_add)
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)
+apply (rule eq_Abs_hypreal [of x])
+apply (rule eq_Abs_hypreal [of y])
+apply (rule eq_Abs_hypreal [of z])
+apply (auto simp add: hypreal_zero_def hypreal_le hypreal_mult
+ linorder_not_le [symmetric])
+apply ultra
done
+
subsection{*The Hyperreals Form an Ordered Field*}
instance hypreal :: ordered_field
proof
fix x y z :: hypreal
show "0 < (1::hypreal)"
- by (unfold hypreal_one_def hypreal_zero_def hypreal_less_def, force)
+ by (simp add: hypreal_zero_def hypreal_one_def linorder_not_le [symmetric],
+ simp add: hypreal_le)
show "x \<le> y ==> z + x \<le> z + y"
- by (rule hypreal_add_left_le_mono1)
+ by (rule hypreal_add_left_mono)
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)"
@@ -783,17 +666,17 @@
lemma hypreal_of_real_zero [simp]: "hypreal_of_real 0 = 0"
by (unfold hypreal_of_real_def hypreal_zero_def, simp)
-lemma hypreal_of_real_less_iff [simp]:
- "(hypreal_of_real w < hypreal_of_real z) = (w < z)"
-apply (unfold hypreal_less_def hypreal_of_real_def, auto)
+lemma hypreal_of_real_le_iff [simp]:
+ "(hypreal_of_real w \<le> hypreal_of_real z) = (w \<le> z)"
+apply (unfold hypreal_le_def hypreal_of_real_def, auto)
apply (rule_tac [2] x = "%n. w" in exI, safe)
apply (rule_tac [3] x = "%n. z" in exI, auto)
apply (rule FreeUltrafilterNat_P, ultra)
done
-lemma hypreal_of_real_le_iff [simp]:
- "(hypreal_of_real w \<le> hypreal_of_real z) = (w \<le> z)"
-by (force simp add: hypreal_less hypreal_le_def linorder_not_less[symmetric])
+lemma hypreal_of_real_less_iff [simp]:
+ "(hypreal_of_real w < hypreal_of_real z) = (w < z)"
+by (simp add: linorder_not_le [symmetric])
lemma hypreal_of_real_eq_iff [simp]:
"(hypreal_of_real w = hypreal_of_real z) = (w = z)"
@@ -821,14 +704,11 @@
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
+by (auto simp add: hypreal_of_real_def hypreal_minus)
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 (case_tac "r=0", simp)
apply (rule_tac c1 = "hypreal_of_real r" in hypreal_mult_left_cancel [THEN iffD1])
apply (auto simp add: hypreal_of_real_mult [symmetric])
done
@@ -840,6 +720,13 @@
subsection{*Misc Others*}
+lemma hypreal_less:
+ "(Abs_hypreal(hyprel``{%n. X n}) < Abs_hypreal(hyprel``{%n. Y n})) =
+ ({n. X n < Y n} \<in> FreeUltrafilterNat)"
+apply (auto simp add: hypreal_le linorder_not_le [symmetric])
+apply ultra+
+done
+
lemma hypreal_zero_num: "0 = Abs_hypreal (hyprel `` {%n. 0})"
by (simp add: hypreal_zero_def [THEN meta_eq_to_obj_eq, symmetric])
@@ -851,7 +738,6 @@
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)})"
@@ -859,6 +745,74 @@
apply (ultra, arith)+
done
+
+
+lemma hypreal_add_zero_less_le_mono: "[|r < x; (0::hypreal) \<le> y|] ==> r < x+y"
+by (auto dest: add_less_le_mono)
+
+text{*The precondition could be weakened to @{term "0\<le>x"}*}
+lemma hypreal_mult_less_mono:
+ "[| u<v; x<y; (0::hypreal) < v; 0 < x |] ==> u*x < v* y"
+ by (simp add: Ring_and_Field.mult_strict_mono order_less_imp_le)
+
+
+subsection{*Existence of Infinite Hyperreal Number*}
+
+lemma Rep_hypreal_omega: "Rep_hypreal(omega) \<in> hypreal"
+apply (unfold omega_def)
+apply (rule Rep_hypreal)
+done
+
+text{*Existence of infinite number not corresponding to any real number.
+Use assumption that member @{term FreeUltrafilterNat} is not finite.*}
+
+
+text{*A few lemmas first*}
+
+lemma lemma_omega_empty_singleton_disj: "{n::nat. x = real n} = {} |
+ (\<exists>y. {n::nat. x = real n} = {y})"
+by (force dest: inj_real_of_nat [THEN injD])
+
+lemma lemma_finite_omega_set: "finite {n::nat. x = real n}"
+by (cut_tac x = x in lemma_omega_empty_singleton_disj, auto)
+
+lemma not_ex_hypreal_of_real_eq_omega:
+ "~ (\<exists>x. hypreal_of_real x = omega)"
+apply (unfold omega_def hypreal_of_real_def)
+apply (auto simp add: real_of_nat_Suc diff_eq_eq [symmetric]
+ lemma_finite_omega_set [THEN FreeUltrafilterNat_finite])
+done
+
+lemma hypreal_of_real_not_eq_omega: "hypreal_of_real x \<noteq> omega"
+by (cut_tac not_ex_hypreal_of_real_eq_omega, auto)
+
+text{*Existence of infinitesimal number also not corresponding to any
+ real number*}
+
+lemma lemma_epsilon_empty_singleton_disj:
+ "{n::nat. x = inverse(real(Suc n))} = {} |
+ (\<exists>y. {n::nat. x = inverse(real(Suc n))} = {y})"
+by (auto simp add: inj_real_of_nat [THEN inj_eq])
+
+lemma lemma_finite_epsilon_set: "finite {n. x = inverse(real(Suc n))}"
+by (cut_tac x = x in lemma_epsilon_empty_singleton_disj, auto)
+
+lemma not_ex_hypreal_of_real_eq_epsilon:
+ "~ (\<exists>x. hypreal_of_real x = epsilon)"
+apply (unfold epsilon_def hypreal_of_real_def)
+apply (auto simp add: lemma_finite_epsilon_set [THEN FreeUltrafilterNat_finite])
+done
+
+lemma hypreal_of_real_not_eq_epsilon: "hypreal_of_real x \<noteq> epsilon"
+by (cut_tac not_ex_hypreal_of_real_eq_epsilon, auto)
+
+lemma hypreal_epsilon_not_zero: "epsilon \<noteq> 0"
+by (unfold epsilon_def hypreal_zero_def, auto)
+
+lemma hypreal_epsilon_inverse_omega: "epsilon = inverse(omega)"
+by (simp add: hypreal_inverse omega_def epsilon_def)
+
+
ML
{*
val hrabs_def = thm "hrabs_def";
@@ -946,23 +900,12 @@
val hypreal_mult_not_0 = thm "hypreal_mult_not_0";
val hypreal_minus_inverse = thm "hypreal_minus_inverse";
val hypreal_inverse_distrib = thm "hypreal_inverse_distrib";
-val hypreal_less_not_refl = thm "hypreal_less_not_refl";
-val hypreal_less_irrefl = thm"hypreal_less_irrefl";
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_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_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";
val hypreal_le = thm "hypreal_le";
-val hypreal_le_imp_less_or_eq = thm "hypreal_le_imp_less_or_eq";
-val hypreal_le_eq_less_or_eq = thm "hypreal_le_eq_less_or_eq";
val hypreal_le_refl = thm "hypreal_le_refl";
val hypreal_le_linear = thm "hypreal_le_linear";
val hypreal_le_trans = thm "hypreal_le_trans";
@@ -984,6 +927,17 @@
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";
+
+val hypreal_add_zero_less_le_mono = thm"hypreal_add_zero_less_le_mono";
+val Rep_hypreal_omega = thm"Rep_hypreal_omega";
+val lemma_omega_empty_singleton_disj = thm"lemma_omega_empty_singleton_disj";
+val lemma_finite_omega_set = thm"lemma_finite_omega_set";
+val not_ex_hypreal_of_real_eq_omega = thm"not_ex_hypreal_of_real_eq_omega";
+val hypreal_of_real_not_eq_omega = thm"hypreal_of_real_not_eq_omega";
+val not_ex_hypreal_of_real_eq_epsilon = thm"not_ex_hypreal_of_real_eq_epsilon";
+val hypreal_of_real_not_eq_epsilon = thm"hypreal_of_real_not_eq_epsilon";
+val hypreal_epsilon_not_zero = thm"hypreal_epsilon_not_zero";
+val hypreal_epsilon_inverse_omega = thm"hypreal_epsilon_inverse_omega";
*}
end
--- a/src/HOL/Hyperreal/HyperNat.ML Wed Jan 28 17:01:01 2004 +0100
+++ b/src/HOL/Hyperreal/HyperNat.ML Thu Jan 29 16:51:17 2004 +0100
@@ -5,6 +5,12 @@
ultrafilters
*)
+fun CLAIM_SIMP str thms =
+ prove_goal (the_context()) str
+ (fn prems => [cut_facts_tac prems 1,
+ auto_tac (claset(),simpset() addsimps thms)]);
+fun CLAIM str = CLAIM_SIMP str [];
+
(* blast confuses different versions of < *)
DelIffs [order_less_irrefl];
Addsimps [order_less_irrefl];
--- a/src/HOL/Hyperreal/HyperOrd.thy Wed Jan 28 17:01:01 2004 +0100
+++ b/src/HOL/Hyperreal/HyperOrd.thy Thu Jan 29 16:51:17 2004 +0100
@@ -1,124 +1,9 @@
-(* Title: Real/Hyperreal/HyperOrd.thy
- Author: Jacques D. Fleuriot
- Copyright: 2000 University of Edinburgh
- Description: Type "hypreal" is a linear order and also
- satisfies plus_ac0: + is an AC-operator with zero
-*)
-
theory HyperOrd = HyperDef:
-(*** Monotonicity results ***)
-
-lemma hypreal_add_less_le_mono: "[|(i::hypreal)<j; k\<le>l |] ==> i + k < j + l"
-by (auto dest!: order_le_imp_less_or_eq intro: hypreal_add_less_mono1 add_strict_mono)
-
-lemma hypreal_add_less_mono2: "!!(A::hypreal). A < B ==> C + A < C + B"
-by (auto intro: hypreal_add_less_mono1 simp add: hypreal_add_commute)
-
-lemma hypreal_add_le_less_mono: "[|(i::hypreal)\<le>j; k<l |] ==> i + k < j + l"
-apply (erule add_right_mono [THEN order_le_less_trans])
-apply (erule add_strict_left_mono)
-done
-
-lemma hypreal_add_zero_less_le_mono: "[|r < x; (0::hypreal) \<le> y|] ==> r < x + y"
-by (auto dest: hypreal_add_less_le_mono)
-
-lemma hypreal_add_order: "[| 0 < x; 0 < y |] ==> (0::hypreal) < x + y"
-apply (erule order_less_trans)
-apply (drule hypreal_add_less_mono2, simp)
-done
-
-lemma hypreal_le_add_order: "[| 0 \<le> x; 0 \<le> y |] ==> (0::hypreal) \<le> x + y"
-apply (drule order_le_imp_less_or_eq)+
-apply (auto intro: hypreal_add_order order_less_imp_le)
-done
-
-text{*The precondition could be weakened to @{term "0\<le>x"}*}
-lemma hypreal_mult_less_mono:
- "[| u<v; x<y; (0::hypreal) < v; 0 < x |] ==> u*x < v* y"
- by (simp add: Ring_and_Field.mult_strict_mono order_less_imp_le)
-
-
-subsection{*Existence of Infinite Hyperreal Number*}
-
-lemma Rep_hypreal_omega: "Rep_hypreal(omega) \<in> hypreal"
-apply (unfold omega_def)
-apply (rule Rep_hypreal)
-done
-
-text{*Existence of infinite number not corresponding to any real number.
-Use assumption that member @{term FreeUltrafilterNat} is not finite.*}
-
-
-text{*A few lemmas first*}
-
-lemma lemma_omega_empty_singleton_disj: "{n::nat. x = real n} = {} |
- (\<exists>y. {n::nat. x = real n} = {y})"
-by (force dest: inj_real_of_nat [THEN injD])
-
-lemma lemma_finite_omega_set: "finite {n::nat. x = real n}"
-by (cut_tac x = x in lemma_omega_empty_singleton_disj, auto)
-
-lemma not_ex_hypreal_of_real_eq_omega:
- "~ (\<exists>x. hypreal_of_real x = omega)"
-apply (unfold omega_def hypreal_of_real_def)
-apply (auto simp add: real_of_nat_Suc diff_eq_eq [symmetric]
- lemma_finite_omega_set [THEN FreeUltrafilterNat_finite])
-done
-
-lemma hypreal_of_real_not_eq_omega: "hypreal_of_real x \<noteq> omega"
-by (cut_tac not_ex_hypreal_of_real_eq_omega, auto)
-
-text{*Existence of infinitesimal number also not corresponding to any
- real number*}
-
-lemma lemma_epsilon_empty_singleton_disj:
- "{n::nat. x = inverse(real(Suc n))} = {} |
- (\<exists>y. {n::nat. x = inverse(real(Suc n))} = {y})"
-apply (auto simp add: inj_real_of_nat [THEN inj_eq])
-done
-
-lemma lemma_finite_epsilon_set: "finite {n. x = inverse(real(Suc n))}"
-by (cut_tac x = x in lemma_epsilon_empty_singleton_disj, auto)
-
-lemma not_ex_hypreal_of_real_eq_epsilon:
- "~ (\<exists>x. hypreal_of_real x = epsilon)"
-apply (unfold epsilon_def hypreal_of_real_def)
-apply (auto simp add: lemma_finite_epsilon_set [THEN FreeUltrafilterNat_finite])
-done
-
-lemma hypreal_of_real_not_eq_epsilon: "hypreal_of_real x \<noteq> epsilon"
-by (cut_tac not_ex_hypreal_of_real_eq_epsilon, auto)
-
-lemma hypreal_epsilon_not_zero: "epsilon \<noteq> 0"
-by (unfold epsilon_def hypreal_zero_def, auto)
-
-lemma hypreal_epsilon_inverse_omega: "epsilon = inverse(omega)"
-by (simp add: hypreal_inverse omega_def epsilon_def)
-
ML
{*
-val hypreal_add_less_mono1 = thm"hypreal_add_less_mono1";
-val hypreal_mult_order = thm"hypreal_mult_order";
-val hypreal_le_add_order = thm"hypreal_le_add_order";
-val hypreal_add_left_le_mono1 = thm"hypreal_add_left_le_mono1";
-val hypreal_add_less_le_mono = thm"hypreal_add_less_le_mono";
-val hypreal_add_le_less_mono = thm"hypreal_add_le_less_mono";
-val hypreal_add_zero_less_le_mono = thm"hypreal_add_zero_less_le_mono";
-val hypreal_mult_less_mono1 = thm"hypreal_mult_less_mono1";
-val hypreal_mult_less_mono2 = thm"hypreal_mult_less_mono2";
-val hypreal_mult_less_mono = thm"hypreal_mult_less_mono";
-val Rep_hypreal_omega = thm"Rep_hypreal_omega";
-val lemma_omega_empty_singleton_disj = thm"lemma_omega_empty_singleton_disj";
-val lemma_finite_omega_set = thm"lemma_finite_omega_set";
-val not_ex_hypreal_of_real_eq_omega = thm"not_ex_hypreal_of_real_eq_omega";
-val hypreal_of_real_not_eq_omega = thm"hypreal_of_real_not_eq_omega";
-val not_ex_hypreal_of_real_eq_epsilon = thm"not_ex_hypreal_of_real_eq_epsilon";
-val hypreal_of_real_not_eq_epsilon = thm"hypreal_of_real_not_eq_epsilon";
-val hypreal_epsilon_not_zero = thm"hypreal_epsilon_not_zero";
-val hypreal_epsilon_inverse_omega = thm"hypreal_epsilon_inverse_omega";
*}
end
--- a/src/HOL/Hyperreal/NSA.ML Wed Jan 28 17:01:01 2004 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,2384 +0,0 @@
-(* Title : NSA.ML
- Author : Jacques D. Fleuriot
- Copyright : 1998 University of Cambridge
- Description : Infinite numbers, Infinitesimals,
- infinitely close relation etc.
-*)
-
-fun CLAIM_SIMP str thms =
- prove_goal (the_context()) str
- (fn prems => [cut_facts_tac prems 1,
- auto_tac (claset(),simpset() addsimps thms)]);
-fun CLAIM str = CLAIM_SIMP str [];
-
-(*--------------------------------------------------------------------
- Closure laws for members of (embedded) set standard real Reals
- --------------------------------------------------------------------*)
-
-Goalw [SReal_def] "[| (x::hypreal): Reals; y: Reals |] ==> x + y: Reals";
-by (Step_tac 1);
-by (res_inst_tac [("x","r + ra")] exI 1);
-by (Simp_tac 1);
-qed "SReal_add";
-
-Goalw [SReal_def] "[| (x::hypreal): Reals; y: Reals |] ==> x * y: Reals";
-by (Step_tac 1);
-by (res_inst_tac [("x","r * ra")] exI 1);
-by (simp_tac (simpset() addsimps [hypreal_of_real_mult]) 1);
-qed "SReal_mult";
-
-Goalw [SReal_def] "(x::hypreal): Reals ==> inverse x : Reals";
-by (blast_tac (claset() addIs [hypreal_of_real_inverse RS sym]) 1);
-qed "SReal_inverse";
-
-Goal "[| (x::hypreal): Reals; y: Reals |] ==> x/y: Reals";
-by (asm_simp_tac (simpset() addsimps [SReal_mult,SReal_inverse,
- hypreal_divide_def]) 1);
-qed "SReal_divide";
-
-Goalw [SReal_def] "(x::hypreal): Reals ==> -x : Reals";
-by (blast_tac (claset() addIs [hypreal_of_real_minus RS sym]) 1);
-qed "SReal_minus";
-
-Goal "(-x : Reals) = ((x::hypreal): Reals)";
-by Auto_tac;
-by (etac SReal_minus 2);
-by (dtac SReal_minus 1);
-by Auto_tac;
-qed "SReal_minus_iff";
-Addsimps [SReal_minus_iff];
-
-Goal "[| (x::hypreal) + y : Reals; y: Reals |] ==> x: Reals";
-by (dres_inst_tac [("x","y")] SReal_minus 1);
-by (dtac SReal_add 1);
-by (assume_tac 1);
-by Auto_tac;
-qed "SReal_add_cancel";
-
-Goalw [SReal_def] "(x::hypreal): Reals ==> abs x : Reals";
-by (auto_tac (claset(), simpset() addsimps [hypreal_of_real_hrabs]));
-qed "SReal_hrabs";
-
-Goalw [SReal_def] "hypreal_of_real x: Reals";
-by (Blast_tac 1);
-qed "SReal_hypreal_of_real";
-Addsimps [SReal_hypreal_of_real];
-
-Goalw [hypreal_number_of_def] "(number_of w ::hypreal) : Reals";
-by (rtac SReal_hypreal_of_real 1);
-qed "SReal_number_of";
-Addsimps [SReal_number_of];
-
-(** As always with numerals, 0 and 1 are special cases **)
-
-Goal "(0::hypreal) : Reals";
-by (stac (hypreal_numeral_0_eq_0 RS sym) 1);
-by (rtac SReal_number_of 1);
-qed "Reals_0";
-Addsimps [Reals_0];
-
-Goal "(1::hypreal) : Reals";
-by (stac (hypreal_numeral_1_eq_1 RS sym) 1);
-by (rtac SReal_number_of 1);
-qed "Reals_1";
-Addsimps [Reals_1];
-
-Goalw [hypreal_divide_def] "r : Reals ==> r/(number_of w::hypreal) : Reals";
-by (blast_tac (claset() addSIs [SReal_number_of, SReal_mult,
- SReal_inverse]) 1);
-qed "SReal_divide_number_of";
-
-(* Infinitesimal epsilon not in Reals *)
-
-Goalw [SReal_def] "epsilon ~: Reals";
-by (auto_tac (claset(),
- simpset() addsimps [hypreal_of_real_not_eq_epsilon RS not_sym]));
-qed "SReal_epsilon_not_mem";
-
-Goalw [SReal_def] "omega ~: Reals";
-by (auto_tac (claset(),
- simpset() addsimps [hypreal_of_real_not_eq_omega RS not_sym]));
-qed "SReal_omega_not_mem";
-
-Goalw [SReal_def] "{x. hypreal_of_real x : Reals} = (UNIV::real set)";
-by Auto_tac;
-qed "SReal_UNIV_real";
-
-Goalw [SReal_def] "(x: Reals) = (EX y. x = hypreal_of_real y)";
-by Auto_tac;
-qed "SReal_iff";
-
-Goalw [SReal_def] "hypreal_of_real `(UNIV::real set) = Reals";
-by Auto_tac;
-qed "hypreal_of_real_image";
-
-Goalw [SReal_def] "inv hypreal_of_real `Reals = (UNIV::real set)";
-by Auto_tac;
-by (rtac (inj_hypreal_of_real RS inv_f_f RS subst) 1);
-by (Blast_tac 1);
-qed "inv_hypreal_of_real_image";
-
-Goalw [SReal_def]
- "[| EX x. x: P; P <= Reals |] ==> EX Q. P = hypreal_of_real ` Q";
-by (Best_tac 1);
-qed "SReal_hypreal_of_real_image";
-
-Goal "[| (x::hypreal): Reals; y: Reals; x<y |] ==> EX r: Reals. x<r & r<y";
-by (auto_tac (claset(), simpset() addsimps [SReal_iff]));
-by (dtac real_dense 1 THEN Step_tac 1);
-by (res_inst_tac [("x","hypreal_of_real r")] bexI 1);
-by Auto_tac;
-qed "SReal_dense";
-
-(*------------------------------------------------------------------
- Completeness of Reals
- ------------------------------------------------------------------*)
-Goal "P <= Reals ==> ((EX x:P. y < x) = \
-\ (EX X. hypreal_of_real X : P & y < hypreal_of_real X))";
-by (blast_tac (claset() addSDs [SReal_iff RS iffD1]) 1);
-by (flexflex_tac );
-qed "SReal_sup_lemma";
-
-Goal "[| P <= Reals; EX x. x: P; EX y : Reals. ALL x: P. x < y |] \
-\ ==> (EX X. X: {w. hypreal_of_real w : P}) & \
-\ (EX Y. ALL X: {w. hypreal_of_real w : P}. X < Y)";
-by (rtac conjI 1);
-by (fast_tac (claset() addSDs [SReal_iff RS iffD1]) 1);
-by (Auto_tac THEN ftac subsetD 1 THEN assume_tac 1);
-by (dtac (SReal_iff RS iffD1) 1);
-by (Auto_tac THEN res_inst_tac [("x","ya")] exI 1);
-by Auto_tac;
-qed "SReal_sup_lemma2";
-
-(*------------------------------------------------------
- lifting of ub and property of lub
- -------------------------------------------------------*)
-Goalw [isUb_def,setle_def]
- "(isUb (Reals) (hypreal_of_real ` Q) (hypreal_of_real Y)) = \
-\ (isUb (UNIV :: real set) Q Y)";
-by Auto_tac;
-qed "hypreal_of_real_isUb_iff";
-
-Goalw [isLub_def,leastP_def]
- "isLub Reals (hypreal_of_real ` Q) (hypreal_of_real Y) \
-\ ==> isLub (UNIV :: real set) Q Y";
-by (auto_tac (claset() addIs [hypreal_of_real_isUb_iff RS iffD2],
- simpset() addsimps [hypreal_of_real_isUb_iff, setge_def]));
-qed "hypreal_of_real_isLub1";
-
-Goalw [isLub_def,leastP_def]
- "isLub (UNIV :: real set) Q Y \
-\ ==> isLub Reals (hypreal_of_real ` Q) (hypreal_of_real Y)";
-by (auto_tac (claset(),
- simpset() addsimps [hypreal_of_real_isUb_iff, setge_def]));
-by (forw_inst_tac [("x2","x")] (isUbD2a RS (SReal_iff RS iffD1) RS exE) 1);
-by (assume_tac 2);
-by (dres_inst_tac [("x","xa")] spec 1);
-by (auto_tac (claset(), simpset() addsimps [hypreal_of_real_isUb_iff]));
-qed "hypreal_of_real_isLub2";
-
-Goal "(isLub Reals (hypreal_of_real ` Q) (hypreal_of_real Y)) = \
-\ (isLub (UNIV :: real set) Q Y)";
-by (blast_tac (claset() addIs [hypreal_of_real_isLub1,
- hypreal_of_real_isLub2]) 1);
-qed "hypreal_of_real_isLub_iff";
-
-(* lemmas *)
-Goalw [isUb_def]
- "isUb Reals P Y ==> EX Yo. isUb Reals P (hypreal_of_real Yo)";
-by (auto_tac (claset(), simpset() addsimps [SReal_iff]));
-qed "lemma_isUb_hypreal_of_real";
-
-Goalw [isLub_def,leastP_def,isUb_def]
- "isLub Reals P Y ==> EX Yo. isLub Reals P (hypreal_of_real Yo)";
-by (auto_tac (claset(), simpset() addsimps [SReal_iff]));
-qed "lemma_isLub_hypreal_of_real";
-
-Goalw [isLub_def,leastP_def,isUb_def]
- "EX Yo. isLub Reals P (hypreal_of_real Yo) ==> EX Y. isLub Reals P Y";
-by Auto_tac;
-qed "lemma_isLub_hypreal_of_real2";
-
-Goal "[| P <= Reals; EX x. x: P; EX Y. isUb Reals P Y |] \
-\ ==> EX t::hypreal. isLub Reals P t";
-by (ftac SReal_hypreal_of_real_image 1);
-by (Auto_tac THEN dtac lemma_isUb_hypreal_of_real 1);
-by (auto_tac (claset() addSIs [reals_complete, lemma_isLub_hypreal_of_real2],
- simpset() addsimps [hypreal_of_real_isLub_iff,hypreal_of_real_isUb_iff]));
-qed "SReal_complete";
-
-(*--------------------------------------------------------------------
- Set of finite elements is a subring of the extended reals
- --------------------------------------------------------------------*)
-Goalw [HFinite_def] "[|x : HFinite; y : HFinite|] ==> (x+y) : HFinite";
-by (blast_tac (claset() addSIs [SReal_add,hrabs_add_less]) 1);
-qed "HFinite_add";
-
-Goalw [HFinite_def] "[|x : HFinite; y : HFinite|] ==> x*y : HFinite";
-by (Asm_full_simp_tac 1);
-by (blast_tac (claset() addSIs [SReal_mult,abs_mult_less]) 1);
-qed "HFinite_mult";
-
-Goalw [HFinite_def] "(-x : HFinite) = (x : HFinite)";
-by (Simp_tac 1);
-qed "HFinite_minus_iff";
-
-Goalw [SReal_def,HFinite_def] "Reals <= HFinite";
-by Auto_tac;
-by (res_inst_tac [("x","1 + abs(hypreal_of_real r)")] exI 1);
-by (auto_tac (claset(), simpset() addsimps [hypreal_of_real_hrabs]));
-by (res_inst_tac [("x","1 + abs r")] exI 1);
-by (Simp_tac 1);
-qed "SReal_subset_HFinite";
-
-Goal "hypreal_of_real x : HFinite";
-by (auto_tac (claset() addIs [(SReal_subset_HFinite RS subsetD)],
- simpset()));
-qed "HFinite_hypreal_of_real";
-
-Addsimps [HFinite_hypreal_of_real];
-
-Goalw [HFinite_def] "x : HFinite ==> EX t: Reals. abs x < t";
-by Auto_tac;
-qed "HFiniteD";
-
-Goalw [HFinite_def] "(abs x : HFinite) = (x : HFinite)";
-by Auto_tac;
-qed "HFinite_hrabs_iff";
-AddIffs [HFinite_hrabs_iff];
-
-Goal "number_of w : HFinite";
-by (rtac (SReal_number_of RS (SReal_subset_HFinite RS subsetD)) 1);
-qed "HFinite_number_of";
-Addsimps [HFinite_number_of];
-
-(** As always with numerals, 0 and 1 are special cases **)
-
-Goal "0 : HFinite";
-by (stac (hypreal_numeral_0_eq_0 RS sym) 1);
-by (rtac HFinite_number_of 1);
-qed "HFinite_0";
-Addsimps [HFinite_0];
-
-Goal "1 : HFinite";
-by (stac (hypreal_numeral_1_eq_1 RS sym) 1);
-by (rtac HFinite_number_of 1);
-qed "HFinite_1";
-Addsimps [HFinite_1];
-
-Goal "[|x : HFinite; y <= x; 0 <= y |] ==> y: HFinite";
-by (case_tac "x <= 0" 1);
-by (dres_inst_tac [("y","x")] order_trans 1);
-by (dtac hypreal_le_anti_sym 2);
-by (auto_tac (claset(), simpset() addsimps [linorder_not_le]));
-by (auto_tac (claset() addSIs [bexI] addIs [order_le_less_trans],
- simpset() addsimps [hrabs_eqI1,hrabs_eqI2,hrabs_minus_eqI1,HFinite_def]));
-qed "HFinite_bounded";
-
-(*------------------------------------------------------------------
- Set of infinitesimals is a subring of the hyperreals
- ------------------------------------------------------------------*)
-Goalw [Infinitesimal_def]
- "x : Infinitesimal ==> ALL r: Reals. 0 < r --> abs x < r";
-by Auto_tac;
-qed "InfinitesimalD";
-
-Goalw [Infinitesimal_def] "0 : Infinitesimal";
-by (Simp_tac 1);
-qed "Infinitesimal_zero";
-AddIffs [Infinitesimal_zero];
-
-Goal "x/(2::hypreal) + x/(2::hypreal) = x";
-by Auto_tac;
-qed "hypreal_sum_of_halves";
-
-Goal "0 < r ==> 0 < r/(2::hypreal)";
-by Auto_tac;
-qed "hypreal_half_gt_zero";
-
-Goalw [Infinitesimal_def]
- "[| x : Infinitesimal; y : Infinitesimal |] ==> (x+y) : Infinitesimal";
-by Auto_tac;
-by (rtac (hypreal_sum_of_halves RS subst) 1);
-by (dtac hypreal_half_gt_zero 1);
-by (blast_tac (claset() addIs [hrabs_add_less, hrabs_add_less,
- SReal_divide_number_of]) 1);
-qed "Infinitesimal_add";
-
-Goalw [Infinitesimal_def] "(-x:Infinitesimal) = (x:Infinitesimal)";
-by (Full_simp_tac 1);
-qed "Infinitesimal_minus_iff";
-Addsimps [Infinitesimal_minus_iff];
-
-Goal "[| x : Infinitesimal; y : Infinitesimal |] ==> x-y : Infinitesimal";
-by (asm_simp_tac
- (simpset() addsimps [hypreal_diff_def, Infinitesimal_add]) 1);
-qed "Infinitesimal_diff";
-
-Goalw [Infinitesimal_def]
- "[| x : Infinitesimal; y : Infinitesimal |] ==> (x * y) : Infinitesimal";
-by Auto_tac;
-by (case_tac "y=0" 1);
-by (cut_inst_tac [("u","abs x"),("v","1"),("x","abs y"),("y","r")]
- hypreal_mult_less_mono 2);
-by Auto_tac;
-qed "Infinitesimal_mult";
-
-Goal "[| x : Infinitesimal; y : HFinite |] ==> (x * y) : Infinitesimal";
-by (auto_tac (claset() addSDs [HFiniteD],
- simpset() addsimps [Infinitesimal_def]));
-by (ftac hrabs_less_gt_zero 1);
-by (dres_inst_tac [("x","r/t")] bspec 1);
-by (blast_tac (claset() addIs [SReal_divide]) 1);
-by (asm_full_simp_tac (simpset() addsimps [thm"Ring_and_Field.zero_less_divide_iff"]) 1);
-by (case_tac "x=0 | y=0" 1);
-by (cut_inst_tac [("u","abs x"),("v","r/t"),("x","abs y")]
- hypreal_mult_less_mono 2);
-by (auto_tac (claset(), simpset() addsimps [thm"Ring_and_Field.zero_less_divide_iff"]));
-qed "Infinitesimal_HFinite_mult";
-
-Goal "[| x : Infinitesimal; y : HFinite |] ==> (y * x) : Infinitesimal";
-by (auto_tac (claset() addDs [Infinitesimal_HFinite_mult],
- simpset() addsimps [hypreal_mult_commute]));
-qed "Infinitesimal_HFinite_mult2";
-
-(*** rather long proof ***)
-Goalw [HInfinite_def,Infinitesimal_def]
- "x: HInfinite ==> inverse x: Infinitesimal";
-by Auto_tac;
-by (eres_inst_tac [("x","inverse r")] ballE 1);
-by (forw_inst_tac [("a1","r"),("z","abs x")]
- (positive_imp_inverse_positive RS order_less_trans) 1);
-by (assume_tac 1);
-by (dtac ((inverse_inverse_eq RS sym) RS subst) 1);
-by (rtac (inverse_less_iff_less RS iffD1) 1);
-by (auto_tac (claset(), simpset() addsimps [SReal_inverse]));
-qed "HInfinite_inverse_Infinitesimal";
-
-
-
-Goalw [HInfinite_def] "[|x: HInfinite;y: HInfinite|] ==> (x*y): HInfinite";
-by Auto_tac;
-by (eres_inst_tac [("x","1")] ballE 1);
-by (eres_inst_tac [("x","r")] ballE 1);
-by (case_tac "y=0" 1);
-by (cut_inst_tac [("x","1"),("y","abs x"),
- ("u","r"),("v","abs y")] hypreal_mult_less_mono 2);
-by (auto_tac (claset(), simpset() addsimps mult_ac));
-qed "HInfinite_mult";
-
-Goalw [HInfinite_def]
- "[|x: HInfinite; 0 <= y; 0 <= x|] ==> (x + y): HInfinite";
-by (auto_tac (claset() addSIs [hypreal_add_zero_less_le_mono],
- simpset() addsimps [hrabs_eqI1, hypreal_add_commute,
- hypreal_le_add_order]));
-qed "HInfinite_add_ge_zero";
-
-Goal "[|x: HInfinite; 0 <= y; 0 <= x|] ==> (y + x): HInfinite";
-by (auto_tac (claset() addSIs [HInfinite_add_ge_zero],
- simpset() addsimps [hypreal_add_commute]));
-qed "HInfinite_add_ge_zero2";
-
-Goal "[|x: HInfinite; 0 < y; 0 < x|] ==> (x + y): HInfinite";
-by (blast_tac (claset() addIs [HInfinite_add_ge_zero,
- order_less_imp_le]) 1);
-qed "HInfinite_add_gt_zero";
-
-Goalw [HInfinite_def] "(-x: HInfinite) = (x: HInfinite)";
-by Auto_tac;
-qed "HInfinite_minus_iff";
-
-Goal "[|x: HInfinite; y <= 0; x <= 0|] ==> (x + y): HInfinite";
-by (dtac (HInfinite_minus_iff RS iffD2) 1);
-by (rtac (HInfinite_minus_iff RS iffD1) 1);
-by (auto_tac (claset() addIs [HInfinite_add_ge_zero],
- simpset()));
-qed "HInfinite_add_le_zero";
-
-Goal "[|x: HInfinite; y < 0; x < 0|] ==> (x + y): HInfinite";
-by (blast_tac (claset() addIs [HInfinite_add_le_zero,
- order_less_imp_le]) 1);
-qed "HInfinite_add_lt_zero";
-
-Goal "[|a: HFinite; b: HFinite; c: HFinite|] \
-\ ==> a*a + b*b + c*c : HFinite";
-by (auto_tac (claset() addIs [HFinite_mult,HFinite_add], simpset()));
-qed "HFinite_sum_squares";
-
-Goal "x ~: Infinitesimal ==> x ~= 0";
-by Auto_tac;
-qed "not_Infinitesimal_not_zero";
-
-Goal "x: HFinite - Infinitesimal ==> x ~= 0";
-by Auto_tac;
-qed "not_Infinitesimal_not_zero2";
-
-Goal "(abs x : Infinitesimal) = (x : Infinitesimal)";
-by (auto_tac (claset(), simpset() addsimps [hrabs_def]));
-qed "Infinitesimal_hrabs_iff";
-AddIffs [Infinitesimal_hrabs_iff];
-
-Goal "x : HFinite - Infinitesimal ==> abs x : HFinite - Infinitesimal";
-by (Blast_tac 1);
-qed "HFinite_diff_Infinitesimal_hrabs";
-
-Goalw [Infinitesimal_def]
- "[| e : Infinitesimal; abs x < e |] ==> x : Infinitesimal";
-by (auto_tac (claset(), simpset() addsimps [abs_less_iff]));
-qed "hrabs_less_Infinitesimal";
-
-Goal "[| e : Infinitesimal; abs x <= e |] ==> x : Infinitesimal";
-by (blast_tac (claset() addDs [order_le_imp_less_or_eq]
- addIs [hrabs_less_Infinitesimal]) 1);
-qed "hrabs_le_Infinitesimal";
-
-Goalw [Infinitesimal_def]
- "[| e : Infinitesimal; \
-\ e' : Infinitesimal; \
-\ e' < x ; x < e |] ==> x : Infinitesimal";
-by (auto_tac (claset(), simpset() addsimps [abs_less_iff]));
-qed "Infinitesimal_interval";
-
-Goal "[| e : Infinitesimal; e' : Infinitesimal; \
-\ e' <= x ; x <= e |] ==> x : Infinitesimal";
-by (auto_tac (claset() addIs [Infinitesimal_interval],
- simpset() addsimps [hypreal_le_eq_less_or_eq]));
-qed "Infinitesimal_interval2";
-
-Goalw [Infinitesimal_def]
- "[| x ~: Infinitesimal; y ~: Infinitesimal|] ==> (x*y) ~:Infinitesimal";
-by (Clarify_tac 1);
-by (asm_full_simp_tac (simpset() addsimps [linorder_not_less]) 1);
-by (eres_inst_tac [("x","r*ra")] ballE 1);
-by (fast_tac (claset() addIs [SReal_mult]) 2);
-by (auto_tac (claset(), simpset() addsimps [zero_less_mult_iff]));
-by (cut_inst_tac [("c","ra"),("d","abs y"),
- ("a","r"),("b","abs x")] mult_mono 1);
-by Auto_tac;
-qed "not_Infinitesimal_mult";
-
-Goal "x*y : Infinitesimal ==> x : Infinitesimal | y : Infinitesimal";
-by (rtac ccontr 1);
-by (dtac (de_Morgan_disj RS iffD1) 1);
-by (fast_tac (claset() addDs [not_Infinitesimal_mult]) 1);
-qed "Infinitesimal_mult_disj";
-
-Goal "x: HFinite-Infinitesimal ==> x ~= 0";
-by (Blast_tac 1);
-qed "HFinite_Infinitesimal_not_zero";
-
-Goal "[| x : HFinite - Infinitesimal; \
-\ y : HFinite - Infinitesimal \
-\ |] ==> (x*y) : HFinite - Infinitesimal";
-by (Clarify_tac 1);
-by (blast_tac (claset() addDs [HFinite_mult,not_Infinitesimal_mult]) 1);
-qed "HFinite_Infinitesimal_diff_mult";
-
-Goalw [Infinitesimal_def,HFinite_def]
- "Infinitesimal <= HFinite";
-by Auto_tac;
-by (res_inst_tac [("x","1")] bexI 1);
-by Auto_tac;
-qed "Infinitesimal_subset_HFinite";
-
-Goal "x: Infinitesimal ==> x * hypreal_of_real r : Infinitesimal";
-by (etac (HFinite_hypreal_of_real RSN
- (2,Infinitesimal_HFinite_mult)) 1);
-qed "Infinitesimal_hypreal_of_real_mult";
-
-Goal "x: Infinitesimal ==> hypreal_of_real r * x: Infinitesimal";
-by (etac (HFinite_hypreal_of_real RSN
- (2,Infinitesimal_HFinite_mult2)) 1);
-qed "Infinitesimal_hypreal_of_real_mult2";
-
-(*----------------------------------------------------------------------
- Infinitely close relation @=
- ----------------------------------------------------------------------*)
-
-Goalw [Infinitesimal_def,approx_def]
- "(x:Infinitesimal) = (x @= 0)";
-by (Simp_tac 1);
-qed "mem_infmal_iff";
-
-Goalw [approx_def]" (x @= y) = (x + -y @= 0)";
-by (Simp_tac 1);
-qed "approx_minus_iff";
-
-Goalw [approx_def]" (x @= y) = (-y + x @= 0)";
-by (simp_tac (simpset() addsimps [hypreal_add_commute]) 1);
-qed "approx_minus_iff2";
-
-Goalw [approx_def,Infinitesimal_def] "x @= x";
-by (Simp_tac 1);
-qed "approx_refl";
-AddIffs [approx_refl];
-
-Goalw [approx_def] "x @= y ==> y @= x";
-by (rtac (hypreal_minus_distrib1 RS subst) 1);
-by (etac (Infinitesimal_minus_iff RS iffD2) 1);
-qed "approx_sym";
-
-Goalw [approx_def] "[| x @= y; y @= z |] ==> x @= z";
-by (dtac Infinitesimal_add 1);
-by (assume_tac 1);
-by Auto_tac;
-qed "approx_trans";
-
-Goal "[| r @= x; s @= x |] ==> r @= s";
-by (blast_tac (claset() addIs [approx_sym, approx_trans]) 1);
-qed "approx_trans2";
-
-Goal "[| x @= r; x @= s|] ==> r @= s";
-by (blast_tac (claset() addIs [approx_sym, approx_trans]) 1);
-qed "approx_trans3";
-
-Goal "(number_of w @= x) = (x @= number_of w)";
-by (blast_tac (claset() addIs [approx_sym]) 1);
-qed "number_of_approx_reorient";
-
-Goal "(0 @= x) = (x @= 0)";
-by (blast_tac (claset() addIs [approx_sym]) 1);
-qed "zero_approx_reorient";
-
-Goal "(1 @= x) = (x @= 1)";
-by (blast_tac (claset() addIs [approx_sym]) 1);
-qed "one_approx_reorient";
-
-
-(*** re-orientation, following HOL/Integ/Bin.ML
- We re-orient x @=y where x is 0, 1 or a numeral, unless y is as well!
- ***)
-
-(*reorientation simprules using ==, for the following simproc*)
-val meta_zero_approx_reorient = zero_approx_reorient RS eq_reflection;
-val meta_one_approx_reorient = one_approx_reorient RS eq_reflection;
-val meta_number_of_approx_reorient = number_of_approx_reorient RS eq_reflection;
-
-(*reorientation simplification procedure: reorients (polymorphic)
- 0 = x, 1 = x, nnn = x provided x isn't 0, 1 or a numeral.*)
-fun reorient_proc sg _ (_ $ t $ u) =
- case u of
- Const("0", _) => None
- | Const("1", _) => None
- | Const("Numeral.number_of", _) $ _ => None
- | _ => Some (case t of
- Const("0", _) => meta_zero_approx_reorient
- | Const("1", _) => meta_one_approx_reorient
- | Const("Numeral.number_of", _) $ _ =>
- meta_number_of_approx_reorient);
-
-val approx_reorient_simproc =
- Bin_Simprocs.prep_simproc
- ("reorient_simproc", ["0@=x", "1@=x", "number_of w @= x"], reorient_proc);
-
-Addsimprocs [approx_reorient_simproc];
-
-
-Goal "(x-y : Infinitesimal) = (x @= y)";
-by (auto_tac (claset(),
- simpset() addsimps [hypreal_diff_def, approx_minus_iff RS sym,
- mem_infmal_iff]));
-qed "Infinitesimal_approx_minus";
-
-Goalw [monad_def] "(x @= y) = (monad(x)=monad(y))";
-by (auto_tac (claset() addDs [approx_sym]
- addSEs [approx_trans,equalityCE],
- simpset()));
-qed "approx_monad_iff";
-
-Goal "[| x: Infinitesimal; y: Infinitesimal |] ==> x @= y";
-by (asm_full_simp_tac (simpset() addsimps [mem_infmal_iff]) 1);
-by (blast_tac (claset() addIs [approx_trans, approx_sym]) 1);
-qed "Infinitesimal_approx";
-
-val prem1::prem2::rest =
-goalw thy [approx_def] "[| a @= b; c @= d |] ==> a+c @= b+d";
-by (stac minus_add_distrib 1);
-by (stac hypreal_add_assoc 1);
-by (res_inst_tac [("b1","c")] (add_left_commute RS subst) 1);
-by (rtac (hypreal_add_assoc RS subst) 1);
-by (rtac ([prem1,prem2] MRS Infinitesimal_add) 1);
-qed "approx_add";
-
-Goal "a @= b ==> -a @= -b";
-by (rtac ((approx_minus_iff RS iffD2) RS approx_sym) 1);
-by (dtac (approx_minus_iff RS iffD1) 1);
-by (simp_tac (simpset() addsimps [hypreal_add_commute]) 1);
-qed "approx_minus";
-
-Goal "-a @= -b ==> a @= b";
-by (auto_tac (claset() addDs [approx_minus], simpset()));
-qed "approx_minus2";
-
-Goal "(-a @= -b) = (a @= b)";
-by (blast_tac (claset() addIs [approx_minus,approx_minus2]) 1);
-qed "approx_minus_cancel";
-
-Addsimps [approx_minus_cancel];
-
-Goal "[| a @= b; c @= d |] ==> a + -c @= b + -d";
-by (blast_tac (claset() addSIs [approx_add,approx_minus]) 1);
-qed "approx_add_minus";
-
-Goalw [approx_def] "[| a @= b; c: HFinite|] ==> a*c @= b*c";
-by (asm_full_simp_tac (simpset() addsimps [Infinitesimal_HFinite_mult,
- minus_mult_left,left_distrib RS sym]
- delsimps [minus_mult_left RS sym]) 1);
-qed "approx_mult1";
-
-Goal "[|a @= b; c: HFinite|] ==> c*a @= c*b";
-by (asm_simp_tac (simpset() addsimps [approx_mult1,hypreal_mult_commute]) 1);
-qed "approx_mult2";
-
-Goal "[|u @= v*x; x @= y; v: HFinite|] ==> u @= v*y";
-by (fast_tac (claset() addIs [approx_mult2,approx_trans]) 1);
-qed "approx_mult_subst";
-
-Goal "[| u @= x*v; x @= y; v: HFinite |] ==> u @= y*v";
-by (fast_tac (claset() addIs [approx_mult1,approx_trans]) 1);
-qed "approx_mult_subst2";
-
-Goal "[| u @= x*hypreal_of_real v; x @= y |] ==> u @= y*hypreal_of_real v";
-by (auto_tac (claset() addIs [approx_mult_subst2], simpset()));
-qed "approx_mult_subst_SReal";
-
-Goalw [approx_def] "a = b ==> a @= b";
-by (Asm_simp_tac 1);
-qed "approx_eq_imp";
-
-Goal "x: Infinitesimal ==> -x @= x";
-by (fast_tac (HOL_cs addIs [Infinitesimal_minus_iff RS iffD2,
- mem_infmal_iff RS iffD1,approx_trans2]) 1);
-qed "Infinitesimal_minus_approx";
-
-Goalw [approx_def] "(EX y: Infinitesimal. x + -z = y) = (x @= z)";
-by (Blast_tac 1);
-qed "bex_Infinitesimal_iff";
-
-Goal "(EX y: Infinitesimal. x = z + y) = (x @= z)";
-by (asm_full_simp_tac (simpset() addsimps [bex_Infinitesimal_iff RS sym]) 1);
-by (Force_tac 1);
-qed "bex_Infinitesimal_iff2";
-
-Goal "[| y: Infinitesimal; x + y = z |] ==> x @= z";
-by (rtac (bex_Infinitesimal_iff RS iffD1) 1);
-by (dtac (Infinitesimal_minus_iff RS iffD2) 1);
-by (auto_tac (claset(), simpset() addsimps [minus_add_distrib,
- hypreal_add_assoc RS sym]));
-qed "Infinitesimal_add_approx";
-
-Goal "y: Infinitesimal ==> x @= x + y";
-by (rtac (bex_Infinitesimal_iff RS iffD1) 1);
-by (dtac (Infinitesimal_minus_iff RS iffD2) 1);
-by (auto_tac (claset(), simpset() addsimps [minus_add_distrib,
- hypreal_add_assoc RS sym]));
-qed "Infinitesimal_add_approx_self";
-
-Goal "y: Infinitesimal ==> x @= y + x";
-by (auto_tac (claset() addDs [Infinitesimal_add_approx_self],
- simpset() addsimps [hypreal_add_commute]));
-qed "Infinitesimal_add_approx_self2";
-
-Goal "y: Infinitesimal ==> x @= x + -y";
-by (blast_tac (claset() addSIs [Infinitesimal_add_approx_self,
- Infinitesimal_minus_iff RS iffD2]) 1);
-qed "Infinitesimal_add_minus_approx_self";
-
-Goal "[| y: Infinitesimal; x+y @= z|] ==> x @= z";
-by (dres_inst_tac [("x","x")] (Infinitesimal_add_approx_self RS approx_sym) 1);
-by (etac (approx_trans3 RS approx_sym) 1);
-by (assume_tac 1);
-qed "Infinitesimal_add_cancel";
-
-Goal "[| y: Infinitesimal; x @= z + y|] ==> x @= z";
-by (dres_inst_tac [("x","z")] (Infinitesimal_add_approx_self2 RS approx_sym) 1);
-by (etac (approx_trans3 RS approx_sym) 1);
-by (asm_full_simp_tac (simpset() addsimps [hypreal_add_commute]) 1);
-by (etac approx_sym 1);
-qed "Infinitesimal_add_right_cancel";
-
-Goal "d + b @= d + c ==> b @= c";
-by (dtac (approx_minus_iff RS iffD1) 1);
-by (asm_full_simp_tac (simpset() addsimps
- [minus_add_distrib,approx_minus_iff RS sym]
- @ add_ac) 1);
-qed "approx_add_left_cancel";
-
-Goal "b + d @= c + d ==> b @= c";
-by (rtac approx_add_left_cancel 1);
-by (asm_full_simp_tac (simpset() addsimps
- [hypreal_add_commute]) 1);
-qed "approx_add_right_cancel";
-
-Goal "b @= c ==> d + b @= d + c";
-by (rtac (approx_minus_iff RS iffD2) 1);
-by (asm_full_simp_tac (simpset() addsimps
- [minus_add_distrib,approx_minus_iff RS sym]
- @ add_ac) 1);
-qed "approx_add_mono1";
-
-Goal "b @= c ==> b + a @= c + a";
-by (asm_simp_tac (simpset() addsimps
- [hypreal_add_commute,approx_add_mono1]) 1);
-qed "approx_add_mono2";
-
-Goal "(a + b @= a + c) = (b @= c)";
-by (fast_tac (claset() addEs [approx_add_left_cancel,
- approx_add_mono1]) 1);
-qed "approx_add_left_iff";
-
-Addsimps [approx_add_left_iff];
-
-Goal "(b + a @= c + a) = (b @= c)";
-by (simp_tac (simpset() addsimps [hypreal_add_commute]) 1);
-qed "approx_add_right_iff";
-
-Addsimps [approx_add_right_iff];
-
-Goal "[| x: HFinite; x @= y |] ==> y: HFinite";
-by (dtac (bex_Infinitesimal_iff2 RS iffD2) 1);
-by (Step_tac 1);
-by (dtac (Infinitesimal_subset_HFinite RS subsetD
- RS (HFinite_minus_iff RS iffD2)) 1);
-by (dtac HFinite_add 1);
-by (auto_tac (claset(), simpset() addsimps [hypreal_add_assoc]));
-qed "approx_HFinite";
-
-Goal "x @= hypreal_of_real D ==> x: HFinite";
-by (rtac (approx_sym RSN (2,approx_HFinite)) 1);
-by Auto_tac;
-qed "approx_hypreal_of_real_HFinite";
-
-Goal "[|a @= b; c @= d; b: HFinite; d: HFinite|] ==> a*c @= b*d";
-by (rtac approx_trans 1);
-by (rtac approx_mult2 2);
-by (rtac approx_mult1 1);
-by (blast_tac (claset() addIs [approx_HFinite, approx_sym]) 2);
-by Auto_tac;
-qed "approx_mult_HFinite";
-
-Goal "[|a @= hypreal_of_real b; c @= hypreal_of_real d |] \
-\ ==> a*c @= hypreal_of_real b*hypreal_of_real d";
-by (blast_tac (claset() addSIs [approx_mult_HFinite,
- approx_hypreal_of_real_HFinite,HFinite_hypreal_of_real]) 1);
-qed "approx_mult_hypreal_of_real";
-
-Goal "[| a: Reals; a ~= 0; a*x @= 0 |] ==> x @= 0";
-by (dtac (SReal_inverse RS (SReal_subset_HFinite RS subsetD)) 1);
-by (auto_tac (claset() addDs [approx_mult2],
- simpset() addsimps [hypreal_mult_assoc RS sym]));
-qed "approx_SReal_mult_cancel_zero";
-
-(* REM comments: newly added *)
-Goal "[| a: Reals; x @= 0 |] ==> x*a @= 0";
-by (auto_tac (claset() addDs [(SReal_subset_HFinite RS subsetD),
- approx_mult1], simpset()));
-qed "approx_mult_SReal1";
-
-Goal "[| a: Reals; x @= 0 |] ==> a*x @= 0";
-by (auto_tac (claset() addDs [(SReal_subset_HFinite RS subsetD),
- approx_mult2], simpset()));
-qed "approx_mult_SReal2";
-
-Goal "[|a : Reals; a ~= 0 |] ==> (a*x @= 0) = (x @= 0)";
-by (blast_tac (claset() addIs [approx_SReal_mult_cancel_zero,
- approx_mult_SReal2]) 1);
-qed "approx_mult_SReal_zero_cancel_iff";
-Addsimps [approx_mult_SReal_zero_cancel_iff];
-
-Goal "[| a: Reals; a ~= 0; a* w @= a*z |] ==> w @= z";
-by (dtac (SReal_inverse RS (SReal_subset_HFinite RS subsetD)) 1);
-by (auto_tac (claset() addDs [approx_mult2],
- simpset() addsimps [hypreal_mult_assoc RS sym]));
-qed "approx_SReal_mult_cancel";
-
-Goal "[| a: Reals; a ~= 0|] ==> (a* w @= a*z) = (w @= z)";
-by (auto_tac (claset() addSIs [approx_mult2,SReal_subset_HFinite RS subsetD]
- addIs [approx_SReal_mult_cancel], simpset()));
-qed "approx_SReal_mult_cancel_iff1";
-Addsimps [approx_SReal_mult_cancel_iff1];
-
-Goal "[| z <= f; f @= g; g <= z |] ==> f @= z";
-by (asm_full_simp_tac (simpset() addsimps [bex_Infinitesimal_iff2 RS sym]) 1);
-by Auto_tac;
-by (res_inst_tac [("x","g+y-z")] bexI 1);
-by (Simp_tac 1);
-by (rtac Infinitesimal_interval2 1);
-by (rtac Infinitesimal_zero 2);
-by Auto_tac;
-qed "approx_le_bound";
-
-(*-----------------------------------------------------------------
- Zero is the only infinitesimal that is also a real
- -----------------------------------------------------------------*)
-
-Goalw [Infinitesimal_def]
- "[| x: Reals; y: Infinitesimal; 0 < x |] ==> y < x";
-by (rtac (abs_ge_self RS order_le_less_trans) 1);
-by Auto_tac;
-qed "Infinitesimal_less_SReal";
-
-Goal "y: Infinitesimal ==> ALL r: Reals. 0 < r --> y < r";
-by (blast_tac (claset() addIs [Infinitesimal_less_SReal]) 1);
-qed "Infinitesimal_less_SReal2";
-
-Goalw [Infinitesimal_def]
- "[| 0 < y; y: Reals|] ==> y ~: Infinitesimal";
-by (auto_tac (claset(), simpset() addsimps [hrabs_def]));
-qed "SReal_not_Infinitesimal";
-
-Goal "[| y < 0; y : Reals |] ==> y ~: Infinitesimal";
-by (stac (Infinitesimal_minus_iff RS sym) 1);
-by (rtac SReal_not_Infinitesimal 1);
-by Auto_tac;
-qed "SReal_minus_not_Infinitesimal";
-
-Goal "Reals Int Infinitesimal = {0}";
-by Auto_tac;
-by (cut_inst_tac [("x","x"),("y","0")] hypreal_linear 1);
-by (blast_tac (claset() addDs [SReal_not_Infinitesimal,
- SReal_minus_not_Infinitesimal]) 1);
-qed "SReal_Int_Infinitesimal_zero";
-
-Goal "[| x: Reals; x: Infinitesimal|] ==> x = 0";
-by (cut_facts_tac [SReal_Int_Infinitesimal_zero] 1);
-by (Blast_tac 1);
-qed "SReal_Infinitesimal_zero";
-
-Goal "[| x : Reals; x ~= 0 |] ==> x : HFinite - Infinitesimal";
-by (auto_tac (claset() addDs [SReal_Infinitesimal_zero,
- SReal_subset_HFinite RS subsetD],
- simpset()));
-qed "SReal_HFinite_diff_Infinitesimal";
-
-Goal "hypreal_of_real x ~= 0 ==> hypreal_of_real x : HFinite - Infinitesimal";
-by (rtac SReal_HFinite_diff_Infinitesimal 1);
-by Auto_tac;
-qed "hypreal_of_real_HFinite_diff_Infinitesimal";
-
-Goal "(hypreal_of_real x : Infinitesimal) = (x=0)";
-by Auto_tac;
-by (rtac ccontr 1);
-by (rtac (hypreal_of_real_HFinite_diff_Infinitesimal RS DiffD2) 1);
-by Auto_tac;
-qed "hypreal_of_real_Infinitesimal_iff_0";
-AddIffs [hypreal_of_real_Infinitesimal_iff_0];
-
-Goal "number_of w ~= (0::hypreal) ==> number_of w ~: Infinitesimal";
-by (fast_tac (claset() addDs [SReal_number_of RS SReal_Infinitesimal_zero]) 1);
-qed "number_of_not_Infinitesimal";
-Addsimps [number_of_not_Infinitesimal];
-
-(*again: 1 is a special case, but not 0 this time*)
-Goal "1 ~: Infinitesimal";
-by (stac (hypreal_numeral_1_eq_1 RS sym) 1);
-by (rtac number_of_not_Infinitesimal 1);
-by (Simp_tac 1);
-qed "one_not_Infinitesimal";
-Addsimps [one_not_Infinitesimal];
-
-Goal "[| y: Reals; x @= y; y~= 0 |] ==> x ~= 0";
-by (cut_inst_tac [("x","y")] hypreal_trichotomy 1);
-by (Asm_full_simp_tac 1);
-by (blast_tac (claset() addDs
- [approx_sym RS (mem_infmal_iff RS iffD2),
- SReal_not_Infinitesimal, SReal_minus_not_Infinitesimal]) 1);
-qed "approx_SReal_not_zero";
-
-Goal "[| x @= y; y : HFinite - Infinitesimal |] \
-\ ==> x : HFinite - Infinitesimal";
-by (auto_tac (claset() addIs [approx_sym RSN (2,approx_HFinite)],
- simpset() addsimps [mem_infmal_iff]));
-by (dtac approx_trans3 1 THEN assume_tac 1);
-by (blast_tac (claset() addDs [approx_sym]) 1);
-qed "HFinite_diff_Infinitesimal_approx";
-
-(*The premise y~=0 is essential; otherwise x/y =0 and we lose the
- HFinite premise.*)
-Goal "[| y ~= 0; y: Infinitesimal; x/y : HFinite |] ==> x : Infinitesimal";
-by (dtac Infinitesimal_HFinite_mult2 1);
-by (assume_tac 1);
-by (asm_full_simp_tac
- (simpset() addsimps [hypreal_divide_def, hypreal_mult_assoc]) 1);
-qed "Infinitesimal_ratio";
-
-(*------------------------------------------------------------------
- Standard Part Theorem: Every finite x: R* is infinitely
- close to a unique real number (i.e a member of Reals)
- ------------------------------------------------------------------*)
-(*------------------------------------------------------------------
- Uniqueness: Two infinitely close reals are equal
- ------------------------------------------------------------------*)
-
-Goal "[|x: Reals; y: Reals|] ==> (x @= y) = (x = y)";
-by Auto_tac;
-by (rewtac approx_def);
-by (dres_inst_tac [("x","y")] SReal_minus 1);
-by (dtac SReal_add 1 THEN assume_tac 1);
-by (dtac SReal_Infinitesimal_zero 1 THEN assume_tac 1);
-by (dtac sym 1);
-by (asm_full_simp_tac (simpset() addsimps [hypreal_eq_minus_iff RS sym]) 1);
-qed "SReal_approx_iff";
-
-Goal "(number_of v @= number_of w) = (number_of v = (number_of w :: hypreal))";
-by (rtac SReal_approx_iff 1);
-by Auto_tac;
-qed "number_of_approx_iff";
-Addsimps [number_of_approx_iff];
-
-(*And also for 0 @= #nn and 1 @= #nn, #nn @= 0 and #nn @= 1.*)
-Addsimps
- (map (simplify (simpset()))
- [inst "v" "bin.Pls" number_of_approx_iff,
- inst "v" "bin.Pls BIT True" number_of_approx_iff,
- inst "w" "bin.Pls" number_of_approx_iff,
- inst "w" "bin.Pls BIT True" number_of_approx_iff]);
-
-
-Goal "~ (0 @= 1)";
-by (stac SReal_approx_iff 1);
-by Auto_tac;
-qed "not_0_approx_1";
-Addsimps [not_0_approx_1];
-
-Goal "~ (1 @= 0)";
-by (stac SReal_approx_iff 1);
-by Auto_tac;
-qed "not_1_approx_0";
-Addsimps [not_1_approx_0];
-
-Goal "(hypreal_of_real k @= hypreal_of_real m) = (k = m)";
-by Auto_tac;
-by (rtac (inj_hypreal_of_real RS injD) 1);
-by (rtac (SReal_approx_iff RS iffD1) 1);
-by Auto_tac;
-qed "hypreal_of_real_approx_iff";
-Addsimps [hypreal_of_real_approx_iff];
-
-Goal "(hypreal_of_real k @= number_of w) = (k = number_of w)";
-by (stac (hypreal_of_real_approx_iff RS sym) 1);
-by Auto_tac;
-qed "hypreal_of_real_approx_number_of_iff";
-Addsimps [hypreal_of_real_approx_number_of_iff];
-
-(*And also for 0 and 1.*)
-Addsimps
- (map (simplify (simpset()))
- [inst "w" "bin.Pls" hypreal_of_real_approx_number_of_iff,
- inst "w" "bin.Pls BIT True" hypreal_of_real_approx_number_of_iff]);
-
-Goal "[| r: Reals; s: Reals; r @= x; s @= x|] ==> r = s";
-by (blast_tac (claset() addIs [(SReal_approx_iff RS iffD1),
- approx_trans2]) 1);
-qed "approx_unique_real";
-
-(*------------------------------------------------------------------
- Existence of unique real infinitely close
- ------------------------------------------------------------------*)
-(* lemma about lubs *)
-Goal "!!(x::hypreal). [| isLub R S x; isLub R S y |] ==> x = y";
-by (ftac isLub_isUb 1);
-by (forw_inst_tac [("x","y")] isLub_isUb 1);
-by (blast_tac (claset() addSIs [hypreal_le_anti_sym]
- addSDs [isLub_le_isUb]) 1);
-qed "hypreal_isLub_unique";
-
-Goal "x: HFinite ==> EX u. isUb Reals {s. s: Reals & s < x} u";
-by (dtac HFiniteD 1 THEN Step_tac 1);
-by (rtac exI 1 THEN rtac isUbI 1 THEN assume_tac 2);
-by (auto_tac (claset() addIs [setleI,isUbI], simpset() addsimps [abs_less_iff]));
-qed "lemma_st_part_ub";
-
-Goal "x: HFinite ==> EX y. y: {s. s: Reals & s < x}";
-by (dtac HFiniteD 1 THEN Step_tac 1);
-by (dtac SReal_minus 1);
-by (res_inst_tac [("x","-t")] exI 1);
-by (auto_tac (claset(), simpset() addsimps [abs_less_iff]));
-qed "lemma_st_part_nonempty";
-
-Goal "{s. s: Reals & s < x} <= Reals";
-by Auto_tac;
-qed "lemma_st_part_subset";
-
-Goal "x: HFinite ==> EX t. isLub Reals {s. s: Reals & s < x} t";
-by (blast_tac (claset() addSIs [SReal_complete,lemma_st_part_ub,
- lemma_st_part_nonempty, lemma_st_part_subset]) 1);
-qed "lemma_st_part_lub";
-
-Goal "((t::hypreal) + r <= t) = (r <= 0)";
-by (Step_tac 1);
-by (dres_inst_tac [("x","-t")] hypreal_add_left_le_mono1 1);
-by (dres_inst_tac [("x","t")] hypreal_add_left_le_mono1 2);
-by (auto_tac (claset(), simpset() addsimps [hypreal_add_assoc RS sym]));
-qed "lemma_hypreal_le_left_cancel";
-
-Goal "[| x: HFinite; isLub Reals {s. s: Reals & s < x} t; \
-\ r: Reals; 0 < r |] ==> x <= t + r";
-by (ftac isLubD1a 1);
-by (rtac ccontr 1 THEN dtac (linorder_not_le RS iffD2) 1);
-by (dres_inst_tac [("x","t")] SReal_add 1 THEN assume_tac 1);
-by (dres_inst_tac [("y","t + r")] (isLubD1 RS setleD) 1);
-by Auto_tac;
-qed "lemma_st_part_le1";
-
-Goalw [setle_def] "!!x::hypreal. [| S *<= x; x < y |] ==> S *<= y";
-by (auto_tac (claset() addSDs [bspec,order_le_less_trans]
- addIs [order_less_imp_le],
- simpset()));
-qed "hypreal_setle_less_trans";
-
-Goalw [isUb_def]
- "!!x::hypreal. [| isUb R S x; x < y; y: R |] ==> isUb R S y";
-by (blast_tac (claset() addIs [hypreal_setle_less_trans]) 1);
-qed "hypreal_gt_isUb";
-
-Goal "[| x: HFinite; x < y; y: Reals |] \
-\ ==> isUb Reals {s. s: Reals & s < x} y";
-by (auto_tac (claset() addDs [order_less_trans]
- addIs [order_less_imp_le] addSIs [isUbI,setleI], simpset()));
-qed "lemma_st_part_gt_ub";
-
-Goal "t <= t + -r ==> r <= (0::hypreal)";
-by (dres_inst_tac [("x","-t")] hypreal_add_left_le_mono1 1);
-by (auto_tac (claset(), simpset() addsimps [hypreal_add_assoc RS sym]));
-qed "lemma_minus_le_zero";
-
-Goal "[| x: HFinite; \
-\ isLub Reals {s. s: Reals & s < x} t; \
-\ r: Reals; 0 < r |] \
-\ ==> t + -r <= x";
-by (ftac isLubD1a 1);
-by (rtac ccontr 1 THEN dtac (linorder_not_le RS iffD1) 1);
-by (dtac SReal_minus 1 THEN dres_inst_tac [("x","t")]
- SReal_add 1 THEN assume_tac 1);
-by (dtac lemma_st_part_gt_ub 1 THEN REPEAT(assume_tac 1));
-by (dtac isLub_le_isUb 1 THEN assume_tac 1);
-by (dtac lemma_minus_le_zero 1);
-by (auto_tac (claset() addDs [order_less_le_trans], simpset()));
-qed "lemma_st_part_le2";
-
-Goal "((x::hypreal) <= t + r) = (x + -t <= r)";
-by Auto_tac;
-qed "lemma_hypreal_le_swap";
-
-Goal "[| x: HFinite; \
-\ isLub Reals {s. s: Reals & s < x} t; \
-\ r: Reals; 0 < r |] \
-\ ==> x + -t <= r";
-by (blast_tac (claset() addSIs [lemma_hypreal_le_swap RS iffD1,
- lemma_st_part_le1]) 1);
-qed "lemma_st_part1a";
-
-Goal "(t + -r <= x) = (-(x + -t) <= (r::hypreal))";
-by Auto_tac;
-qed "lemma_hypreal_le_swap2";
-
-Goal "[| x: HFinite; \
-\ isLub Reals {s. s: Reals & s < x} t; \
-\ r: Reals; 0 < r |] \
-\ ==> -(x + -t) <= r";
-by (blast_tac (claset() addSIs [lemma_hypreal_le_swap2 RS iffD1,
- lemma_st_part_le2]) 1);
-qed "lemma_st_part2a";
-
-Goal "(x::hypreal): Reals ==> isUb Reals {s. s: Reals & s < x} x";
-by (auto_tac (claset() addIs [isUbI, setleI,order_less_imp_le], simpset()));
-qed "lemma_SReal_ub";
-
-Goal "(x::hypreal): Reals ==> isLub Reals {s. s: Reals & s < x} x";
-by (auto_tac (claset() addSIs [isLubI2,lemma_SReal_ub,setgeI], simpset()));
-by (ftac isUbD2a 1);
-by (res_inst_tac [("x","x"),("y","y")] linorder_cases 1);
-by (auto_tac (claset() addSIs [order_less_imp_le], simpset()));
-by (EVERY1[dtac SReal_dense, assume_tac, assume_tac, Step_tac]);
-by (dres_inst_tac [("y","r")] isUbD 1);
-by (auto_tac (claset() addDs [order_less_le_trans], simpset()));
-qed "lemma_SReal_lub";
-
-Goal "[| x: HFinite; \
-\ isLub Reals {s. s: Reals & s < x} t; \
-\ r: Reals; 0 < r |] \
-\ ==> x + -t ~= r";
-by Auto_tac;
-by (forward_tac [isLubD1a RS SReal_minus] 1);
-by (dtac SReal_add_cancel 1 THEN assume_tac 1);
-by (dres_inst_tac [("x","x")] lemma_SReal_lub 1);
-by (dtac hypreal_isLub_unique 1 THEN assume_tac 1);
-by Auto_tac;
-qed "lemma_st_part_not_eq1";
-
-Goal "[| x: HFinite; \
-\ isLub Reals {s. s: Reals & s < x} t; \
-\ r: Reals; 0 < r |] \
-\ ==> -(x + -t) ~= r";
-by (auto_tac (claset(), simpset() addsimps [minus_add_distrib]));
-by (ftac isLubD1a 1);
-by (dtac SReal_add_cancel 1 THEN assume_tac 1);
-by (dres_inst_tac [("x","-x")] SReal_minus 1);
-by (Asm_full_simp_tac 1);
-by (dres_inst_tac [("x","x")] lemma_SReal_lub 1);
-by (dtac hypreal_isLub_unique 1 THEN assume_tac 1);
-by Auto_tac;
-qed "lemma_st_part_not_eq2";
-
-Goal "[| x: HFinite; \
-\ isLub Reals {s. s: Reals & s < x} t; \
-\ r: Reals; 0 < r |] \
-\ ==> abs (x + -t) < r";
-by (ftac lemma_st_part1a 1);
-by (ftac lemma_st_part2a 4);
-by Auto_tac;
-by (REPEAT(dtac order_le_imp_less_or_eq 1));
-by (auto_tac (claset() addDs [lemma_st_part_not_eq1, lemma_st_part_not_eq2],
- simpset() addsimps [abs_less_iff]));
-qed "lemma_st_part_major";
-
-Goal "[| x: HFinite; \
-\ isLub Reals {s. s: Reals & s < x} t |] \
-\ ==> ALL r: Reals. 0 < r --> abs (x + -t) < r";
-by (blast_tac (claset() addSDs [lemma_st_part_major]) 1);
-qed "lemma_st_part_major2";
-
-(*----------------------------------------------
- Existence of real and Standard Part Theorem
- ----------------------------------------------*)
-Goal "x: HFinite ==> \
-\ EX t: Reals. ALL r: Reals. 0 < r --> abs (x + -t) < r";
-by (ftac lemma_st_part_lub 1 THEN Step_tac 1);
-by (ftac isLubD1a 1);
-by (blast_tac (claset() addDs [lemma_st_part_major2]) 1);
-qed "lemma_st_part_Ex";
-
-Goalw [approx_def,Infinitesimal_def]
- "x:HFinite ==> EX t: Reals. x @= t";
-by (dtac lemma_st_part_Ex 1);
-by Auto_tac;
-qed "st_part_Ex";
-
-(*--------------------------------
- Unique real infinitely close
- -------------------------------*)
-Goal "x:HFinite ==> EX! t. t: Reals & x @= t";
-by (dtac st_part_Ex 1 THEN Step_tac 1);
-by (dtac approx_sym 2 THEN dtac approx_sym 2
- THEN dtac approx_sym 2);
-by (auto_tac (claset() addSIs [approx_unique_real], simpset()));
-qed "st_part_Ex1";
-
-(*------------------------------------------------------------------
- Finite and Infinite --- more theorems
- ------------------------------------------------------------------*)
-
-Goalw [HFinite_def,HInfinite_def] "HFinite Int HInfinite = {}";
-by (auto_tac (claset() addIs [hypreal_less_irrefl] addDs [order_less_trans],
- simpset()));
-qed "HFinite_Int_HInfinite_empty";
-Addsimps [HFinite_Int_HInfinite_empty];
-
-Goal "x: HFinite ==> x ~: HInfinite";
-by (EVERY1[Step_tac, dtac IntI, assume_tac]);
-by Auto_tac;
-qed "HFinite_not_HInfinite";
-
-Goalw [HInfinite_def, HFinite_def] "x~: HFinite ==> x: HInfinite";
-by Auto_tac;
-by (dres_inst_tac [("x","r + 1")] bspec 1);
-by (auto_tac (claset(), simpset() addsimps [SReal_add]));
-qed "not_HFinite_HInfinite";
-
-Goal "x : HInfinite | x : HFinite";
-by (blast_tac (claset() addIs [not_HFinite_HInfinite]) 1);
-qed "HInfinite_HFinite_disj";
-
-Goal "(x : HInfinite) = (x ~: HFinite)";
-by (blast_tac (claset() addDs [HFinite_not_HInfinite,
- not_HFinite_HInfinite]) 1);
-qed "HInfinite_HFinite_iff";
-
-Goal "(x : HFinite) = (x ~: HInfinite)";
-by (simp_tac (simpset() addsimps [HInfinite_HFinite_iff]) 1);
-qed "HFinite_HInfinite_iff";
-
-(*------------------------------------------------------------------
- Finite, Infinite and Infinitesimal --- more theorems
- ------------------------------------------------------------------*)
-
-Goal "x ~: Infinitesimal ==> x : HInfinite | x : HFinite - Infinitesimal";
-by (fast_tac (claset() addIs [not_HFinite_HInfinite]) 1);
-qed "HInfinite_diff_HFinite_Infinitesimal_disj";
-
-Goal "[| x : HFinite; x ~: Infinitesimal |] ==> inverse x : HFinite";
-by (cut_inst_tac [("x","inverse x")] HInfinite_HFinite_disj 1);
-by (auto_tac (claset() addSDs [HInfinite_inverse_Infinitesimal], simpset()));
-qed "HFinite_inverse";
-
-Goal "x : HFinite - Infinitesimal ==> inverse x : HFinite";
-by (blast_tac (claset() addIs [HFinite_inverse]) 1);
-qed "HFinite_inverse2";
-
-(* stronger statement possible in fact *)
-Goal "x ~: Infinitesimal ==> inverse(x) : HFinite";
-by (dtac HInfinite_diff_HFinite_Infinitesimal_disj 1);
-by (blast_tac (claset() addIs [HFinite_inverse,
- HInfinite_inverse_Infinitesimal,
- Infinitesimal_subset_HFinite RS subsetD]) 1);
-qed "Infinitesimal_inverse_HFinite";
-
-Goal "x : HFinite - Infinitesimal ==> inverse x : HFinite - Infinitesimal";
-by (auto_tac (claset() addIs [Infinitesimal_inverse_HFinite], simpset()));
-by (dtac Infinitesimal_HFinite_mult2 1);
-by (assume_tac 1);
-by (asm_full_simp_tac
- (simpset() addsimps [not_Infinitesimal_not_zero, hypreal_mult_inverse]) 1);
-qed "HFinite_not_Infinitesimal_inverse";
-
-Goal "[| x @= y; y : HFinite - Infinitesimal |] \
-\ ==> inverse x @= inverse y";
-by (ftac HFinite_diff_Infinitesimal_approx 1);
-by (assume_tac 1);
-by (ftac not_Infinitesimal_not_zero2 1);
-by (forw_inst_tac [("x","x")] not_Infinitesimal_not_zero2 1);
-by (REPEAT(dtac HFinite_inverse2 1));
-by (dtac approx_mult2 1 THEN assume_tac 1);
-by Auto_tac;
-by (dres_inst_tac [("c","inverse x")] approx_mult1 1
- THEN assume_tac 1);
-by (auto_tac (claset() addIs [approx_sym],
- simpset() addsimps [hypreal_mult_assoc]));
-qed "approx_inverse";
-
-(*Used for NSLIM_inverse, NSLIMSEQ_inverse*)
-bind_thm ("hypreal_of_real_approx_inverse",
- hypreal_of_real_HFinite_diff_Infinitesimal RSN (2, approx_inverse));
-
-Goal "[| x: HFinite - Infinitesimal; \
-\ h : Infinitesimal |] ==> inverse(x + h) @= inverse x";
-by (auto_tac (claset() addIs [approx_inverse, approx_sym,
- Infinitesimal_add_approx_self],
- simpset()));
-qed "inverse_add_Infinitesimal_approx";
-
-Goal "[| x: HFinite - Infinitesimal; \
-\ h : Infinitesimal |] ==> inverse(h + x) @= inverse x";
-by (rtac (hypreal_add_commute RS subst) 1);
-by (blast_tac (claset() addIs [inverse_add_Infinitesimal_approx]) 1);
-qed "inverse_add_Infinitesimal_approx2";
-
-Goal "[| x : HFinite - Infinitesimal; \
-\ h : Infinitesimal |] ==> inverse(x + h) + -inverse x @= h";
-by (rtac approx_trans2 1);
-by (auto_tac (claset() addIs [inverse_add_Infinitesimal_approx],
- simpset() addsimps [mem_infmal_iff,
- approx_minus_iff RS sym]));
-qed "inverse_add_Infinitesimal_approx_Infinitesimal";
-
-Goal "(x : Infinitesimal) = (x*x : Infinitesimal)";
-by (auto_tac (claset() addIs [Infinitesimal_mult], simpset()));
-by (rtac ccontr 1 THEN ftac Infinitesimal_inverse_HFinite 1);
-by (ftac not_Infinitesimal_not_zero 1);
-by (auto_tac (claset() addDs [Infinitesimal_HFinite_mult],
- simpset() addsimps [hypreal_mult_assoc]));
-qed "Infinitesimal_square_iff";
-Addsimps [Infinitesimal_square_iff RS sym];
-
-Goal "(x*x : HFinite) = (x : HFinite)";
-by (auto_tac (claset() addIs [HFinite_mult], simpset()));
-by (auto_tac (claset() addDs [HInfinite_mult],
- simpset() addsimps [HFinite_HInfinite_iff]));
-qed "HFinite_square_iff";
-Addsimps [HFinite_square_iff];
-
-Goal "(x*x : HInfinite) = (x : HInfinite)";
-by (auto_tac (claset(), simpset() addsimps
- [HInfinite_HFinite_iff]));
-qed "HInfinite_square_iff";
-Addsimps [HInfinite_square_iff];
-
-Goal "[| a: HFinite-Infinitesimal; a* w @= a*z |] ==> w @= z";
-by (Step_tac 1);
-by (ftac HFinite_inverse 1 THEN assume_tac 1);
-by (dtac not_Infinitesimal_not_zero 1);
-by (auto_tac (claset() addDs [approx_mult2],
- simpset() addsimps [hypreal_mult_assoc RS sym]));
-qed "approx_HFinite_mult_cancel";
-
-Goal "a: HFinite-Infinitesimal ==> (a * w @= a * z) = (w @= z)";
-by (auto_tac (claset() addIs [approx_mult2,
- approx_HFinite_mult_cancel], simpset()));
-qed "approx_HFinite_mult_cancel_iff1";
-
-(*------------------------------------------------------------------
- more about absolute value (hrabs)
- ------------------------------------------------------------------*)
-
-Goal "abs x @= x | abs x @= -x";
-by (cut_inst_tac [("x","x")] hrabs_disj 1);
-by Auto_tac;
-qed "approx_hrabs_disj";
-
-(*------------------------------------------------------------------
- Theorems about monads
- ------------------------------------------------------------------*)
-
-Goal "monad (abs x) <= monad(x) Un monad(-x)";
-by (res_inst_tac [("x1","x")] (hrabs_disj RS disjE) 1);
-by Auto_tac;
-qed "monad_hrabs_Un_subset";
-
-Goal "e : Infinitesimal ==> monad (x+e) = monad x";
-by (fast_tac (claset() addSIs [Infinitesimal_add_approx_self RS approx_sym,
- approx_monad_iff RS iffD1]) 1);
-qed "Infinitesimal_monad_eq";
-
-Goalw [monad_def] "(u:monad x) = (-u:monad (-x))";
-by Auto_tac;
-qed "mem_monad_iff";
-
-Goalw [monad_def] "(x:Infinitesimal) = (x:monad 0)";
-by (auto_tac (claset() addIs [approx_sym],
- simpset() addsimps [mem_infmal_iff]));
-qed "Infinitesimal_monad_zero_iff";
-
-Goal "(x:monad 0) = (-x:monad 0)";
-by (simp_tac (simpset() addsimps [Infinitesimal_monad_zero_iff RS sym]) 1);
-qed "monad_zero_minus_iff";
-
-Goal "(x:monad 0) = (abs x:monad 0)";
-by (res_inst_tac [("x1","x")] (hrabs_disj RS disjE) 1);
-by (auto_tac (claset(), simpset() addsimps [monad_zero_minus_iff RS sym]));
-qed "monad_zero_hrabs_iff";
-
-Goalw [monad_def] "x:monad x";
-by (Simp_tac 1);
-qed "mem_monad_self";
-Addsimps [mem_monad_self];
-
-(*------------------------------------------------------------------
- Proof that x @= y ==> abs x @= abs y
- ------------------------------------------------------------------*)
-Goal "x @= y ==> {x,y}<=monad x";
-by (Simp_tac 1);
-by (asm_full_simp_tac (simpset() addsimps
- [approx_monad_iff]) 1);
-qed "approx_subset_monad";
-
-Goal "x @= y ==> {x,y}<=monad y";
-by (dtac approx_sym 1);
-by (fast_tac (claset() addDs [approx_subset_monad]) 1);
-qed "approx_subset_monad2";
-
-Goalw [monad_def] "u:monad x ==> x @= u";
-by (Fast_tac 1);
-qed "mem_monad_approx";
-
-Goalw [monad_def] "x @= u ==> u:monad x";
-by (Fast_tac 1);
-qed "approx_mem_monad";
-
-Goalw [monad_def] "x @= u ==> x:monad u";
-by (blast_tac (claset() addSIs [approx_sym]) 1);
-qed "approx_mem_monad2";
-
-Goal "[| x @= y;x:monad 0 |] ==> y:monad 0";
-by (dtac mem_monad_approx 1);
-by (fast_tac (claset() addIs [approx_mem_monad,approx_trans]) 1);
-qed "approx_mem_monad_zero";
-
-Goal "[| x @= y; x: Infinitesimal |] ==> abs x @= abs y";
-by (dtac (Infinitesimal_monad_zero_iff RS iffD1) 1);
-by (blast_tac (claset() addIs [approx_mem_monad_zero,
- monad_zero_hrabs_iff RS iffD1, mem_monad_approx, approx_trans3]) 1);
-qed "Infinitesimal_approx_hrabs";
-
-Goal "[| 0 < x; x ~:Infinitesimal; e :Infinitesimal |] ==> e < x";
-by (rtac ccontr 1);
-by (auto_tac (claset()
- addIs [Infinitesimal_zero RSN (2, Infinitesimal_interval)]
- addSDs [order_le_imp_less_or_eq],
- simpset() addsimps [linorder_not_less]));
-qed "less_Infinitesimal_less";
-
-Goal "[| 0 < x; x ~: Infinitesimal; u: monad x |] ==> 0 < u";
-by (dtac (mem_monad_approx RS approx_sym) 1);
-by (etac (bex_Infinitesimal_iff2 RS iffD2 RS bexE) 1);
-by (dres_inst_tac [("e","-xa")] less_Infinitesimal_less 1);
-by Auto_tac;
-qed "Ball_mem_monad_gt_zero";
-
-Goal "[| x < 0; x ~: Infinitesimal; u: monad x |] ==> u < 0";
-by (dtac (mem_monad_approx RS approx_sym) 1);
-by (etac (bex_Infinitesimal_iff RS iffD2 RS bexE) 1);
-by (cut_inst_tac [("x","-x"),("e","xa")] less_Infinitesimal_less 1);
-by Auto_tac;
-qed "Ball_mem_monad_less_zero";
-
-Goal "[|0 < x; x ~: Infinitesimal; x @= y|] ==> 0 < y";
-by (blast_tac (claset() addDs [Ball_mem_monad_gt_zero,
- approx_subset_monad]) 1);
-qed "lemma_approx_gt_zero";
-
-Goal "[|x < 0; x ~: Infinitesimal; x @= y|] ==> y < 0";
-by (blast_tac (claset() addDs [Ball_mem_monad_less_zero,
- approx_subset_monad]) 1);
-qed "lemma_approx_less_zero";
-
-Goal "[| x @= y; x < 0; x ~: Infinitesimal |] ==> abs x @= abs y";
-by (ftac lemma_approx_less_zero 1);
-by (REPEAT(assume_tac 1));
-by (REPEAT(dtac hrabs_minus_eqI2 1));
-by Auto_tac;
-qed "approx_hrabs1";
-
-Goal "[| x @= y; 0 < x; x ~: Infinitesimal |] ==> abs x @= abs y";
-by (ftac lemma_approx_gt_zero 1);
-by (REPEAT(assume_tac 1));
-by (REPEAT(dtac hrabs_eqI2 1));
-by Auto_tac;
-qed "approx_hrabs2";
-
-Goal "x @= y ==> abs x @= abs y";
-by (res_inst_tac [("Q","x:Infinitesimal")] (excluded_middle RS disjE) 1);
-by (res_inst_tac [("x1","x"),("y1","0")] (hypreal_linear RS disjE) 1);
-by (auto_tac (claset() addIs [approx_hrabs1,approx_hrabs2,
- Infinitesimal_approx_hrabs], simpset()));
-qed "approx_hrabs";
-
-Goal "abs(x) @= 0 ==> x @= 0";
-by (cut_inst_tac [("x","x")] hrabs_disj 1);
-by (auto_tac (claset() addDs [approx_minus], simpset()));
-qed "approx_hrabs_zero_cancel";
-
-Goal "e: Infinitesimal ==> abs x @= abs(x+e)";
-by (fast_tac (claset() addIs [approx_hrabs,
- Infinitesimal_add_approx_self]) 1);
-qed "approx_hrabs_add_Infinitesimal";
-
-Goal "e: Infinitesimal ==> abs x @= abs(x + -e)";
-by (fast_tac (claset() addIs [approx_hrabs,
- Infinitesimal_add_minus_approx_self]) 1);
-qed "approx_hrabs_add_minus_Infinitesimal";
-
-Goal "[| e: Infinitesimal; e': Infinitesimal; \
-\ abs(x+e) = abs(y+e')|] ==> abs x @= abs y";
-by (dres_inst_tac [("x","x")] approx_hrabs_add_Infinitesimal 1);
-by (dres_inst_tac [("x","y")] approx_hrabs_add_Infinitesimal 1);
-by (auto_tac (claset() addIs [approx_trans2], simpset()));
-qed "hrabs_add_Infinitesimal_cancel";
-
-Goal "[| e: Infinitesimal; e': Infinitesimal; \
-\ abs(x + -e) = abs(y + -e')|] ==> abs x @= abs y";
-by (dres_inst_tac [("x","x")] approx_hrabs_add_minus_Infinitesimal 1);
-by (dres_inst_tac [("x","y")] approx_hrabs_add_minus_Infinitesimal 1);
-by (auto_tac (claset() addIs [approx_trans2], simpset()));
-qed "hrabs_add_minus_Infinitesimal_cancel";
-
-(* interesting slightly counterintuitive theorem: necessary
- for proving that an open interval is an NS open set
-*)
-Goalw [Infinitesimal_def]
- "[| x < y; u: Infinitesimal |] \
-\ ==> hypreal_of_real x + u < hypreal_of_real y";
-by (dtac (hypreal_of_real_less_iff RS iffD2) 1);
-by (dtac (hypreal_less_minus_iff RS iffD1) 1 THEN Step_tac 1);
-by (rtac (hypreal_less_minus_iff RS iffD2) 1);
-by (dres_inst_tac [("x","hypreal_of_real y + -hypreal_of_real x")] bspec 1);
-by (auto_tac (claset(),
- simpset() addsimps [hypreal_add_commute,
- abs_less_iff,
- SReal_add, SReal_minus]));
-qed "Infinitesimal_add_hypreal_of_real_less";
-
-Goal "[| x: Infinitesimal; abs(hypreal_of_real r) < hypreal_of_real y |] \
-\ ==> abs (hypreal_of_real r + x) < hypreal_of_real y";
-by (dres_inst_tac [("x","hypreal_of_real r")]
- approx_hrabs_add_Infinitesimal 1);
-by (dtac (approx_sym RS (bex_Infinitesimal_iff2 RS iffD2)) 1);
-by (auto_tac (claset() addSIs [Infinitesimal_add_hypreal_of_real_less],
- simpset() addsimps [hypreal_of_real_hrabs]));
-qed "Infinitesimal_add_hrabs_hypreal_of_real_less";
-
-Goal "[| x: Infinitesimal; abs(hypreal_of_real r) < hypreal_of_real y |] \
-\ ==> abs (x + hypreal_of_real r) < hypreal_of_real y";
-by (rtac (hypreal_add_commute RS subst) 1);
-by (etac Infinitesimal_add_hrabs_hypreal_of_real_less 1);
-by (assume_tac 1);
-qed "Infinitesimal_add_hrabs_hypreal_of_real_less2";
-
-Goalw [hypreal_le_def]
- "[| u: Infinitesimal; hypreal_of_real x + u <= hypreal_of_real y |] \
-\ ==> hypreal_of_real x <= hypreal_of_real y";
-by (EVERY1 [rtac notI, rtac contrapos_np, assume_tac]);
-by (res_inst_tac [("c1","-u")] (add_less_cancel_right RS iffD1) 1);
-by (Asm_full_simp_tac 1);
-by (dtac (Infinitesimal_minus_iff RS iffD2) 1);
-by (dtac Infinitesimal_add_hypreal_of_real_less 1);
-by (assume_tac 1);
-by Auto_tac;
-qed "Infinitesimal_add_cancel_hypreal_of_real_le";
-
-Goal "[| u: Infinitesimal; hypreal_of_real x + u <= hypreal_of_real y |] \
-\ ==> x <= y";
-by (blast_tac (claset() addSIs [hypreal_of_real_le_iff RS iffD1,
- Infinitesimal_add_cancel_hypreal_of_real_le]) 1);
-qed "Infinitesimal_add_cancel_real_le";
-
-Goal "[| u: Infinitesimal; v: Infinitesimal; \
-\ hypreal_of_real x + u <= hypreal_of_real y + v |] \
-\ ==> hypreal_of_real x <= hypreal_of_real y";
-by (asm_full_simp_tac (simpset() addsimps [linorder_not_less RS sym]) 1);
-by Auto_tac;
-by (dres_inst_tac [("u","v-u")] Infinitesimal_add_hypreal_of_real_less 1);
-by (auto_tac (claset(), simpset() addsimps [Infinitesimal_diff]));
-qed "hypreal_of_real_le_add_Infininitesimal_cancel";
-
-Goal "[| u: Infinitesimal; v: Infinitesimal; \
-\ hypreal_of_real x + u <= hypreal_of_real y + v |] \
-\ ==> x <= y";
-by (blast_tac (claset() addSIs [hypreal_of_real_le_iff RS iffD1,
- hypreal_of_real_le_add_Infininitesimal_cancel]) 1);
-qed "hypreal_of_real_le_add_Infininitesimal_cancel2";
-
-Goal "[| hypreal_of_real x < e; e: Infinitesimal |] ==> hypreal_of_real x <= 0";
-by (rtac (linorder_not_less RS iffD1) 1 THEN Safe_tac);
-by (dtac Infinitesimal_interval 1);
-by (dtac (SReal_hypreal_of_real RS SReal_Infinitesimal_zero) 4);
-by Auto_tac;
-qed "hypreal_of_real_less_Infinitesimal_le_zero";
-
-(*used once, in Lim/NSDERIV_inverse*)
-Goal "[| h: Infinitesimal; x ~= 0 |] ==> hypreal_of_real x + h ~= 0";
-by Auto_tac;
-by (subgoal_tac "h = - hypreal_of_real x" 1);
-by Auto_tac;
-qed "Infinitesimal_add_not_zero";
-
-Goal "x*x + y*y : Infinitesimal ==> x*x : Infinitesimal";
-by (rtac Infinitesimal_interval2 1);
-by (rtac zero_le_square 3);
-by (assume_tac 1);
-by (auto_tac (claset(), simpset() addsimps [zero_le_square]));
-qed "Infinitesimal_square_cancel";
-Addsimps [Infinitesimal_square_cancel];
-
-Goal "x*x + y*y : HFinite ==> x*x : HFinite";
-by (rtac HFinite_bounded 1);
-by (assume_tac 1);
-by (auto_tac (claset(), simpset() addsimps [zero_le_square]));
-qed "HFinite_square_cancel";
-Addsimps [HFinite_square_cancel];
-
-Goal "x*x + y*y : Infinitesimal ==> y*y : Infinitesimal";
-by (rtac Infinitesimal_square_cancel 1);
-by (rtac (hypreal_add_commute RS subst) 1);
-by (Simp_tac 1);
-qed "Infinitesimal_square_cancel2";
-Addsimps [Infinitesimal_square_cancel2];
-
-Goal "x*x + y*y : HFinite ==> y*y : HFinite";
-by (rtac HFinite_square_cancel 1);
-by (rtac (hypreal_add_commute RS subst) 1);
-by (Simp_tac 1);
-qed "HFinite_square_cancel2";
-Addsimps [HFinite_square_cancel2];
-
-Goal "x*x + y*y + z*z : Infinitesimal ==> x*x : Infinitesimal";
-by (rtac Infinitesimal_interval2 1);
-by (assume_tac 1);
-by (rtac zero_le_square 2);
-by (Asm_full_simp_tac 1);
-by (cut_inst_tac [("a","y")] (thm"zero_le_square") 1);
-by (cut_inst_tac [("a","z")] (thm"zero_le_square") 1);
-by (asm_simp_tac (simpset() addsimps []) 1);
-qed "Infinitesimal_sum_square_cancel";
-Addsimps [Infinitesimal_sum_square_cancel];
-
-Goal "x*x + y*y + z*z : HFinite ==> x*x : HFinite";
-by (rtac HFinite_bounded 1);
-by (assume_tac 1);
-by (rtac zero_le_square 2);
-by (cut_inst_tac [("a","y")] (thm"zero_le_square") 1);
-by (cut_inst_tac [("a","z")] (thm"zero_le_square") 1);
-by (asm_simp_tac (simpset() addsimps []) 1);
-qed "HFinite_sum_square_cancel";
-Addsimps [HFinite_sum_square_cancel];
-
-Goal "y*y + x*x + z*z : Infinitesimal ==> x*x : Infinitesimal";
-by (rtac Infinitesimal_sum_square_cancel 1);
-by (asm_full_simp_tac (simpset() addsimps add_ac) 1);
-qed "Infinitesimal_sum_square_cancel2";
-Addsimps [Infinitesimal_sum_square_cancel2];
-
-Goal "y*y + x*x + z*z : HFinite ==> x*x : HFinite";
-by (rtac HFinite_sum_square_cancel 1);
-by (asm_full_simp_tac (simpset() addsimps add_ac) 1);
-qed "HFinite_sum_square_cancel2";
-Addsimps [HFinite_sum_square_cancel2];
-
-Goal "z*z + y*y + x*x : Infinitesimal ==> x*x : Infinitesimal";
-by (rtac Infinitesimal_sum_square_cancel 1);
-by (asm_full_simp_tac (simpset() addsimps add_ac) 1);
-qed "Infinitesimal_sum_square_cancel3";
-Addsimps [Infinitesimal_sum_square_cancel3];
-
-Goal "z*z + y*y + x*x : HFinite ==> x*x : HFinite";
-by (rtac HFinite_sum_square_cancel 1);
-by (asm_full_simp_tac (simpset() addsimps add_ac) 1);
-qed "HFinite_sum_square_cancel3";
-Addsimps [HFinite_sum_square_cancel3];
-
-Goal "[| y: monad x; 0 < hypreal_of_real e |] \
-\ ==> abs (y + -x) < hypreal_of_real e";
-by (dtac (mem_monad_approx RS approx_sym) 1);
-by (dtac (bex_Infinitesimal_iff RS iffD2) 1);
-by (auto_tac (claset() addSDs [InfinitesimalD], simpset()));
-qed "monad_hrabs_less";
-
-Goal "x: monad (hypreal_of_real a) ==> x: HFinite";
-by (dtac (mem_monad_approx RS approx_sym) 1);
-by (dtac (bex_Infinitesimal_iff2 RS iffD2) 1);
-by (step_tac (claset() addSDs
- [Infinitesimal_subset_HFinite RS subsetD]) 1);
-by (etac (SReal_hypreal_of_real RS (SReal_subset_HFinite
- RS subsetD) RS HFinite_add) 1);
-qed "mem_monad_SReal_HFinite";
-
-(*------------------------------------------------------------------
- Theorems about standard part
- ------------------------------------------------------------------*)
-
-Goalw [st_def] "x: HFinite ==> st x @= x";
-by (ftac st_part_Ex 1 THEN Step_tac 1);
-by (rtac someI2 1);
-by (auto_tac (claset() addIs [approx_sym], simpset()));
-qed "st_approx_self";
-
-Goalw [st_def] "x: HFinite ==> st x: Reals";
-by (ftac st_part_Ex 1 THEN Step_tac 1);
-by (rtac someI2 1);
-by (auto_tac (claset() addIs [approx_sym], simpset()));
-qed "st_SReal";
-
-Goal "x: HFinite ==> st x: HFinite";
-by (etac (st_SReal RS (SReal_subset_HFinite RS subsetD)) 1);
-qed "st_HFinite";
-
-Goalw [st_def] "x: Reals ==> st x = x";
-by (rtac some_equality 1);
-by (fast_tac (claset() addIs [(SReal_subset_HFinite RS subsetD)]) 1);
-by (blast_tac (claset() addDs [SReal_approx_iff RS iffD1]) 1);
-qed "st_SReal_eq";
-
-(* should be added to simpset *)
-Goal "st (hypreal_of_real x) = hypreal_of_real x";
-by (rtac (SReal_hypreal_of_real RS st_SReal_eq) 1);
-qed "st_hypreal_of_real";
-
-Goal "[| x: HFinite; y: HFinite; st x = st y |] ==> x @= y";
-by (auto_tac (claset() addSDs [st_approx_self]
- addSEs [approx_trans3], simpset()));
-qed "st_eq_approx";
-
-Goal "[| x: HFinite; y: HFinite; x @= y |] ==> st x = st y";
-by (EVERY1 [ftac st_approx_self ,
- forw_inst_tac [("x","y")] st_approx_self,
- dtac st_SReal,dtac st_SReal]);
-by (fast_tac (claset() addEs [approx_trans,
- approx_trans2,SReal_approx_iff RS iffD1]) 1);
-qed "approx_st_eq";
-
-Goal "[| x: HFinite; y: HFinite|] \
-\ ==> (x @= y) = (st x = st y)";
-by (blast_tac (claset() addIs [approx_st_eq,
- st_eq_approx]) 1);
-qed "st_eq_approx_iff";
-
-Goal "[| x: Reals; e: Infinitesimal |] ==> st(x + e) = x";
-by (forward_tac [st_SReal_eq RS subst] 1);
-by (assume_tac 2);
-by (forward_tac [SReal_subset_HFinite RS subsetD] 1);
-by (forward_tac [Infinitesimal_subset_HFinite RS subsetD] 1);
-by (dtac st_SReal_eq 1);
-by (rtac approx_st_eq 1);
-by (auto_tac (claset() addIs [HFinite_add],
- simpset() addsimps [Infinitesimal_add_approx_self
- RS approx_sym]));
-qed "st_Infinitesimal_add_SReal";
-
-Goal "[| x: Reals; e: Infinitesimal |] \
-\ ==> st(e + x) = x";
-by (rtac (hypreal_add_commute RS subst) 1);
-by (blast_tac (claset() addSIs [st_Infinitesimal_add_SReal]) 1);
-qed "st_Infinitesimal_add_SReal2";
-
-Goal "x: HFinite ==> \
-\ EX e: Infinitesimal. x = st(x) + e";
-by (blast_tac (claset() addSDs [(st_approx_self RS
- approx_sym),bex_Infinitesimal_iff2 RS iffD2]) 1);
-qed "HFinite_st_Infinitesimal_add";
-
-Goal "[| x: HFinite; y: HFinite |] ==> st (x + y) = st(x) + st(y)";
-by (ftac HFinite_st_Infinitesimal_add 1);
-by (forw_inst_tac [("x","y")] HFinite_st_Infinitesimal_add 1);
-by (Step_tac 1);
-by (subgoal_tac "st (x + y) = st ((st x + e) + (st y + ea))" 1);
-by (dtac sym 2 THEN dtac sym 2);
-by (Asm_full_simp_tac 2);
-by (asm_simp_tac (simpset() addsimps add_ac) 1);
-by (REPEAT(dtac st_SReal 1));
-by (dtac SReal_add 1 THEN assume_tac 1);
-by (dtac Infinitesimal_add 1 THEN assume_tac 1);
-by (rtac (hypreal_add_assoc RS subst) 1);
-by (blast_tac (claset() addSIs [st_Infinitesimal_add_SReal2]) 1);
-qed "st_add";
-
-Goal "st (number_of w) = number_of w";
-by (rtac (SReal_number_of RS st_SReal_eq) 1);
-qed "st_number_of";
-Addsimps [st_number_of];
-
-(*the theorem above for the special cases of zero and one*)
-Addsimps
- (map (simplify (simpset()))
- [inst "w" "bin.Pls" st_number_of, inst "w" "bin.Pls BIT True" st_number_of]);
-
-Goal "y: HFinite ==> st(-y) = -st(y)";
-by (forward_tac [HFinite_minus_iff RS iffD2] 1);
-by (rtac hypreal_add_minus_eq_minus 1);
-by (dtac (st_add RS sym) 1 THEN assume_tac 1);
-by Auto_tac;
-qed "st_minus";
-
-Goalw [hypreal_diff_def]
- "[| x: HFinite; y: HFinite |] ==> st (x-y) = st(x) - st(y)";
-by (forw_inst_tac [("y1","y")] (st_minus RS sym) 1);
-by (dres_inst_tac [("x1","y")] (HFinite_minus_iff RS iffD2) 1);
-by (asm_simp_tac (simpset() addsimps [st_add]) 1);
-qed "st_diff";
-
-(* lemma *)
-Goal "[| x: HFinite; y: HFinite; \
-\ e: Infinitesimal; \
-\ ea : Infinitesimal |] \
-\ ==> e*y + x*ea + e*ea: Infinitesimal";
-by (forw_inst_tac [("x","e"),("y","y")] Infinitesimal_HFinite_mult 1);
-by (forw_inst_tac [("x","ea"),("y","x")] Infinitesimal_HFinite_mult 2);
-by (dtac Infinitesimal_mult 3);
-by (auto_tac (claset() addIs [Infinitesimal_add],
- simpset() addsimps add_ac @ mult_ac));
-qed "lemma_st_mult";
-
-Goal "[| x: HFinite; y: HFinite |] \
-\ ==> st (x * y) = st(x) * st(y)";
-by (ftac HFinite_st_Infinitesimal_add 1);
-by (forw_inst_tac [("x","y")] HFinite_st_Infinitesimal_add 1);
-by (Step_tac 1);
-by (subgoal_tac "st (x * y) = st ((st x + e) * (st y + ea))" 1);
-by (dtac sym 2 THEN dtac sym 2);
-by (Asm_full_simp_tac 2);
-by (thin_tac "x = st x + e" 1);
-by (thin_tac "y = st y + ea" 1);
-by (asm_full_simp_tac (simpset() addsimps [left_distrib,right_distrib]) 1);
-by (REPEAT(dtac st_SReal 1));
-by (full_simp_tac (simpset() addsimps [hypreal_add_assoc]) 1);
-by (rtac st_Infinitesimal_add_SReal 1);
-by (blast_tac (claset() addSIs [SReal_mult]) 1);
-by (REPEAT(dtac (SReal_subset_HFinite RS subsetD) 1));
-by (rtac (hypreal_add_assoc RS subst) 1);
-by (blast_tac (claset() addSIs [lemma_st_mult]) 1);
-qed "st_mult";
-
-Goal "x: Infinitesimal ==> st x = 0";
-by (stac (hypreal_numeral_0_eq_0 RS sym) 1);
-by (rtac (st_number_of RS subst) 1);
-by (rtac approx_st_eq 1);
-by (auto_tac (claset() addIs [Infinitesimal_subset_HFinite RS subsetD],
- simpset() addsimps [mem_infmal_iff RS sym]));
-qed "st_Infinitesimal";
-
-Goal "st(x) ~= 0 ==> x ~: Infinitesimal";
-by (fast_tac (claset() addIs [st_Infinitesimal]) 1);
-qed "st_not_Infinitesimal";
-
-Goal "[| x: HFinite; st x ~= 0 |] \
-\ ==> st(inverse x) = inverse (st x)";
-by (res_inst_tac [("c1","st x")] (hypreal_mult_left_cancel RS iffD1) 1);
-by (auto_tac (claset(),
- simpset() addsimps [st_mult RS sym, st_not_Infinitesimal,
- HFinite_inverse]));
-by (stac hypreal_mult_inverse 1);
-by Auto_tac;
-qed "st_inverse";
-
-Goal "[| x: HFinite; y: HFinite; st y ~= 0 |] \
-\ ==> st(x/y) = (st x) / (st y)";
-by (auto_tac (claset(),
- simpset() addsimps [hypreal_divide_def, st_mult, st_not_Infinitesimal,
- HFinite_inverse, st_inverse]));
-qed "st_divide";
-Addsimps [st_divide];
-
-Goal "x: HFinite ==> st(st(x)) = st(x)";
-by (blast_tac (claset() addIs [st_HFinite, st_approx_self,
- approx_st_eq]) 1);
-qed "st_idempotent";
-Addsimps [st_idempotent];
-
-(*** lemmas ***)
-Goal "[| x: HFinite; y: HFinite; \
-\ u: Infinitesimal; st x < st y \
-\ |] ==> st x + u < st y";
-by (REPEAT(dtac st_SReal 1));
-by (auto_tac (claset() addSIs [Infinitesimal_add_hypreal_of_real_less],
- simpset() addsimps [SReal_iff]));
-qed "Infinitesimal_add_st_less";
-
-Goalw [hypreal_le_def]
- "[| x: HFinite; y: HFinite; \
-\ u: Infinitesimal; st x <= st y + u\
-\ |] ==> st x <= st y";
-by (auto_tac (claset() addDs [Infinitesimal_add_st_less],
- simpset()));
-qed "Infinitesimal_add_st_le_cancel";
-
-Goal "[| x: HFinite; y: HFinite; x <= y |] ==> st(x) <= st(y)";
-by (ftac HFinite_st_Infinitesimal_add 1);
-by (rotate_tac 1 1);
-by (ftac HFinite_st_Infinitesimal_add 1);
-by (Step_tac 1);
-by (rtac Infinitesimal_add_st_le_cancel 1);
-by (res_inst_tac [("x","ea"),("y","e")]
- Infinitesimal_diff 3);
-by (auto_tac (claset(), simpset() addsimps [hypreal_add_assoc RS sym]));
-qed "st_le";
-
-Goal "[| 0 <= x; x: HFinite |] ==> 0 <= st x";
-by (stac (hypreal_numeral_0_eq_0 RS sym) 1);
-by (rtac (st_number_of RS subst) 1);
-by (rtac st_le 1);
-by Auto_tac;
-qed "st_zero_le";
-
-Goal "[| x <= 0; x: HFinite |] ==> st x <= 0";
-by (stac (hypreal_numeral_0_eq_0 RS sym) 1);
-by (rtac (st_number_of RS subst) 1);
-by (rtac st_le 1);
-by Auto_tac;
-qed "st_zero_ge";
-
-Goal "x: HFinite ==> abs(st x) = st(abs x)";
-by (case_tac "0 <= x" 1);
-by (auto_tac (claset() addSDs [order_less_imp_le],
- simpset() addsimps [linorder_not_le,st_zero_le,hrabs_eqI1, hrabs_minus_eqI1,
- st_zero_ge,st_minus]));
-qed "st_hrabs";
-
-(*--------------------------------------------------------------------
- Alternative definitions for HFinite using Free ultrafilter
- --------------------------------------------------------------------*)
-
-Goal "[| X: Rep_hypreal x; Y: Rep_hypreal x |] \
-\ ==> {n. X n = Y n} : FreeUltrafilterNat";
-by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
-by Auto_tac;
-by (Ultra_tac 1);
-qed "FreeUltrafilterNat_Rep_hypreal";
-
-Goal "{n. Yb n < Y n} Int {n. -y = Yb n} <= {n. -y < Y n}";
-by Auto_tac;
-qed "lemma_free1";
-
-Goal "{n. Xa n < Yc n} Int {n. y = Yc n} <= {n. Xa n < y}";
-by Auto_tac;
-qed "lemma_free2";
-
-Goalw [HFinite_def]
- "x : HFinite ==> EX X: Rep_hypreal x. \
-\ EX u. {n. abs (X n) < u}: FreeUltrafilterNat";
-by (auto_tac (claset(), simpset() addsimps [abs_less_iff, inst "a" "x" minus_less_iff]));
-by (auto_tac (claset(), simpset() addsimps
- [hypreal_less_def,SReal_iff,hypreal_minus,
- hypreal_of_real_def]));
-by (dtac FreeUltrafilterNat_Rep_hypreal 1 THEN assume_tac 1);
-by (rename_tac "Z" 1);
-by (res_inst_tac [("x","Z")] bexI 1 THEN assume_tac 2);
-by (res_inst_tac [("x","y")] exI 1);
-by (Ultra_tac 1);
-qed "HFinite_FreeUltrafilterNat";
-
-Goalw [HFinite_def]
- "EX X: Rep_hypreal x. \
-\ EX u. {n. abs (X n) < u}: FreeUltrafilterNat\
-\ ==> x : HFinite";
-by (auto_tac (claset(), simpset() addsimps [abs_less_iff, inst "a" "x" minus_less_iff]));
-by (res_inst_tac [("x","hypreal_of_real u")] bexI 1);
-by (auto_tac (claset() addSIs [exI],
- simpset() addsimps
- [hypreal_less_def,SReal_iff,hypreal_minus, hypreal_of_real_def]));
-by (ALLGOALS Ultra_tac);
-qed "FreeUltrafilterNat_HFinite";
-
-Goal "(x : HFinite) = (EX X: Rep_hypreal x. \
-\ EX u. {n. abs (X n) < u}: FreeUltrafilterNat)";
-by (blast_tac (claset() addSIs [HFinite_FreeUltrafilterNat,
- FreeUltrafilterNat_HFinite]) 1);
-qed "HFinite_FreeUltrafilterNat_iff";
-
-(*--------------------------------------------------------------------
- Alternative definitions for HInfinite using Free ultrafilter
- --------------------------------------------------------------------*)
-Goal "- {n. (u::real) < abs (xa n)} = {n. abs (xa n) <= u}";
-by Auto_tac;
-qed "lemma_Compl_eq";
-
-Goal "- {n. abs (xa n) < (u::real)} = {n. u <= abs (xa n)}";
-by Auto_tac;
-qed "lemma_Compl_eq2";
-
-Goal "{n. abs (xa n) <= (u::real)} Int {n. u <= abs (xa n)} \
-\ = {n. abs(xa n) = u}";
-by Auto_tac;
-qed "lemma_Int_eq1";
-
-Goal "{n. abs (xa n) = u} <= {n. abs (xa n) < u + (1::real)}";
-by Auto_tac;
-qed "lemma_FreeUltrafilterNat_one";
-
-(*-------------------------------------
- Exclude this type of sets from free
- ultrafilter for Infinite numbers!
- -------------------------------------*)
-Goal "[| xa: Rep_hypreal x; \
-\ {n. abs (xa n) = u} : FreeUltrafilterNat \
-\ |] ==> x: HFinite";
-by (rtac FreeUltrafilterNat_HFinite 1);
-by (res_inst_tac [("x","xa")] bexI 1);
-by (res_inst_tac [("x","u + 1")] exI 1);
-by (Ultra_tac 1 THEN assume_tac 1);
-qed "FreeUltrafilterNat_const_Finite";
-
-val [prem] = goal thy "x : HInfinite ==> EX X: Rep_hypreal x. \
-\ ALL u. {n. u < abs (X n)}: FreeUltrafilterNat";
-by (cut_facts_tac [(prem RS (HInfinite_HFinite_iff RS iffD1))] 1);
-by (cut_inst_tac [("x","x")] Rep_hypreal_nonempty 1);
-by (auto_tac (claset(), simpset() delsimps [Rep_hypreal_nonempty]
- addsimps [HFinite_FreeUltrafilterNat_iff,Bex_def]));
-by (REPEAT(dtac spec 1));
-by Auto_tac;
-by (dres_inst_tac [("x","u")] spec 1 THEN
- REPEAT(dtac FreeUltrafilterNat_Compl_mem 1));
-by (dtac FreeUltrafilterNat_Int 1 THEN assume_tac 1);
-by (asm_full_simp_tac (simpset() addsimps [lemma_Compl_eq,
- lemma_Compl_eq2,lemma_Int_eq1]) 1);
-by (auto_tac (claset() addDs [FreeUltrafilterNat_const_Finite],
- simpset() addsimps [(prem RS (HInfinite_HFinite_iff RS iffD1))]));
-qed "HInfinite_FreeUltrafilterNat";
-
-(* yet more lemmas! *)
-Goal "{n. abs (Xa n) < u} Int {n. X n = Xa n} \
-\ <= {n. abs (X n) < (u::real)}";
-by Auto_tac;
-qed "lemma_Int_HI";
-
-Goal "{n. u < abs (X n)} Int {n. abs (X n) < (u::real)} = {}";
-by (auto_tac (claset() addIs [order_less_asym], simpset()));
-qed "lemma_Int_HIa";
-
-Goal "EX X: Rep_hypreal x. ALL u. \
-\ {n. u < abs (X n)}: FreeUltrafilterNat\
-\ ==> x : HInfinite";
-by (rtac (HInfinite_HFinite_iff RS iffD2) 1);
-by (Step_tac 1 THEN dtac HFinite_FreeUltrafilterNat 1);
-by Auto_tac;
-by (dres_inst_tac [("x","u")] spec 1);
-by (dtac FreeUltrafilterNat_Rep_hypreal 1 THEN assume_tac 1);
-by (dres_inst_tac [("Y","{n. X n = Xa n}")] FreeUltrafilterNat_Int 1);
-by (dtac (lemma_Int_HI RSN (2,FreeUltrafilterNat_subset)) 2);
-by (dres_inst_tac [("Y","{n. abs (X n) < u}")] FreeUltrafilterNat_Int 2);
-by (auto_tac (claset(), simpset() addsimps [lemma_Int_HIa,
- FreeUltrafilterNat_empty]));
-qed "FreeUltrafilterNat_HInfinite";
-
-Goal "(x : HInfinite) = (EX X: Rep_hypreal x. \
-\ ALL u. {n. u < abs (X n)}: FreeUltrafilterNat)";
-by (blast_tac (claset() addSIs [HInfinite_FreeUltrafilterNat,
- FreeUltrafilterNat_HInfinite]) 1);
-qed "HInfinite_FreeUltrafilterNat_iff";
-
-(*--------------------------------------------------------------------
- Alternative definitions for Infinitesimal using Free ultrafilter
- --------------------------------------------------------------------*)
-
-Goal "{n. - u < Yd n} Int {n. xa n = Yd n} <= {n. -u < xa n}";
-by Auto_tac;
-qed "lemma_free4";
-
-Goal "{n. Yb n < u} Int {n. xa n = Yb n} <= {n. xa n < u}";
-by Auto_tac;
-qed "lemma_free5";
-
-Goalw [Infinitesimal_def]
- "x : Infinitesimal ==> EX X: Rep_hypreal x. \
-\ ALL u. 0 < u --> {n. abs (X n) < u}: FreeUltrafilterNat";
-by (auto_tac (claset(), simpset() addsimps [abs_less_iff, inst "a" "x" minus_less_iff]));
-by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
-by (EVERY[Auto_tac, rtac bexI 1, rtac lemma_hyprel_refl 2, Step_tac 1]);
-by (dtac (hypreal_of_real_less_iff RS iffD2) 1);
-by (dres_inst_tac [("x","hypreal_of_real u")] bspec 1);
-by Auto_tac;
-by (auto_tac (claset(),
- simpset() addsimps [hypreal_less_def, hypreal_minus,
- hypreal_of_real_def]));
-by (Ultra_tac 1);
-qed "Infinitesimal_FreeUltrafilterNat";
-
-Goalw [Infinitesimal_def]
- "EX X: Rep_hypreal x. \
-\ ALL u. 0 < u --> {n. abs (X n) < u} : FreeUltrafilterNat \
-\ ==> x : Infinitesimal";
-by (auto_tac (claset(),
- simpset() addsimps [abs_less_iff,abs_interval_iff, inst "a" "x" minus_less_iff]));
-by (auto_tac (claset(),
- simpset() addsimps [SReal_iff]));
-by (auto_tac (claset() addSIs [exI]
- addIs [FreeUltrafilterNat_subset],
- simpset() addsimps [hypreal_less_def, hypreal_minus,hypreal_of_real_def]));
-qed "FreeUltrafilterNat_Infinitesimal";
-
-Goal "(x : Infinitesimal) = (EX X: Rep_hypreal x. \
-\ ALL u. 0 < u --> {n. abs (X n) < u}: FreeUltrafilterNat)";
-by (blast_tac (claset() addSIs [Infinitesimal_FreeUltrafilterNat,
- FreeUltrafilterNat_Infinitesimal]) 1);
-qed "Infinitesimal_FreeUltrafilterNat_iff";
-
-(*------------------------------------------------------------------------
- Infinitesimals as smaller than 1/n for all n::nat (> 0)
- ------------------------------------------------------------------------*)
-
-Goal "(ALL r. 0 < r --> x < r) = (ALL n. x < inverse(real (Suc n)))";
-by (auto_tac (claset(), simpset() addsimps [real_of_nat_Suc_gt_zero]));
-by (blast_tac (claset() addSDs [reals_Archimedean]
- addIs [order_less_trans]) 1);
-qed "lemma_Infinitesimal";
-
-Goal "(ALL r: Reals. 0 < r --> x < r) = \
-\ (ALL n. x < inverse(hypreal_of_nat (Suc n)))";
-by (Step_tac 1);
-by (dres_inst_tac [("x","inverse (hypreal_of_real(real (Suc n)))")]
- bspec 1);
-by (full_simp_tac (simpset() addsimps [SReal_inverse]) 1);
-by (rtac (real_of_nat_Suc_gt_zero RS positive_imp_inverse_positive RS
- (hypreal_of_real_less_iff RS iffD2) RSN(2,impE)) 1);
-by (assume_tac 2);
-by (asm_full_simp_tac (simpset() addsimps
- [real_of_nat_Suc_gt_zero, hypreal_of_nat_def]) 1);
-by (auto_tac (claset() addSDs [reals_Archimedean],
- simpset() addsimps [SReal_iff]));
-by (dtac (hypreal_of_real_less_iff RS iffD2) 1);
-by (asm_full_simp_tac (simpset() addsimps
- [real_of_nat_Suc_gt_zero, hypreal_of_nat_def])1);
-by (blast_tac (claset() addIs [order_less_trans]) 1);
-qed "lemma_Infinitesimal2";
-
-Goalw [Infinitesimal_def]
- "Infinitesimal = {x. ALL n. abs x < inverse (hypreal_of_nat (Suc n))}";
-by (auto_tac (claset(), simpset() addsimps [lemma_Infinitesimal2]));
-qed "Infinitesimal_hypreal_of_nat_iff";
-
-
-(*-------------------------------------------------------------------------
- Proof that omega is an infinite number and
- hence that epsilon is an infinitesimal number.
- -------------------------------------------------------------------------*)
-Goal "{n. n < Suc m} = {n. n < m} Un {n. n = m}";
-by (auto_tac (claset(), simpset() addsimps [less_Suc_eq]));
-qed "Suc_Un_eq";
-
-(*-------------------------------------------
- Prove that any segment is finite and
- hence cannot belong to FreeUltrafilterNat
- -------------------------------------------*)
-Goal "finite {n::nat. n < m}";
-by (induct_tac "m" 1);
-by (auto_tac (claset(), simpset() addsimps [Suc_Un_eq]));
-qed "finite_nat_segment";
-
-Goal "finite {n::nat. real n < real (m::nat)}";
-by (auto_tac (claset() addIs [finite_nat_segment], simpset()));
-qed "finite_real_of_nat_segment";
-
-Goal "finite {n::nat. real n < u}";
-by (cut_inst_tac [("x","u")] reals_Archimedean2 1);
-by (Step_tac 1);
-by (rtac (finite_real_of_nat_segment RSN (2,finite_subset)) 1);
-by (auto_tac (claset() addDs [order_less_trans], simpset()));
-qed "finite_real_of_nat_less_real";
-
-Goal "{n. f n <= u} = {n. f n < u} Un {n. u = (f n :: real)}";
-by (auto_tac (claset() addDs [order_le_imp_less_or_eq],
- simpset() addsimps [order_less_imp_le]));
-qed "lemma_real_le_Un_eq";
-
-Goal "finite {n::nat. real n <= u}";
-by (auto_tac (claset(),
- simpset() addsimps [lemma_real_le_Un_eq,lemma_finite_omega_set,
- finite_real_of_nat_less_real]));
-qed "finite_real_of_nat_le_real";
-
-Goal "finite {n::nat. abs(real n) <= u}";
-by (simp_tac (simpset() addsimps [real_of_nat_Suc_gt_zero,
- finite_real_of_nat_le_real]) 1);
-qed "finite_rabs_real_of_nat_le_real";
-
-Goal "{n. abs(real n) <= u} ~: FreeUltrafilterNat";
-by (blast_tac (claset() addSIs [FreeUltrafilterNat_finite,
- finite_rabs_real_of_nat_le_real]) 1);
-qed "rabs_real_of_nat_le_real_FreeUltrafilterNat";
-
-Goal "{n. u < real n} : FreeUltrafilterNat";
-by (rtac ccontr 1 THEN dtac FreeUltrafilterNat_Compl_mem 1);
-by (subgoal_tac "- {n::nat. u < real n} = {n. real n <= u}" 1);
-by (Force_tac 2);
-by (asm_full_simp_tac (simpset() addsimps
- [finite_real_of_nat_le_real RS FreeUltrafilterNat_finite]) 1);
-qed "FreeUltrafilterNat_nat_gt_real";
-
-(*--------------------------------------------------------------
- The complement of {n. abs(real n) <= u} =
- {n. u < abs (real n)} is in FreeUltrafilterNat
- by property of (free) ultrafilters
- --------------------------------------------------------------*)
-Goal "- {n::nat. real n <= u} = {n. u < real n}";
-by (auto_tac (claset() addSDs [order_le_less_trans],
- simpset() addsimps [linorder_not_le]));
-val lemma = result();
-
-(*-----------------------------------------------
- Omega is a member of HInfinite
- -----------------------------------------------*)
-
-Goal "hyprel``{%n::nat. real (Suc n)} : hypreal";
-by Auto_tac;
-qed "hypreal_omega";
-
-Goal "{n. u < real n} : FreeUltrafilterNat";
-by (cut_inst_tac [("u","u")] rabs_real_of_nat_le_real_FreeUltrafilterNat 1);
-by (auto_tac (claset() addDs [FreeUltrafilterNat_Compl_mem],
- simpset() addsimps [lemma]));
-qed "FreeUltrafilterNat_omega";
-
-Goalw [omega_def] "omega: HInfinite";
-by (auto_tac (claset() addSIs [FreeUltrafilterNat_HInfinite],
- simpset()));
-by (rtac bexI 1);
-by (rtac lemma_hyprel_refl 2);
-by Auto_tac;
-by (simp_tac (simpset() addsimps [real_of_nat_Suc, diff_less_eq RS sym,
- FreeUltrafilterNat_omega]) 1);
-qed "HInfinite_omega";
-Addsimps [HInfinite_omega];
-
-(*-----------------------------------------------
- Epsilon is a member of Infinitesimal
- -----------------------------------------------*)
-
-Goal "epsilon : Infinitesimal";
-by (auto_tac (claset() addSIs [HInfinite_inverse_Infinitesimal,HInfinite_omega],
- simpset() addsimps [hypreal_epsilon_inverse_omega]));
-qed "Infinitesimal_epsilon";
-Addsimps [Infinitesimal_epsilon];
-
-Goal "epsilon : HFinite";
-by (auto_tac (claset() addIs [Infinitesimal_subset_HFinite RS subsetD],
- simpset()));
-qed "HFinite_epsilon";
-Addsimps [HFinite_epsilon];
-
-Goal "epsilon @= 0";
-by (simp_tac (simpset() addsimps [mem_infmal_iff RS sym]) 1);
-qed "epsilon_approx_zero";
-Addsimps [epsilon_approx_zero];
-
-(*------------------------------------------------------------------------
- Needed for proof that we define a hyperreal [<X(n)] @= hypreal_of_real a given
- that ALL n. |X n - a| < 1/n. Used in proof of NSLIM => LIM.
- -----------------------------------------------------------------------*)
-
-Goal "0 < u ==> \
-\ (u < inverse (real(Suc n))) = (real(Suc n) < inverse u)";
-by (asm_full_simp_tac (simpset() addsimps [inverse_eq_divide]) 1);
-by (stac pos_less_divide_eq 1);
-by (assume_tac 1);
-by (stac pos_less_divide_eq 1);
-by (simp_tac (simpset() addsimps [real_mult_commute]) 2);
-by (simp_tac (simpset() addsimps [real_of_nat_Suc_gt_zero]) 1);
-qed "real_of_nat_less_inverse_iff";
-
-Goal "0 < u ==> finite {n. u < inverse(real(Suc n))}";
-by (asm_simp_tac (simpset() addsimps [real_of_nat_less_inverse_iff]) 1);
-by (asm_simp_tac (simpset() addsimps [real_of_nat_Suc,
- less_diff_eq RS sym]) 1);
-by (rtac finite_real_of_nat_less_real 1);
-qed "finite_inverse_real_of_posnat_gt_real";
-
-Goal "{n. u <= inverse(real(Suc n))} = \
-\ {n. u < inverse(real(Suc n))} Un {n. u = inverse(real(Suc n))}";
-by (auto_tac (claset() addDs [order_le_imp_less_or_eq],
- simpset() addsimps [order_less_imp_le]));
-qed "lemma_real_le_Un_eq2";
-
-Goal "(inverse (real(Suc n)) <= r) = (1 <= r * real(Suc n))";
-by (simp_tac (simpset() addsimps [linorder_not_less RS sym]) 1);
-by (simp_tac (simpset() addsimps [inverse_eq_divide]) 1);
-by (stac pos_less_divide_eq 1);
-by (simp_tac (simpset() addsimps [real_of_nat_Suc_gt_zero]) 1);
-by (simp_tac (simpset() addsimps [real_mult_commute]) 1);
-qed "real_of_nat_inverse_le_iff";
-
-Goal "(u = inverse (real(Suc n))) = (real(Suc n) = inverse u)";
-by (auto_tac (claset(),
- simpset() addsimps [inverse_inverse_eq, real_of_nat_Suc_gt_zero,
- real_not_refl2 RS not_sym]));
-qed "real_of_nat_inverse_eq_iff";
-
-Goal "finite {n::nat. u = inverse(real(Suc n))}";
-by (asm_simp_tac (simpset() addsimps [real_of_nat_inverse_eq_iff]) 1);
-by (cut_inst_tac [("x","inverse u - 1")] lemma_finite_omega_set 1);
-by (asm_full_simp_tac (simpset() addsimps [real_of_nat_Suc,
- diff_eq_eq RS sym, eq_commute]) 1);
-qed "lemma_finite_omega_set2";
-
-Goal "0 < u ==> finite {n. u <= inverse(real(Suc n))}";
-by (auto_tac (claset(),
- simpset() addsimps [lemma_real_le_Un_eq2,lemma_finite_omega_set2,
- finite_inverse_real_of_posnat_gt_real]));
-qed "finite_inverse_real_of_posnat_ge_real";
-
-Goal "0 < u ==> \
-\ {n. u <= inverse(real(Suc n))} ~: FreeUltrafilterNat";
-by (blast_tac (claset() addSIs [FreeUltrafilterNat_finite,
- finite_inverse_real_of_posnat_ge_real]) 1);
-qed "inverse_real_of_posnat_ge_real_FreeUltrafilterNat";
-
-(*--------------------------------------------------------------
- The complement of {n. u <= inverse(real(Suc n))} =
- {n. inverse(real(Suc n)) < u} is in FreeUltrafilterNat
- by property of (free) ultrafilters
- --------------------------------------------------------------*)
-Goal "- {n. u <= inverse(real(Suc n))} = \
-\ {n. inverse(real(Suc n)) < u}";
-by (auto_tac (claset() addSDs [order_le_less_trans],
- simpset() addsimps [linorder_not_le]));
-val lemma = result();
-
-Goal "0 < u ==> \
-\ {n. inverse(real(Suc n)) < u} : FreeUltrafilterNat";
-by (cut_inst_tac [("u","u")] inverse_real_of_posnat_ge_real_FreeUltrafilterNat 1);
-by (auto_tac (claset() addDs [FreeUltrafilterNat_Compl_mem],
- simpset() addsimps [lemma]));
-qed "FreeUltrafilterNat_inverse_real_of_posnat";
-
-(*--------------------------------------------------------------
- Example where we get a hyperreal from a real sequence
- for which a particular property holds. The theorem is
- used in proofs about equivalence of nonstandard and
- standard neighbourhoods. Also used for equivalence of
- nonstandard ans standard definitions of pointwise
- limit (the theorem was previously in REALTOPOS.thy).
- -------------------------------------------------------------*)
-(*-----------------------------------------------------
- |X(n) - x| < 1/n ==> [<X n>] - hypreal_of_real x|: Infinitesimal
- -----------------------------------------------------*)
-Goal "ALL n. abs(X n + -x) < inverse(real(Suc n)) \
-\ ==> Abs_hypreal(hyprel``{X}) + -hypreal_of_real x : Infinitesimal";
-by (auto_tac (claset() addSIs [bexI]
- addDs [FreeUltrafilterNat_inverse_real_of_posnat,
- FreeUltrafilterNat_all,FreeUltrafilterNat_Int]
- addIs [order_less_trans, FreeUltrafilterNat_subset],
- simpset() addsimps [hypreal_minus,
- hypreal_of_real_def,hypreal_add,
- Infinitesimal_FreeUltrafilterNat_iff,hypreal_inverse]));
-qed "real_seq_to_hypreal_Infinitesimal";
-
-Goal "ALL n. abs(X n + -x) < inverse(real(Suc n)) \
-\ ==> Abs_hypreal(hyprel``{X}) @= hypreal_of_real x";
-by (stac approx_minus_iff 1);
-by (rtac (mem_infmal_iff RS subst) 1);
-by (etac real_seq_to_hypreal_Infinitesimal 1);
-qed "real_seq_to_hypreal_approx";
-
-Goal "ALL n. abs(x + -X n) < inverse(real(Suc n)) \
-\ ==> Abs_hypreal(hyprel``{X}) @= hypreal_of_real x";
-by (asm_full_simp_tac (simpset() addsimps [abs_minus_add_cancel,
- real_seq_to_hypreal_approx]) 1);
-qed "real_seq_to_hypreal_approx2";
-
-Goal "ALL n. abs(X n + -Y n) < inverse(real(Suc n)) \
-\ ==> Abs_hypreal(hyprel``{X}) + \
-\ -Abs_hypreal(hyprel``{Y}) : Infinitesimal";
-by (auto_tac (claset() addSIs [bexI]
- addDs [FreeUltrafilterNat_inverse_real_of_posnat,
- FreeUltrafilterNat_all,FreeUltrafilterNat_Int]
- addIs [order_less_trans, FreeUltrafilterNat_subset],
- simpset() addsimps
- [Infinitesimal_FreeUltrafilterNat_iff,hypreal_minus,hypreal_add,
- hypreal_inverse]));
-qed "real_seq_to_hypreal_Infinitesimal2";
-
-
-
-
-
-(*MOVE UP*)
-Goal "[| x + y : HInfinite; y: HFinite |] ==> x : HInfinite";
-by (rtac ccontr 1);
-by (dtac (HFinite_HInfinite_iff RS iffD2) 1);
-by (auto_tac (claset() addDs [HFinite_add],simpset()
- addsimps [HInfinite_HFinite_iff]));
-qed "HInfinite_HFinite_add_cancel";
-
-Goal "[| x : HInfinite; y : HFinite |] ==> x + y : HInfinite";
-by (res_inst_tac [("y","-y")] HInfinite_HFinite_add_cancel 1);
-by (auto_tac (claset(),simpset() addsimps [hypreal_add_assoc,
- HFinite_minus_iff]));
-qed "HInfinite_HFinite_add";
-
-Goal "[| x : HInfinite; x <= y; 0 <= x |] ==> y : HInfinite";
-by (auto_tac (claset() addIs [HFinite_bounded],simpset()
- addsimps [HInfinite_HFinite_iff]));
-qed "HInfinite_ge_HInfinite";
-
-Goal "[| x : Infinitesimal; x ~= 0 |] ==> inverse x : HInfinite";
-by (rtac ccontr 1 THEN dtac (HFinite_HInfinite_iff RS iffD2) 1);
-by (auto_tac (claset() addDs [Infinitesimal_HFinite_mult2],simpset()));
-qed "Infinitesimal_inverse_HInfinite";
-
-Goal "[| x : HInfinite; y : HFinite - Infinitesimal |] \
-\ ==> x * y : HInfinite";
-by (rtac ccontr 1 THEN dtac (HFinite_HInfinite_iff RS iffD2) 1);
-by (ftac HFinite_Infinitesimal_not_zero 1);
-by (dtac HFinite_not_Infinitesimal_inverse 1);
-by (Step_tac 1 THEN dtac HFinite_mult 1);
-by (auto_tac (claset(),simpset() addsimps [hypreal_mult_assoc,
- HFinite_HInfinite_iff]));
-qed "HInfinite_HFinite_not_Infinitesimal_mult";
-
-Goal "[| x : HInfinite; y : HFinite - Infinitesimal |] \
-\ ==> y * x : HInfinite";
-by (auto_tac (claset(),simpset() addsimps [hypreal_mult_commute,
- HInfinite_HFinite_not_Infinitesimal_mult]));
-qed "HInfinite_HFinite_not_Infinitesimal_mult2";
-
-Goal "[| x : HInfinite; 0 < x; y : Reals |] ==> y < x";
-by (auto_tac (claset() addSDs [bspec],simpset() addsimps
- [HInfinite_def,hrabs_def,order_less_imp_le]));
-qed "HInfinite_gt_SReal";
-
-Goal "[| x : HInfinite; 0 < x |] ==> 1 < x";
-by (auto_tac (claset() addIs [HInfinite_gt_SReal],simpset()));
-qed "HInfinite_gt_zero_gt_one";
-
-
-Goal "1 ~: HInfinite";
-by (simp_tac (simpset() addsimps [HInfinite_HFinite_iff]) 1);
-qed "not_HInfinite_one";
-Addsimps [not_HInfinite_one];
--- a/src/HOL/Hyperreal/NSA.thy Wed Jan 28 17:01:01 2004 +0100
+++ b/src/HOL/Hyperreal/NSA.thy Thu Jan 29 16:51:17 2004 +0100
@@ -1,46 +1,2346 @@
(* Title : NSA.thy
Author : Jacques D. Fleuriot
Copyright : 1998 University of Cambridge
- Description : Infinite numbers, Infinitesimals,
- infinitely close relation etc.
-*)
+*)
-NSA = HRealAbs + RComplete +
+header{*Infinite Numbers, Infinitesimals, Infinitely Close Relation*}
+
+theory NSA = HRealAbs + RComplete:
constdefs
Infinitesimal :: "hypreal set"
- "Infinitesimal == {x. ALL r: Reals. 0 < r --> abs x < r}"
+ "Infinitesimal == {x. \<forall>r \<in> Reals. 0 < r --> abs x < r}"
HFinite :: "hypreal set"
- "HFinite == {x. EX r: Reals. abs x < r}"
+ "HFinite == {x. \<exists>r \<in> Reals. abs x < r}"
HInfinite :: "hypreal set"
- "HInfinite == {x. ALL r: Reals. r < abs x}"
+ "HInfinite == {x. \<forall>r \<in> Reals. r < abs x}"
(* standard part map *)
- st :: hypreal => hypreal
- "st == (%x. @r. x : HFinite & r:Reals & r @= x)"
+ st :: "hypreal => hypreal"
+ "st == (%x. @r. x \<in> HFinite & r \<in> Reals & r @= x)"
- monad :: hypreal => hypreal set
+ monad :: "hypreal => hypreal set"
"monad x == {y. x @= y}"
- galaxy :: hypreal => hypreal set
- "galaxy x == {y. (x + -y) : HFinite}"
-
+ galaxy :: "hypreal => hypreal set"
+ "galaxy x == {y. (x + -y) \<in> HFinite}"
+
(* infinitely close *)
approx :: "[hypreal, hypreal] => bool" (infixl "@=" 50)
- "x @= y == (x + -y) : Infinitesimal"
+ "x @= y == (x + -y) \<in> Infinitesimal"
+
+
+defs
+
+ (*standard real numbers as a subset of the hyperreals*)
+ SReal_def: "Reals == {x. \<exists>r. x = hypreal_of_real r}"
+
+syntax (xsymbols)
+ approx :: "[hypreal, hypreal] => bool" (infixl "\<approx>" 50)
+
+
+
+(*--------------------------------------------------------------------
+ Closure laws for members of (embedded) set standard real Reals
+ --------------------------------------------------------------------*)
+
+lemma SReal_add: "[| (x::hypreal) \<in> Reals; y \<in> Reals |] ==> x + y \<in> Reals"
+apply (auto simp add: SReal_def)
+apply (rule_tac x = "r + ra" in exI, simp)
+done
+
+lemma SReal_mult: "[| (x::hypreal) \<in> Reals; y \<in> Reals |] ==> x * y \<in> Reals"
+apply (simp add: SReal_def, safe)
+apply (rule_tac x = "r * ra" in exI)
+apply (simp (no_asm) add: hypreal_of_real_mult)
+done
+
+lemma SReal_inverse: "(x::hypreal) \<in> Reals ==> inverse x \<in> Reals"
+apply (simp add: SReal_def)
+apply (blast intro: hypreal_of_real_inverse [symmetric])
+done
+
+lemma SReal_divide: "[| (x::hypreal) \<in> Reals; y \<in> Reals |] ==> x/y \<in> Reals"
+apply (simp (no_asm_simp) add: SReal_mult SReal_inverse hypreal_divide_def)
+done
+
+lemma SReal_minus: "(x::hypreal) \<in> Reals ==> -x \<in> Reals"
+apply (simp add: SReal_def)
+apply (blast intro: hypreal_of_real_minus [symmetric])
+done
+
+lemma SReal_minus_iff: "(-x \<in> Reals) = ((x::hypreal) \<in> Reals)"
+apply auto
+apply (erule_tac [2] SReal_minus)
+apply (drule SReal_minus, auto)
+done
+declare SReal_minus_iff [simp]
+
+lemma SReal_add_cancel: "[| (x::hypreal) + y \<in> Reals; y \<in> Reals |] ==> x \<in> Reals"
+apply (drule_tac x = y in SReal_minus)
+apply (drule SReal_add, assumption, auto)
+done
+
+lemma SReal_hrabs: "(x::hypreal) \<in> Reals ==> abs x \<in> Reals"
+apply (simp add: SReal_def)
+apply (auto simp add: hypreal_of_real_hrabs)
+done
+
+lemma SReal_hypreal_of_real: "hypreal_of_real x \<in> Reals"
+by (simp add: SReal_def)
+declare SReal_hypreal_of_real [simp]
+
+lemma SReal_number_of: "(number_of w ::hypreal) \<in> Reals"
+apply (unfold hypreal_number_of_def)
+apply (rule SReal_hypreal_of_real)
+done
+declare SReal_number_of [simp]
+
+(** As always with numerals, 0 and 1 are special cases **)
+
+lemma Reals_0: "(0::hypreal) \<in> Reals"
+apply (subst hypreal_numeral_0_eq_0 [symmetric])
+apply (rule SReal_number_of)
+done
+declare Reals_0 [simp]
+
+lemma Reals_1: "(1::hypreal) \<in> Reals"
+apply (subst hypreal_numeral_1_eq_1 [symmetric])
+apply (rule SReal_number_of)
+done
+declare Reals_1 [simp]
+
+lemma SReal_divide_number_of: "r \<in> Reals ==> r/(number_of w::hypreal) \<in> Reals"
+apply (unfold hypreal_divide_def)
+apply (blast intro!: SReal_number_of SReal_mult SReal_inverse)
+done
+
+(* Infinitesimal epsilon not in Reals *)
+
+lemma SReal_epsilon_not_mem: "epsilon \<notin> Reals"
+apply (simp add: SReal_def)
+apply (auto simp add: hypreal_of_real_not_eq_epsilon [THEN not_sym])
+done
+
+lemma SReal_omega_not_mem: "omega \<notin> Reals"
+apply (simp add: SReal_def)
+apply (auto simp add: hypreal_of_real_not_eq_omega [THEN not_sym])
+done
+
+lemma SReal_UNIV_real: "{x. hypreal_of_real x \<in> Reals} = (UNIV::real set)"
+by (simp add: SReal_def)
+
+lemma SReal_iff: "(x \<in> Reals) = (\<exists>y. x = hypreal_of_real y)"
+by (simp add: SReal_def)
+
+lemma hypreal_of_real_image: "hypreal_of_real `(UNIV::real set) = Reals"
+by (auto simp add: SReal_def)
+
+lemma inv_hypreal_of_real_image: "inv hypreal_of_real ` Reals = UNIV"
+apply (auto simp add: SReal_def)
+apply (rule inj_hypreal_of_real [THEN inv_f_f, THEN subst], blast)
+done
+
+lemma SReal_hypreal_of_real_image:
+ "[| \<exists>x. x: P; P <= Reals |] ==> \<exists>Q. P = hypreal_of_real ` Q"
+apply (simp add: SReal_def, blast)
+done
+
+lemma SReal_dense: "[| (x::hypreal) \<in> Reals; y \<in> Reals; x<y |] ==> \<exists>r \<in> Reals. x<r & r<y"
+apply (auto simp add: SReal_iff)
+apply (drule real_dense, safe)
+apply (rule_tac x = "hypreal_of_real r" in bexI, auto)
+done
+
+(*------------------------------------------------------------------
+ Completeness of Reals
+ ------------------------------------------------------------------*)
+lemma SReal_sup_lemma: "P <= Reals ==> ((\<exists>x \<in> P. y < x) =
+ (\<exists>X. hypreal_of_real X \<in> P & y < hypreal_of_real X))"
+by (blast dest!: SReal_iff [THEN iffD1])
+
+lemma SReal_sup_lemma2:
+ "[| P <= Reals; \<exists>x. x \<in> P; \<exists>y \<in> Reals. \<forall>x \<in> P. x < y |]
+ ==> (\<exists>X. X \<in> {w. hypreal_of_real w \<in> P}) &
+ (\<exists>Y. \<forall>X \<in> {w. hypreal_of_real w \<in> P}. X < Y)"
+apply (rule conjI)
+apply (fast dest!: SReal_iff [THEN iffD1])
+apply (auto, frule subsetD, assumption)
+apply (drule SReal_iff [THEN iffD1])
+apply (auto, rule_tac x = ya in exI, auto)
+done
+
+(*------------------------------------------------------
+ lifting of ub and property of lub
+ -------------------------------------------------------*)
+lemma hypreal_of_real_isUb_iff:
+ "(isUb (Reals) (hypreal_of_real ` Q) (hypreal_of_real Y)) =
+ (isUb (UNIV :: real set) Q Y)"
+apply (simp add: isUb_def setle_def)
+done
+
+lemma hypreal_of_real_isLub1:
+ "isLub Reals (hypreal_of_real ` Q) (hypreal_of_real Y)
+ ==> isLub (UNIV :: real set) Q Y"
+apply (simp add: isLub_def leastP_def)
+apply (auto intro: hypreal_of_real_isUb_iff [THEN iffD2]
+ simp add: hypreal_of_real_isUb_iff setge_def)
+done
+
+lemma hypreal_of_real_isLub2:
+ "isLub (UNIV :: real set) Q Y
+ ==> isLub Reals (hypreal_of_real ` Q) (hypreal_of_real Y)"
+apply (simp add: isLub_def leastP_def)
+apply (auto simp add: hypreal_of_real_isUb_iff setge_def)
+apply (frule_tac x2 = x in isUbD2a [THEN SReal_iff [THEN iffD1], THEN exE])
+ prefer 2 apply assumption
+apply (drule_tac x = xa in spec)
+apply (auto simp add: hypreal_of_real_isUb_iff)
+done
+
+lemma hypreal_of_real_isLub_iff: "(isLub Reals (hypreal_of_real ` Q) (hypreal_of_real Y)) =
+ (isLub (UNIV :: real set) Q Y)"
+apply (blast intro: hypreal_of_real_isLub1 hypreal_of_real_isLub2)
+done
+
+(* lemmas *)
+lemma lemma_isUb_hypreal_of_real:
+ "isUb Reals P Y ==> \<exists>Yo. isUb Reals P (hypreal_of_real Yo)"
+by (auto simp add: SReal_iff isUb_def)
+
+lemma lemma_isLub_hypreal_of_real:
+ "isLub Reals P Y ==> \<exists>Yo. isLub Reals P (hypreal_of_real Yo)"
+by (auto simp add: isLub_def leastP_def isUb_def SReal_iff)
+
+lemma lemma_isLub_hypreal_of_real2:
+ "\<exists>Yo. isLub Reals P (hypreal_of_real Yo) ==> \<exists>Y. isLub Reals P Y"
+by (auto simp add: isLub_def leastP_def isUb_def)
+
+lemma SReal_complete: "[| P <= Reals; \<exists>x. x \<in> P; \<exists>Y. isUb Reals P Y |]
+ ==> \<exists>t::hypreal. isLub Reals P t"
+apply (frule SReal_hypreal_of_real_image)
+apply (auto, drule lemma_isUb_hypreal_of_real)
+apply (auto intro!: reals_complete lemma_isLub_hypreal_of_real2 simp add: hypreal_of_real_isLub_iff hypreal_of_real_isUb_iff)
+done
+
+(*--------------------------------------------------------------------
+ Set of finite elements is a subring of the extended reals
+ --------------------------------------------------------------------*)
+lemma HFinite_add: "[|x \<in> HFinite; y \<in> HFinite|] ==> (x+y) \<in> HFinite"
+apply (simp add: HFinite_def)
+apply (blast intro!: SReal_add hrabs_add_less)
+done
+
+lemma HFinite_mult: "[|x \<in> HFinite; y \<in> HFinite|] ==> x*y \<in> HFinite"
+apply (simp add: HFinite_def)
+apply (blast intro!: SReal_mult abs_mult_less)
+done
+
+lemma HFinite_minus_iff: "(-x \<in> HFinite) = (x \<in> HFinite)"
+by (simp add: HFinite_def)
+
+lemma SReal_subset_HFinite: "Reals <= HFinite"
+apply (auto simp add: SReal_def HFinite_def)
+apply (rule_tac x = "1 + abs (hypreal_of_real r) " in exI)
+apply (auto simp add: hypreal_of_real_hrabs)
+apply (rule_tac x = "1 + abs r" in exI, simp)
+done
+
+lemma HFinite_hypreal_of_real [simp]: "hypreal_of_real x \<in> HFinite"
+by (auto intro: SReal_subset_HFinite [THEN subsetD])
+
+lemma HFiniteD: "x \<in> HFinite ==> \<exists>t \<in> Reals. abs x < t"
+by (simp add: HFinite_def)
+
+lemma HFinite_hrabs_iff: "(abs x \<in> HFinite) = (x \<in> HFinite)"
+by (simp add: HFinite_def)
+declare HFinite_hrabs_iff [iff]
+
+lemma HFinite_number_of: "number_of w \<in> HFinite"
+by (rule SReal_number_of [THEN SReal_subset_HFinite [THEN subsetD]])
+declare HFinite_number_of [simp]
+
+(** As always with numerals, 0 and 1 are special cases **)
+
+lemma HFinite_0: "0 \<in> HFinite"
+apply (subst hypreal_numeral_0_eq_0 [symmetric])
+apply (rule HFinite_number_of)
+done
+declare HFinite_0 [simp]
+
+lemma HFinite_1: "1 \<in> HFinite"
+apply (subst hypreal_numeral_1_eq_1 [symmetric])
+apply (rule HFinite_number_of)
+done
+declare HFinite_1 [simp]
+
+lemma HFinite_bounded: "[|x \<in> HFinite; y <= x; 0 <= y |] ==> y \<in> HFinite"
+apply (case_tac "x <= 0")
+apply (drule_tac y = x in order_trans)
+apply (drule_tac [2] hypreal_le_anti_sym)
+apply (auto simp add: linorder_not_le)
+apply (auto intro: order_le_less_trans simp add: abs_if HFinite_def)
+done
+
+(*------------------------------------------------------------------
+ Set of infinitesimals is a subring of the hyperreals
+ ------------------------------------------------------------------*)
+lemma InfinitesimalD:
+ "x \<in> Infinitesimal ==> \<forall>r \<in> Reals. 0 < r --> abs x < r"
+apply (simp add: Infinitesimal_def)
+done
+
+lemma Infinitesimal_zero: "0 \<in> Infinitesimal"
+by (simp add: Infinitesimal_def)
+declare Infinitesimal_zero [iff]
+
+lemma hypreal_sum_of_halves: "x/(2::hypreal) + x/(2::hypreal) = x"
+by auto
+
+lemma hypreal_half_gt_zero: "0 < r ==> 0 < r/(2::hypreal)"
+by auto
+
+lemma Infinitesimal_add:
+ "[| x \<in> Infinitesimal; y \<in> Infinitesimal |] ==> (x+y) \<in> Infinitesimal"
+apply (auto simp add: Infinitesimal_def)
+apply (rule hypreal_sum_of_halves [THEN subst])
+apply (drule hypreal_half_gt_zero)
+apply (blast intro: hrabs_add_less hrabs_add_less SReal_divide_number_of)
+done
+
+lemma Infinitesimal_minus_iff: "(-x:Infinitesimal) = (x:Infinitesimal)"
+by (simp add: Infinitesimal_def)
+declare Infinitesimal_minus_iff [simp]
+
+lemma Infinitesimal_diff: "[| x \<in> Infinitesimal; y \<in> Infinitesimal |] ==> x-y \<in> Infinitesimal"
+by (simp add: hypreal_diff_def Infinitesimal_add)
+
+lemma Infinitesimal_mult:
+ "[| x \<in> Infinitesimal; y \<in> Infinitesimal |] ==> (x * y) \<in> Infinitesimal"
+apply (auto simp add: Infinitesimal_def)
+apply (case_tac "y=0")
+apply (cut_tac [2] a = "abs x" and b = 1 and c = "abs y" and d = r in mult_strict_mono, auto)
+done
+
+lemma Infinitesimal_HFinite_mult: "[| x \<in> Infinitesimal; y \<in> HFinite |] ==> (x * y) \<in> Infinitesimal"
+apply (auto dest!: HFiniteD simp add: Infinitesimal_def)
+apply (frule hrabs_less_gt_zero)
+apply (drule_tac x = "r/t" in bspec)
+apply (blast intro: SReal_divide)
+apply (simp add: zero_less_divide_iff)
+apply (case_tac "x=0 | y=0")
+apply (cut_tac [2] a = "abs x" and b = "r/t" and c = "abs y" in mult_strict_mono)
+apply (auto simp add: zero_less_divide_iff)
+done
+
+lemma Infinitesimal_HFinite_mult2: "[| x \<in> Infinitesimal; y \<in> HFinite |] ==> (y * x) \<in> Infinitesimal"
+by (auto dest: Infinitesimal_HFinite_mult simp add: hypreal_mult_commute)
+
+(*** rather long proof ***)
+lemma HInfinite_inverse_Infinitesimal:
+ "x \<in> HInfinite ==> inverse x: Infinitesimal"
+apply (auto simp add: HInfinite_def Infinitesimal_def)
+apply (erule_tac x = "inverse r" in ballE)
+apply (frule_tac a1 = r and z = "abs x" in positive_imp_inverse_positive [THEN order_less_trans], assumption)
+apply (drule inverse_inverse_eq [symmetric, THEN subst])
+apply (rule inverse_less_iff_less [THEN iffD1])
+apply (auto simp add: SReal_inverse)
+done
+
+
+
+lemma HInfinite_mult: "[|x \<in> HInfinite;y \<in> HInfinite|] ==> (x*y) \<in> HInfinite"
+apply (simp add: HInfinite_def, auto)
+apply (erule_tac x = 1 in ballE)
+apply (erule_tac x = r in ballE)
+apply (case_tac "y=0")
+apply (cut_tac [2] c = 1 and d = "abs x" and a = r and b = "abs y" in mult_strict_mono)
+apply (auto simp add: mult_ac)
+done
+
+lemma HInfinite_add_ge_zero:
+ "[|x \<in> HInfinite; 0 <= y; 0 <= x|] ==> (x + y): HInfinite"
+by (auto intro!: hypreal_add_zero_less_le_mono
+ simp add: abs_if hypreal_add_commute hypreal_le_add_order HInfinite_def)
+
+lemma HInfinite_add_ge_zero2: "[|x \<in> HInfinite; 0 <= y; 0 <= x|] ==> (y + x): HInfinite"
+by (auto intro!: HInfinite_add_ge_zero simp add: hypreal_add_commute)
+
+lemma HInfinite_add_gt_zero: "[|x \<in> HInfinite; 0 < y; 0 < x|] ==> (x + y): HInfinite"
+by (blast intro: HInfinite_add_ge_zero order_less_imp_le)
+
+lemma HInfinite_minus_iff: "(-x \<in> HInfinite) = (x \<in> HInfinite)"
+by (simp add: HInfinite_def)
+
+lemma HInfinite_add_le_zero: "[|x \<in> HInfinite; y <= 0; x <= 0|] ==> (x + y): HInfinite"
+apply (drule HInfinite_minus_iff [THEN iffD2])
+apply (rule HInfinite_minus_iff [THEN iffD1])
+apply (auto intro: HInfinite_add_ge_zero)
+done
+
+lemma HInfinite_add_lt_zero: "[|x \<in> HInfinite; y < 0; x < 0|] ==> (x + y): HInfinite"
+by (blast intro: HInfinite_add_le_zero order_less_imp_le)
+
+lemma HFinite_sum_squares: "[|a: HFinite; b: HFinite; c: HFinite|]
+ ==> a*a + b*b + c*c \<in> HFinite"
+apply (auto intro: HFinite_mult HFinite_add)
+done
+
+lemma not_Infinitesimal_not_zero: "x \<notin> Infinitesimal ==> x \<noteq> 0"
+by auto
+
+lemma not_Infinitesimal_not_zero2: "x \<in> HFinite - Infinitesimal ==> x \<noteq> 0"
+by auto
+
+lemma Infinitesimal_hrabs_iff: "(abs x \<in> Infinitesimal) = (x \<in> Infinitesimal)"
+by (auto simp add: hrabs_def)
+declare Infinitesimal_hrabs_iff [iff]
+
+lemma HFinite_diff_Infinitesimal_hrabs: "x \<in> HFinite - Infinitesimal ==> abs x \<in> HFinite - Infinitesimal"
+by blast
+
+lemma hrabs_less_Infinitesimal:
+ "[| e \<in> Infinitesimal; abs x < e |] ==> x \<in> Infinitesimal"
+apply (auto simp add: Infinitesimal_def abs_less_iff)
+done
+
+lemma hrabs_le_Infinitesimal: "[| e \<in> Infinitesimal; abs x <= e |] ==> x \<in> Infinitesimal"
+by (blast dest: order_le_imp_less_or_eq intro: hrabs_less_Infinitesimal)
+
+lemma Infinitesimal_interval:
+ "[| e \<in> Infinitesimal; e' \<in> Infinitesimal; e' < x ; x < e |]
+ ==> x \<in> Infinitesimal"
+apply (auto simp add: Infinitesimal_def abs_less_iff)
+done
+
+lemma Infinitesimal_interval2: "[| e \<in> Infinitesimal; e' \<in> Infinitesimal;
+ e' <= x ; x <= e |] ==> x \<in> Infinitesimal"
+apply (auto intro: Infinitesimal_interval simp add: order_le_less)
+done
+
+lemma not_Infinitesimal_mult:
+ "[| x \<notin> Infinitesimal; y \<notin> Infinitesimal|] ==> (x*y) \<notin>Infinitesimal"
+apply (unfold Infinitesimal_def, clarify)
+apply (simp add: linorder_not_less)
+apply (erule_tac x = "r*ra" in ballE)
+prefer 2 apply (fast intro: SReal_mult)
+apply (auto simp add: zero_less_mult_iff)
+apply (cut_tac c = ra and d = "abs y" and a = r and b = "abs x" in mult_mono, auto)
+done
+
+lemma Infinitesimal_mult_disj: "x*y \<in> Infinitesimal ==> x \<in> Infinitesimal | y \<in> Infinitesimal"
+apply (rule ccontr)
+apply (drule de_Morgan_disj [THEN iffD1])
+apply (fast dest: not_Infinitesimal_mult)
+done
+
+lemma HFinite_Infinitesimal_not_zero: "x \<in> HFinite-Infinitesimal ==> x \<noteq> 0"
+by blast
+
+lemma HFinite_Infinitesimal_diff_mult: "[| x \<in> HFinite - Infinitesimal;
+ y \<in> HFinite - Infinitesimal
+ |] ==> (x*y) \<in> HFinite - Infinitesimal"
+apply clarify
+apply (blast dest: HFinite_mult not_Infinitesimal_mult)
+done
+
+lemma Infinitesimal_subset_HFinite:
+ "Infinitesimal <= HFinite"
+apply (simp add: Infinitesimal_def HFinite_def, auto)
+apply (rule_tac x = 1 in bexI, auto)
+done
+
+lemma Infinitesimal_hypreal_of_real_mult: "x \<in> Infinitesimal ==> x * hypreal_of_real r \<in> Infinitesimal"
+by (erule HFinite_hypreal_of_real [THEN [2] Infinitesimal_HFinite_mult])
+
+lemma Infinitesimal_hypreal_of_real_mult2: "x \<in> Infinitesimal ==> hypreal_of_real r * x \<in> Infinitesimal"
+by (erule HFinite_hypreal_of_real [THEN [2] Infinitesimal_HFinite_mult2])
+
+(*----------------------------------------------------------------------
+ Infinitely close relation @=
+ ----------------------------------------------------------------------*)
+
+lemma mem_infmal_iff: "(x \<in> Infinitesimal) = (x @= 0)"
+by (simp add: Infinitesimal_def approx_def)
+
+lemma approx_minus_iff: " (x @= y) = (x + -y @= 0)"
+by (simp add: approx_def)
+
+lemma approx_minus_iff2: " (x @= y) = (-y + x @= 0)"
+by (simp add: approx_def hypreal_add_commute)
+
+lemma approx_refl: "x @= x"
+by (simp add: approx_def Infinitesimal_def)
+declare approx_refl [iff]
+
+lemma approx_sym: "x @= y ==> y @= x"
+apply (simp add: approx_def)
+apply (rule hypreal_minus_distrib1 [THEN subst])
+apply (erule Infinitesimal_minus_iff [THEN iffD2])
+done
+
+lemma approx_trans: "[| x @= y; y @= z |] ==> x @= z"
+apply (simp add: approx_def)
+apply (drule Infinitesimal_add, assumption, auto)
+done
+
+lemma approx_trans2: "[| r @= x; s @= x |] ==> r @= s"
+by (blast intro: approx_sym approx_trans)
+
+lemma approx_trans3: "[| x @= r; x @= s|] ==> r @= s"
+by (blast intro: approx_sym approx_trans)
+
+lemma number_of_approx_reorient: "(number_of w @= x) = (x @= number_of w)"
+by (blast intro: approx_sym)
+
+lemma zero_approx_reorient: "(0 @= x) = (x @= 0)"
+by (blast intro: approx_sym)
+
+lemma one_approx_reorient: "(1 @= x) = (x @= 1)"
+by (blast intro: approx_sym)
-defs
+ML
+{*
+val SReal_add = thm "SReal_add";
+val SReal_mult = thm "SReal_mult";
+val SReal_inverse = thm "SReal_inverse";
+val SReal_divide = thm "SReal_divide";
+val SReal_minus = thm "SReal_minus";
+val SReal_minus_iff = thm "SReal_minus_iff";
+val SReal_add_cancel = thm "SReal_add_cancel";
+val SReal_hrabs = thm "SReal_hrabs";
+val SReal_hypreal_of_real = thm "SReal_hypreal_of_real";
+val SReal_number_of = thm "SReal_number_of";
+val Reals_0 = thm "Reals_0";
+val Reals_1 = thm "Reals_1";
+val SReal_divide_number_of = thm "SReal_divide_number_of";
+val SReal_epsilon_not_mem = thm "SReal_epsilon_not_mem";
+val SReal_omega_not_mem = thm "SReal_omega_not_mem";
+val SReal_UNIV_real = thm "SReal_UNIV_real";
+val SReal_iff = thm "SReal_iff";
+val hypreal_of_real_image = thm "hypreal_of_real_image";
+val inv_hypreal_of_real_image = thm "inv_hypreal_of_real_image";
+val SReal_hypreal_of_real_image = thm "SReal_hypreal_of_real_image";
+val SReal_dense = thm "SReal_dense";
+val SReal_sup_lemma = thm "SReal_sup_lemma";
+val SReal_sup_lemma2 = thm "SReal_sup_lemma2";
+val hypreal_of_real_isUb_iff = thm "hypreal_of_real_isUb_iff";
+val hypreal_of_real_isLub1 = thm "hypreal_of_real_isLub1";
+val hypreal_of_real_isLub2 = thm "hypreal_of_real_isLub2";
+val hypreal_of_real_isLub_iff = thm "hypreal_of_real_isLub_iff";
+val lemma_isUb_hypreal_of_real = thm "lemma_isUb_hypreal_of_real";
+val lemma_isLub_hypreal_of_real = thm "lemma_isLub_hypreal_of_real";
+val lemma_isLub_hypreal_of_real2 = thm "lemma_isLub_hypreal_of_real2";
+val SReal_complete = thm "SReal_complete";
+val HFinite_add = thm "HFinite_add";
+val HFinite_mult = thm "HFinite_mult";
+val HFinite_minus_iff = thm "HFinite_minus_iff";
+val SReal_subset_HFinite = thm "SReal_subset_HFinite";
+val HFinite_hypreal_of_real = thm "HFinite_hypreal_of_real";
+val HFiniteD = thm "HFiniteD";
+val HFinite_hrabs_iff = thm "HFinite_hrabs_iff";
+val HFinite_number_of = thm "HFinite_number_of";
+val HFinite_0 = thm "HFinite_0";
+val HFinite_1 = thm "HFinite_1";
+val HFinite_bounded = thm "HFinite_bounded";
+val InfinitesimalD = thm "InfinitesimalD";
+val Infinitesimal_zero = thm "Infinitesimal_zero";
+val hypreal_sum_of_halves = thm "hypreal_sum_of_halves";
+val hypreal_half_gt_zero = thm "hypreal_half_gt_zero";
+val Infinitesimal_add = thm "Infinitesimal_add";
+val Infinitesimal_minus_iff = thm "Infinitesimal_minus_iff";
+val Infinitesimal_diff = thm "Infinitesimal_diff";
+val Infinitesimal_mult = thm "Infinitesimal_mult";
+val Infinitesimal_HFinite_mult = thm "Infinitesimal_HFinite_mult";
+val Infinitesimal_HFinite_mult2 = thm "Infinitesimal_HFinite_mult2";
+val HInfinite_inverse_Infinitesimal = thm "HInfinite_inverse_Infinitesimal";
+val HInfinite_mult = thm "HInfinite_mult";
+val HInfinite_add_ge_zero = thm "HInfinite_add_ge_zero";
+val HInfinite_add_ge_zero2 = thm "HInfinite_add_ge_zero2";
+val HInfinite_add_gt_zero = thm "HInfinite_add_gt_zero";
+val HInfinite_minus_iff = thm "HInfinite_minus_iff";
+val HInfinite_add_le_zero = thm "HInfinite_add_le_zero";
+val HInfinite_add_lt_zero = thm "HInfinite_add_lt_zero";
+val HFinite_sum_squares = thm "HFinite_sum_squares";
+val not_Infinitesimal_not_zero = thm "not_Infinitesimal_not_zero";
+val not_Infinitesimal_not_zero2 = thm "not_Infinitesimal_not_zero2";
+val Infinitesimal_hrabs_iff = thm "Infinitesimal_hrabs_iff";
+val HFinite_diff_Infinitesimal_hrabs = thm "HFinite_diff_Infinitesimal_hrabs";
+val hrabs_less_Infinitesimal = thm "hrabs_less_Infinitesimal";
+val hrabs_le_Infinitesimal = thm "hrabs_le_Infinitesimal";
+val Infinitesimal_interval = thm "Infinitesimal_interval";
+val Infinitesimal_interval2 = thm "Infinitesimal_interval2";
+val not_Infinitesimal_mult = thm "not_Infinitesimal_mult";
+val Infinitesimal_mult_disj = thm "Infinitesimal_mult_disj";
+val HFinite_Infinitesimal_not_zero = thm "HFinite_Infinitesimal_not_zero";
+val HFinite_Infinitesimal_diff_mult = thm "HFinite_Infinitesimal_diff_mult";
+val Infinitesimal_subset_HFinite = thm "Infinitesimal_subset_HFinite";
+val Infinitesimal_hypreal_of_real_mult = thm "Infinitesimal_hypreal_of_real_mult";
+val Infinitesimal_hypreal_of_real_mult2 = thm "Infinitesimal_hypreal_of_real_mult2";
+val mem_infmal_iff = thm "mem_infmal_iff";
+val approx_minus_iff = thm "approx_minus_iff";
+val approx_minus_iff2 = thm "approx_minus_iff2";
+val approx_refl = thm "approx_refl";
+val approx_sym = thm "approx_sym";
+val approx_trans = thm "approx_trans";
+val approx_trans2 = thm "approx_trans2";
+val approx_trans3 = thm "approx_trans3";
+val number_of_approx_reorient = thm "number_of_approx_reorient";
+val zero_approx_reorient = thm "zero_approx_reorient";
+val one_approx_reorient = thm "one_approx_reorient";
+
+(*** re-orientation, following HOL/Integ/Bin.ML
+ We re-orient x @=y where x is 0, 1 or a numeral, unless y is as well!
+ ***)
+
+(*reorientation simprules using ==, for the following simproc*)
+val meta_zero_approx_reorient = zero_approx_reorient RS eq_reflection;
+val meta_one_approx_reorient = one_approx_reorient RS eq_reflection;
+val meta_number_of_approx_reorient = number_of_approx_reorient RS eq_reflection
+
+(*reorientation simplification procedure: reorients (polymorphic)
+ 0 = x, 1 = x, nnn = x provided x isn't 0, 1 or a numeral.*)
+fun reorient_proc sg _ (_ $ t $ u) =
+ case u of
+ Const("0", _) => None
+ | Const("1", _) => None
+ | Const("Numeral.number_of", _) $ _ => None
+ | _ => Some (case t of
+ Const("0", _) => meta_zero_approx_reorient
+ | Const("1", _) => meta_one_approx_reorient
+ | Const("Numeral.number_of", _) $ _ =>
+ meta_number_of_approx_reorient);
+
+val approx_reorient_simproc =
+ Bin_Simprocs.prep_simproc
+ ("reorient_simproc", ["0@=x", "1@=x", "number_of w @= x"], reorient_proc);
+
+Addsimprocs [approx_reorient_simproc];
+*}
+
+lemma Infinitesimal_approx_minus: "(x-y \<in> Infinitesimal) = (x @= y)"
+by (auto simp add: hypreal_diff_def approx_minus_iff [symmetric] mem_infmal_iff)
+
+lemma approx_monad_iff: "(x @= y) = (monad(x)=monad(y))"
+apply (simp add: monad_def)
+apply (auto dest: approx_sym elim!: approx_trans equalityCE)
+done
+
+lemma Infinitesimal_approx: "[| x \<in> Infinitesimal; y \<in> Infinitesimal |] ==> x @= y"
+apply (simp add: mem_infmal_iff)
+apply (blast intro: approx_trans approx_sym)
+done
+
+lemma approx_add: "[| a @= b; c @= d |] ==> a+c @= b+d"
+proof (unfold approx_def)
+ assume inf: "a + - b \<in> Infinitesimal" "c + - d \<in> Infinitesimal"
+ have "a + c + - (b + d) = (a + - b) + (c + - d)" by arith
+ also have "... \<in> Infinitesimal" using inf by (rule Infinitesimal_add)
+ finally show "a + c + - (b + d) \<in> Infinitesimal" .
+qed
+
+lemma approx_minus: "a @= b ==> -a @= -b"
+apply (rule approx_minus_iff [THEN iffD2, THEN approx_sym])
+apply (drule approx_minus_iff [THEN iffD1])
+apply (simp (no_asm) add: hypreal_add_commute)
+done
+
+lemma approx_minus2: "-a @= -b ==> a @= b"
+by (auto dest: approx_minus)
+
+lemma approx_minus_cancel: "(-a @= -b) = (a @= b)"
+by (blast intro: approx_minus approx_minus2)
+
+declare approx_minus_cancel [simp]
+
+lemma approx_add_minus: "[| a @= b; c @= d |] ==> a + -c @= b + -d"
+by (blast intro!: approx_add approx_minus)
+
+lemma approx_mult1: "[| a @= b; c: HFinite|] ==> a*c @= b*c"
+by (simp add: approx_def Infinitesimal_HFinite_mult minus_mult_left
+ left_distrib [symmetric]
+ del: minus_mult_left [symmetric])
+
+lemma approx_mult2: "[|a @= b; c: HFinite|] ==> c*a @= c*b"
+apply (simp (no_asm_simp) add: approx_mult1 hypreal_mult_commute)
+done
+
+lemma approx_mult_subst: "[|u @= v*x; x @= y; v \<in> HFinite|] ==> u @= v*y"
+by (blast intro: approx_mult2 approx_trans)
+
+lemma approx_mult_subst2: "[| u @= x*v; x @= y; v \<in> HFinite |] ==> u @= y*v"
+by (blast intro: approx_mult1 approx_trans)
+
+lemma approx_mult_subst_SReal: "[| u @= x*hypreal_of_real v; x @= y |] ==> u @= y*hypreal_of_real v"
+by (auto intro: approx_mult_subst2)
+
+lemma approx_eq_imp: "a = b ==> a @= b"
+by (simp add: approx_def)
+
+lemma Infinitesimal_minus_approx: "x \<in> Infinitesimal ==> -x @= x"
+by (blast intro: Infinitesimal_minus_iff [THEN iffD2]
+ mem_infmal_iff [THEN iffD1] approx_trans2)
+
+lemma bex_Infinitesimal_iff: "(\<exists>y \<in> Infinitesimal. x + -z = y) = (x @= z)"
+by (simp add: approx_def)
+
+lemma bex_Infinitesimal_iff2: "(\<exists>y \<in> Infinitesimal. x = z + y) = (x @= z)"
+by (force simp add: bex_Infinitesimal_iff [symmetric])
+
+lemma Infinitesimal_add_approx: "[| y \<in> Infinitesimal; x + y = z |] ==> x @= z"
+apply (rule bex_Infinitesimal_iff [THEN iffD1])
+apply (drule Infinitesimal_minus_iff [THEN iffD2])
+apply (auto simp add: minus_add_distrib hypreal_add_assoc [symmetric])
+done
+
+lemma Infinitesimal_add_approx_self: "y \<in> Infinitesimal ==> x @= x + y"
+apply (rule bex_Infinitesimal_iff [THEN iffD1])
+apply (drule Infinitesimal_minus_iff [THEN iffD2])
+apply (auto simp add: minus_add_distrib hypreal_add_assoc [symmetric])
+done
+
+lemma Infinitesimal_add_approx_self2: "y \<in> Infinitesimal ==> x @= y + x"
+by (auto dest: Infinitesimal_add_approx_self simp add: hypreal_add_commute)
+
+lemma Infinitesimal_add_minus_approx_self: "y \<in> Infinitesimal ==> x @= x + -y"
+by (blast intro!: Infinitesimal_add_approx_self Infinitesimal_minus_iff [THEN iffD2])
+
+lemma Infinitesimal_add_cancel: "[| y \<in> Infinitesimal; x+y @= z|] ==> x @= z"
+apply (drule_tac x = x in Infinitesimal_add_approx_self [THEN approx_sym])
+apply (erule approx_trans3 [THEN approx_sym], assumption)
+done
+
+lemma Infinitesimal_add_right_cancel: "[| y \<in> Infinitesimal; x @= z + y|] ==> x @= z"
+apply (drule_tac x = z in Infinitesimal_add_approx_self2 [THEN approx_sym])
+apply (erule approx_trans3 [THEN approx_sym])
+apply (simp add: hypreal_add_commute)
+apply (erule approx_sym)
+done
+
+lemma approx_add_left_cancel: "d + b @= d + c ==> b @= c"
+apply (drule approx_minus_iff [THEN iffD1])
+apply (simp add: minus_add_distrib approx_minus_iff [symmetric] add_ac)
+done
+
+lemma approx_add_right_cancel: "b + d @= c + d ==> b @= c"
+apply (rule approx_add_left_cancel)
+apply (simp add: hypreal_add_commute)
+done
+
+lemma approx_add_mono1: "b @= c ==> d + b @= d + c"
+apply (rule approx_minus_iff [THEN iffD2])
+apply (simp add: minus_add_distrib approx_minus_iff [symmetric] add_ac)
+done
+
+lemma approx_add_mono2: "b @= c ==> b + a @= c + a"
+apply (simp (no_asm_simp) add: hypreal_add_commute approx_add_mono1)
+done
+
+lemma approx_add_left_iff: "(a + b @= a + c) = (b @= c)"
+by (fast elim: approx_add_left_cancel approx_add_mono1)
+
+declare approx_add_left_iff [simp]
+
+lemma approx_add_right_iff: "(b + a @= c + a) = (b @= c)"
+apply (simp (no_asm) add: hypreal_add_commute)
+done
+
+declare approx_add_right_iff [simp]
+
+lemma approx_HFinite: "[| x \<in> HFinite; x @= y |] ==> y \<in> HFinite"
+apply (drule bex_Infinitesimal_iff2 [THEN iffD2], safe)
+apply (drule Infinitesimal_subset_HFinite [THEN subsetD, THEN HFinite_minus_iff [THEN iffD2]])
+apply (drule HFinite_add)
+apply (auto simp add: hypreal_add_assoc)
+done
+
+lemma approx_hypreal_of_real_HFinite: "x @= hypreal_of_real D ==> x \<in> HFinite"
+by (rule approx_sym [THEN [2] approx_HFinite], auto)
+
+lemma approx_mult_HFinite: "[|a @= b; c @= d; b: HFinite; d: HFinite|] ==> a*c @= b*d"
+apply (rule approx_trans)
+apply (rule_tac [2] approx_mult2)
+apply (rule approx_mult1)
+prefer 2 apply (blast intro: approx_HFinite approx_sym, auto)
+done
+
+lemma approx_mult_hypreal_of_real: "[|a @= hypreal_of_real b; c @= hypreal_of_real d |]
+ ==> a*c @= hypreal_of_real b*hypreal_of_real d"
+apply (blast intro!: approx_mult_HFinite approx_hypreal_of_real_HFinite HFinite_hypreal_of_real)
+done
+
+lemma approx_SReal_mult_cancel_zero: "[| a \<in> Reals; a \<noteq> 0; a*x @= 0 |] ==> x @= 0"
+apply (drule SReal_inverse [THEN SReal_subset_HFinite [THEN subsetD]])
+apply (auto dest: approx_mult2 simp add: hypreal_mult_assoc [symmetric])
+done
+
+(* REM comments: newly added *)
+lemma approx_mult_SReal1: "[| a \<in> Reals; x @= 0 |] ==> x*a @= 0"
+by (auto dest: SReal_subset_HFinite [THEN subsetD] approx_mult1)
+
+lemma approx_mult_SReal2: "[| a \<in> Reals; x @= 0 |] ==> a*x @= 0"
+by (auto dest: SReal_subset_HFinite [THEN subsetD] approx_mult2)
+
+lemma approx_mult_SReal_zero_cancel_iff: "[|a \<in> Reals; a \<noteq> 0 |] ==> (a*x @= 0) = (x @= 0)"
+by (blast intro: approx_SReal_mult_cancel_zero approx_mult_SReal2)
+declare approx_mult_SReal_zero_cancel_iff [simp]
+
+lemma approx_SReal_mult_cancel: "[| a \<in> Reals; a \<noteq> 0; a* w @= a*z |] ==> w @= z"
+apply (drule SReal_inverse [THEN SReal_subset_HFinite [THEN subsetD]])
+apply (auto dest: approx_mult2 simp add: hypreal_mult_assoc [symmetric])
+done
+
+lemma approx_SReal_mult_cancel_iff1: "[| a \<in> Reals; a \<noteq> 0|] ==> (a* w @= a*z) = (w @= z)"
+by (auto intro!: approx_mult2 SReal_subset_HFinite [THEN subsetD] intro: approx_SReal_mult_cancel)
+declare approx_SReal_mult_cancel_iff1 [simp]
+
+lemma approx_le_bound: "[| z <= f; f @= g; g <= z |] ==> f @= z"
+apply (simp add: bex_Infinitesimal_iff2 [symmetric], auto)
+apply (rule_tac x = "g+y-z" in bexI)
+apply (simp (no_asm))
+apply (rule Infinitesimal_interval2)
+apply (rule_tac [2] Infinitesimal_zero, auto)
+done
+
+(*-----------------------------------------------------------------
+ Zero is the only infinitesimal that is also a real
+ -----------------------------------------------------------------*)
+
+lemma Infinitesimal_less_SReal:
+ "[| x \<in> Reals; y \<in> Infinitesimal; 0 < x |] ==> y < x"
+apply (simp add: Infinitesimal_def)
+apply (rule abs_ge_self [THEN order_le_less_trans], auto)
+done
+
+lemma Infinitesimal_less_SReal2: "y \<in> Infinitesimal ==> \<forall>r \<in> Reals. 0 < r --> y < r"
+by (blast intro: Infinitesimal_less_SReal)
+
+lemma SReal_not_Infinitesimal:
+ "[| 0 < y; y \<in> Reals|] ==> y \<notin> Infinitesimal"
+apply (simp add: Infinitesimal_def)
+apply (auto simp add: hrabs_def)
+done
+
+lemma SReal_minus_not_Infinitesimal: "[| y < 0; y \<in> Reals |] ==> y \<notin> Infinitesimal"
+apply (subst Infinitesimal_minus_iff [symmetric])
+apply (rule SReal_not_Infinitesimal, auto)
+done
+
+lemma SReal_Int_Infinitesimal_zero: "Reals Int Infinitesimal = {0}"
+apply auto
+apply (cut_tac x = x and y = 0 in linorder_less_linear)
+apply (blast dest: SReal_not_Infinitesimal SReal_minus_not_Infinitesimal)
+done
+
+lemma SReal_Infinitesimal_zero: "[| x \<in> Reals; x \<in> Infinitesimal|] ==> x = 0"
+by (cut_tac SReal_Int_Infinitesimal_zero, blast)
+
+lemma SReal_HFinite_diff_Infinitesimal: "[| x \<in> Reals; x \<noteq> 0 |] ==> x \<in> HFinite - Infinitesimal"
+by (auto dest: SReal_Infinitesimal_zero SReal_subset_HFinite [THEN subsetD])
+
+lemma hypreal_of_real_HFinite_diff_Infinitesimal: "hypreal_of_real x \<noteq> 0 ==> hypreal_of_real x \<in> HFinite - Infinitesimal"
+by (rule SReal_HFinite_diff_Infinitesimal, auto)
+
+lemma hypreal_of_real_Infinitesimal_iff_0: "(hypreal_of_real x \<in> Infinitesimal) = (x=0)"
+apply auto
+apply (rule ccontr)
+apply (rule hypreal_of_real_HFinite_diff_Infinitesimal [THEN DiffD2], auto)
+done
+declare hypreal_of_real_Infinitesimal_iff_0 [iff]
+
+lemma number_of_not_Infinitesimal: "number_of w \<noteq> (0::hypreal) ==> number_of w \<notin> Infinitesimal"
+by (fast dest: SReal_number_of [THEN SReal_Infinitesimal_zero])
+declare number_of_not_Infinitesimal [simp]
+
+(*again: 1 is a special case, but not 0 this time*)
+lemma one_not_Infinitesimal: "1 \<notin> Infinitesimal"
+apply (subst hypreal_numeral_1_eq_1 [symmetric])
+apply (rule number_of_not_Infinitesimal)
+apply (simp (no_asm))
+done
+declare one_not_Infinitesimal [simp]
+
+lemma approx_SReal_not_zero: "[| y \<in> Reals; x @= y; y\<noteq> 0 |] ==> x \<noteq> 0"
+apply (cut_tac x = 0 and y = y in linorder_less_linear, simp)
+apply (blast dest: approx_sym [THEN mem_infmal_iff [THEN iffD2]] SReal_not_Infinitesimal SReal_minus_not_Infinitesimal)
+done
+
+lemma HFinite_diff_Infinitesimal_approx: "[| x @= y; y \<in> HFinite - Infinitesimal |]
+ ==> x \<in> HFinite - Infinitesimal"
+apply (auto intro: approx_sym [THEN [2] approx_HFinite]
+ simp add: mem_infmal_iff)
+apply (drule approx_trans3, assumption)
+apply (blast dest: approx_sym)
+done
+
+(*The premise y\<noteq>0 is essential; otherwise x/y =0 and we lose the
+ HFinite premise.*)
+lemma Infinitesimal_ratio: "[| y \<noteq> 0; y \<in> Infinitesimal; x/y \<in> HFinite |] ==> x \<in> Infinitesimal"
+apply (drule Infinitesimal_HFinite_mult2, assumption)
+apply (simp add: hypreal_divide_def hypreal_mult_assoc)
+done
+
+(*------------------------------------------------------------------
+ Standard Part Theorem: Every finite x: R* is infinitely
+ close to a unique real number (i.e a member of Reals)
+ ------------------------------------------------------------------*)
+(*------------------------------------------------------------------
+ Uniqueness: Two infinitely close reals are equal
+ ------------------------------------------------------------------*)
+
+lemma SReal_approx_iff: "[|x \<in> Reals; y \<in> Reals|] ==> (x @= y) = (x = y)"
+apply auto
+apply (simp add: approx_def)
+apply (drule_tac x = y in SReal_minus)
+apply (drule SReal_add, assumption)
+apply (drule SReal_Infinitesimal_zero, assumption)
+apply (drule sym)
+apply (simp add: hypreal_eq_minus_iff [symmetric])
+done
+
+lemma number_of_approx_iff: "(number_of v @= number_of w) = (number_of v = (number_of w :: hypreal))"
+by (auto simp add: SReal_approx_iff)
+declare number_of_approx_iff [simp]
+
+(*And also for 0 @= #nn and 1 @= #nn, #nn @= 0 and #nn @= 1.*)
+lemma [simp]: "(0 @= number_of w) = ((number_of w :: hypreal) = 0)"
+ "(number_of w @= 0) = ((number_of w :: hypreal) = 0)"
+ "(1 @= number_of w) = ((number_of w :: hypreal) = 1)"
+ "(number_of w @= 1) = ((number_of w :: hypreal) = 1)"
+ "~ (0 @= 1)" "~ (1 @= 0)"
+by (auto simp only: SReal_number_of SReal_approx_iff Reals_0 Reals_1)
+
+lemma hypreal_of_real_approx_iff: "(hypreal_of_real k @= hypreal_of_real m) = (k = m)"
+apply auto
+apply (rule inj_hypreal_of_real [THEN injD])
+apply (rule SReal_approx_iff [THEN iffD1], auto)
+done
+declare hypreal_of_real_approx_iff [simp]
+
+lemma hypreal_of_real_approx_number_of_iff: "(hypreal_of_real k @= number_of w) = (k = number_of w)"
+by (subst hypreal_of_real_approx_iff [symmetric], auto)
+declare hypreal_of_real_approx_number_of_iff [simp]
+
+(*And also for 0 and 1.*)
+(*And also for 0 @= #nn and 1 @= #nn, #nn @= 0 and #nn @= 1.*)
+lemma [simp]: "(hypreal_of_real k @= 0) = (k = 0)"
+ "(hypreal_of_real k @= 1) = (k = 1)"
+ by (simp_all add: hypreal_of_real_approx_iff [symmetric])
+
+lemma approx_unique_real: "[| r \<in> Reals; s \<in> Reals; r @= x; s @= x|] ==> r = s"
+by (blast intro: SReal_approx_iff [THEN iffD1] approx_trans2)
+
+(*------------------------------------------------------------------
+ Existence of unique real infinitely close
+ ------------------------------------------------------------------*)
+(* lemma about lubs *)
+lemma hypreal_isLub_unique:
+ "[| isLub R S x; isLub R S y |] ==> x = (y::hypreal)"
+apply (frule isLub_isUb)
+apply (frule_tac x = y in isLub_isUb)
+apply (blast intro!: hypreal_le_anti_sym dest!: isLub_le_isUb)
+done
+
+lemma lemma_st_part_ub: "x \<in> HFinite ==> \<exists>u. isUb Reals {s. s \<in> Reals & s < x} u"
+apply (drule HFiniteD, safe)
+apply (rule exI, rule isUbI)
+apply (auto intro: setleI isUbI simp add: abs_less_iff)
+done
+
+lemma lemma_st_part_nonempty: "x \<in> HFinite ==> \<exists>y. y \<in> {s. s \<in> Reals & s < x}"
+apply (drule HFiniteD, safe)
+apply (drule SReal_minus)
+apply (rule_tac x = "-t" in exI)
+apply (auto simp add: abs_less_iff)
+done
+
+lemma lemma_st_part_subset: "{s. s \<in> Reals & s < x} <= Reals"
+by auto
+
+lemma lemma_st_part_lub: "x \<in> HFinite ==> \<exists>t. isLub Reals {s. s \<in> Reals & s < x} t"
+by (blast intro!: SReal_complete lemma_st_part_ub lemma_st_part_nonempty lemma_st_part_subset)
+
+lemma lemma_hypreal_le_left_cancel: "((t::hypreal) + r <= t) = (r <= 0)"
+apply safe
+apply (drule_tac c = "-t" in add_left_mono)
+apply (drule_tac [2] c = t in add_left_mono)
+apply (auto simp add: hypreal_add_assoc [symmetric])
+done
+
+lemma lemma_st_part_le1: "[| x \<in> HFinite; isLub Reals {s. s \<in> Reals & s < x} t;
+ r \<in> Reals; 0 < r |] ==> x <= t + r"
+apply (frule isLubD1a)
+apply (rule ccontr, drule linorder_not_le [THEN iffD2])
+apply (drule_tac x = t in SReal_add, assumption)
+apply (drule_tac y = "t + r" in isLubD1 [THEN setleD], auto)
+done
+
+lemma hypreal_setle_less_trans: "!!x::hypreal. [| S *<= x; x < y |] ==> S *<= y"
+apply (simp add: setle_def)
+apply (auto dest!: bspec order_le_less_trans intro: order_less_imp_le)
+done
+
+lemma hypreal_gt_isUb:
+ "!!x::hypreal. [| isUb R S x; x < y; y \<in> R |] ==> isUb R S y"
+apply (simp add: isUb_def)
+apply (blast intro: hypreal_setle_less_trans)
+done
+
+lemma lemma_st_part_gt_ub: "[| x \<in> HFinite; x < y; y \<in> Reals |]
+ ==> isUb Reals {s. s \<in> Reals & s < x} y"
+apply (auto dest: order_less_trans intro: order_less_imp_le intro!: isUbI setleI)
+done
+
+lemma lemma_minus_le_zero: "t <= t + -r ==> r <= (0::hypreal)"
+apply (drule_tac c = "-t" in add_left_mono)
+apply (auto simp add: hypreal_add_assoc [symmetric])
+done
+
+lemma lemma_st_part_le2: "[| x \<in> HFinite;
+ isLub Reals {s. s \<in> Reals & s < x} t;
+ r \<in> Reals; 0 < r |]
+ ==> t + -r <= x"
+apply (frule isLubD1a)
+apply (rule ccontr, drule linorder_not_le [THEN iffD1])
+apply (drule SReal_minus, drule_tac x = t in SReal_add, assumption)
+apply (drule lemma_st_part_gt_ub, assumption+)
+apply (drule isLub_le_isUb, assumption)
+apply (drule lemma_minus_le_zero)
+apply (auto dest: order_less_le_trans)
+done
+
+lemma lemma_hypreal_le_swap: "((x::hypreal) <= t + r) = (x + -t <= r)"
+by auto
+
+lemma lemma_st_part1a: "[| x \<in> HFinite;
+ isLub Reals {s. s \<in> Reals & s < x} t;
+ r \<in> Reals; 0 < r |]
+ ==> x + -t <= r"
+apply (blast intro!: lemma_hypreal_le_swap [THEN iffD1] lemma_st_part_le1)
+done
+
+lemma lemma_hypreal_le_swap2: "(t + -r <= x) = (-(x + -t) <= (r::hypreal))"
+by auto
+
+lemma lemma_st_part2a: "[| x \<in> HFinite;
+ isLub Reals {s. s \<in> Reals & s < x} t;
+ r \<in> Reals; 0 < r |]
+ ==> -(x + -t) <= r"
+apply (blast intro!: lemma_hypreal_le_swap2 [THEN iffD1] lemma_st_part_le2)
+done
+
+lemma lemma_SReal_ub: "(x::hypreal) \<in> Reals ==> isUb Reals {s. s \<in> Reals & s < x} x"
+by (auto intro: isUbI setleI order_less_imp_le)
+
+lemma lemma_SReal_lub: "(x::hypreal) \<in> Reals ==> isLub Reals {s. s \<in> Reals & s < x} x"
+apply (auto intro!: isLubI2 lemma_SReal_ub setgeI)
+apply (frule isUbD2a)
+apply (rule_tac x = x and y = y in linorder_cases)
+apply (auto intro!: order_less_imp_le)
+apply (drule SReal_dense, assumption, assumption, safe)
+apply (drule_tac y = r in isUbD)
+apply (auto dest: order_less_le_trans)
+done
+
+lemma lemma_st_part_not_eq1: "[| x \<in> HFinite;
+ isLub Reals {s. s \<in> Reals & s < x} t;
+ r \<in> Reals; 0 < r |]
+ ==> x + -t \<noteq> r"
+apply auto
+apply (frule isLubD1a [THEN SReal_minus])
+apply (drule SReal_add_cancel, assumption)
+apply (drule_tac x = x in lemma_SReal_lub)
+apply (drule hypreal_isLub_unique, assumption, auto)
+done
+
+lemma lemma_st_part_not_eq2: "[| x \<in> HFinite;
+ isLub Reals {s. s \<in> Reals & s < x} t;
+ r \<in> Reals; 0 < r |]
+ ==> -(x + -t) \<noteq> r"
+apply (auto simp add: minus_add_distrib)
+apply (frule isLubD1a)
+apply (drule SReal_add_cancel, assumption)
+apply (drule_tac x = "-x" in SReal_minus, simp)
+apply (drule_tac x = x in lemma_SReal_lub)
+apply (drule hypreal_isLub_unique, assumption, auto)
+done
+
+lemma lemma_st_part_major: "[| x \<in> HFinite;
+ isLub Reals {s. s \<in> Reals & s < x} t;
+ r \<in> Reals; 0 < r |]
+ ==> abs (x + -t) < r"
+apply (frule lemma_st_part1a)
+apply (frule_tac [4] lemma_st_part2a, auto)
+apply (drule order_le_imp_less_or_eq)+
+apply (auto dest: lemma_st_part_not_eq1 lemma_st_part_not_eq2 simp add: abs_less_iff)
+done
+
+lemma lemma_st_part_major2: "[| x \<in> HFinite;
+ isLub Reals {s. s \<in> Reals & s < x} t |]
+ ==> \<forall>r \<in> Reals. 0 < r --> abs (x + -t) < r"
+apply (blast dest!: lemma_st_part_major)
+done
+
+(*----------------------------------------------
+ Existence of real and Standard Part Theorem
+ ----------------------------------------------*)
+lemma lemma_st_part_Ex: "x \<in> HFinite ==>
+ \<exists>t \<in> Reals. \<forall>r \<in> Reals. 0 < r --> abs (x + -t) < r"
+apply (frule lemma_st_part_lub, safe)
+apply (frule isLubD1a)
+apply (blast dest: lemma_st_part_major2)
+done
+
+lemma st_part_Ex:
+ "x \<in> HFinite ==> \<exists>t \<in> Reals. x @= t"
+apply (simp add: approx_def Infinitesimal_def)
+apply (drule lemma_st_part_Ex, auto)
+done
+
+(*--------------------------------
+ Unique real infinitely close
+ -------------------------------*)
+lemma st_part_Ex1: "x \<in> HFinite ==> EX! t. t \<in> Reals & x @= t"
+apply (drule st_part_Ex, safe)
+apply (drule_tac [2] approx_sym, drule_tac [2] approx_sym, drule_tac [2] approx_sym)
+apply (auto intro!: approx_unique_real)
+done
+
+(*------------------------------------------------------------------
+ Finite and Infinite --- more theorems
+ ------------------------------------------------------------------*)
+
+lemma HFinite_Int_HInfinite_empty: "HFinite Int HInfinite = {}"
+
+apply (simp add: HFinite_def HInfinite_def)
+apply (auto dest: order_less_trans)
+done
+declare HFinite_Int_HInfinite_empty [simp]
+
+lemma HFinite_not_HInfinite:
+ assumes x: "x \<in> HFinite" shows "x \<notin> HInfinite"
+proof
+ assume x': "x \<in> HInfinite"
+ with x have "x \<in> HFinite \<inter> HInfinite" by blast
+ thus False by auto
+qed
+
+lemma not_HFinite_HInfinite: "x\<notin> HFinite ==> x \<in> HInfinite"
+apply (simp add: HInfinite_def HFinite_def, auto)
+apply (drule_tac x = "r + 1" in bspec)
+apply (auto simp add: SReal_add)
+done
+
+lemma HInfinite_HFinite_disj: "x \<in> HInfinite | x \<in> HFinite"
+by (blast intro: not_HFinite_HInfinite)
+
+lemma HInfinite_HFinite_iff: "(x \<in> HInfinite) = (x \<notin> HFinite)"
+by (blast dest: HFinite_not_HInfinite not_HFinite_HInfinite)
+
+lemma HFinite_HInfinite_iff: "(x \<in> HFinite) = (x \<notin> HInfinite)"
+apply (simp (no_asm) add: HInfinite_HFinite_iff)
+done
+
+(*------------------------------------------------------------------
+ Finite, Infinite and Infinitesimal --- more theorems
+ ------------------------------------------------------------------*)
+
+lemma HInfinite_diff_HFinite_Infinitesimal_disj: "x \<notin> Infinitesimal ==> x \<in> HInfinite | x \<in> HFinite - Infinitesimal"
+by (fast intro: not_HFinite_HInfinite)
+
+lemma HFinite_inverse: "[| x \<in> HFinite; x \<notin> Infinitesimal |] ==> inverse x \<in> HFinite"
+apply (cut_tac x = "inverse x" in HInfinite_HFinite_disj)
+apply (auto dest!: HInfinite_inverse_Infinitesimal)
+done
+
+lemma HFinite_inverse2: "x \<in> HFinite - Infinitesimal ==> inverse x \<in> HFinite"
+by (blast intro: HFinite_inverse)
+
+(* stronger statement possible in fact *)
+lemma Infinitesimal_inverse_HFinite: "x \<notin> Infinitesimal ==> inverse(x) \<in> HFinite"
+apply (drule HInfinite_diff_HFinite_Infinitesimal_disj)
+apply (blast intro: HFinite_inverse HInfinite_inverse_Infinitesimal Infinitesimal_subset_HFinite [THEN subsetD])
+done
+
+lemma HFinite_not_Infinitesimal_inverse: "x \<in> HFinite - Infinitesimal ==> inverse x \<in> HFinite - Infinitesimal"
+apply (auto intro: Infinitesimal_inverse_HFinite)
+apply (drule Infinitesimal_HFinite_mult2, assumption)
+apply (simp add: not_Infinitesimal_not_zero hypreal_mult_inverse)
+done
+
+lemma approx_inverse: "[| x @= y; y \<in> HFinite - Infinitesimal |]
+ ==> inverse x @= inverse y"
+apply (frule HFinite_diff_Infinitesimal_approx, assumption)
+apply (frule not_Infinitesimal_not_zero2)
+apply (frule_tac x = x in not_Infinitesimal_not_zero2)
+apply (drule HFinite_inverse2)+
+apply (drule approx_mult2, assumption, auto)
+apply (drule_tac c = "inverse x" in approx_mult1, assumption)
+apply (auto intro: approx_sym simp add: hypreal_mult_assoc)
+done
+
+(*Used for NSLIM_inverse, NSLIMSEQ_inverse*)
+lemmas hypreal_of_real_approx_inverse = hypreal_of_real_HFinite_diff_Infinitesimal [THEN [2] approx_inverse]
+
+lemma inverse_add_Infinitesimal_approx: "[| x \<in> HFinite - Infinitesimal;
+ h \<in> Infinitesimal |] ==> inverse(x + h) @= inverse x"
+apply (auto intro: approx_inverse approx_sym Infinitesimal_add_approx_self)
+done
+
+lemma inverse_add_Infinitesimal_approx2: "[| x \<in> HFinite - Infinitesimal;
+ h \<in> Infinitesimal |] ==> inverse(h + x) @= inverse x"
+apply (rule hypreal_add_commute [THEN subst])
+apply (blast intro: inverse_add_Infinitesimal_approx)
+done
+
+lemma inverse_add_Infinitesimal_approx_Infinitesimal: "[| x \<in> HFinite - Infinitesimal;
+ h \<in> Infinitesimal |] ==> inverse(x + h) + -inverse x @= h"
+apply (rule approx_trans2)
+apply (auto intro: inverse_add_Infinitesimal_approx simp add: mem_infmal_iff approx_minus_iff [symmetric])
+done
+
+lemma Infinitesimal_square_iff: "(x \<in> Infinitesimal) = (x*x \<in> Infinitesimal)"
+apply (auto intro: Infinitesimal_mult)
+apply (rule ccontr, frule Infinitesimal_inverse_HFinite)
+apply (frule not_Infinitesimal_not_zero)
+apply (auto dest: Infinitesimal_HFinite_mult simp add: hypreal_mult_assoc)
+done
+declare Infinitesimal_square_iff [symmetric, simp]
+
+lemma HFinite_square_iff: "(x*x \<in> HFinite) = (x \<in> HFinite)"
+apply (auto intro: HFinite_mult)
+apply (auto dest: HInfinite_mult simp add: HFinite_HInfinite_iff)
+done
+declare HFinite_square_iff [simp]
+
+lemma HInfinite_square_iff: "(x*x \<in> HInfinite) = (x \<in> HInfinite)"
+by (auto simp add: HInfinite_HFinite_iff)
+declare HInfinite_square_iff [simp]
+
+lemma approx_HFinite_mult_cancel: "[| a: HFinite-Infinitesimal; a* w @= a*z |] ==> w @= z"
+apply safe
+apply (frule HFinite_inverse, assumption)
+apply (drule not_Infinitesimal_not_zero)
+apply (auto dest: approx_mult2 simp add: hypreal_mult_assoc [symmetric])
+done
+
+lemma approx_HFinite_mult_cancel_iff1: "a: HFinite-Infinitesimal ==> (a * w @= a * z) = (w @= z)"
+by (auto intro: approx_mult2 approx_HFinite_mult_cancel)
+
+lemma HInfinite_HFinite_add_cancel: "[| x + y \<in> HInfinite; y \<in> HFinite |] ==> x \<in> HInfinite"
+apply (rule ccontr)
+apply (drule HFinite_HInfinite_iff [THEN iffD2])
+apply (auto dest: HFinite_add simp add: HInfinite_HFinite_iff)
+done
+
+lemma HInfinite_HFinite_add: "[| x \<in> HInfinite; y \<in> HFinite |] ==> x + y \<in> HInfinite"
+apply (rule_tac y = "-y" in HInfinite_HFinite_add_cancel)
+apply (auto simp add: hypreal_add_assoc HFinite_minus_iff)
+done
+
+lemma HInfinite_ge_HInfinite: "[| x \<in> HInfinite; x <= y; 0 <= x |] ==> y \<in> HInfinite"
+by (auto intro: HFinite_bounded simp add: HInfinite_HFinite_iff)
+
+lemma Infinitesimal_inverse_HInfinite: "[| x \<in> Infinitesimal; x \<noteq> 0 |] ==> inverse x \<in> HInfinite"
+apply (rule ccontr, drule HFinite_HInfinite_iff [THEN iffD2])
+apply (auto dest: Infinitesimal_HFinite_mult2)
+done
+
+lemma HInfinite_HFinite_not_Infinitesimal_mult: "[| x \<in> HInfinite; y \<in> HFinite - Infinitesimal |]
+ ==> x * y \<in> HInfinite"
+apply (rule ccontr, drule HFinite_HInfinite_iff [THEN iffD2])
+apply (frule HFinite_Infinitesimal_not_zero)
+apply (drule HFinite_not_Infinitesimal_inverse)
+apply (safe, drule HFinite_mult)
+apply (auto simp add: hypreal_mult_assoc HFinite_HInfinite_iff)
+done
+
+lemma HInfinite_HFinite_not_Infinitesimal_mult2: "[| x \<in> HInfinite; y \<in> HFinite - Infinitesimal |]
+ ==> y * x \<in> HInfinite"
+apply (auto simp add: hypreal_mult_commute HInfinite_HFinite_not_Infinitesimal_mult)
+done
+
+lemma HInfinite_gt_SReal: "[| x \<in> HInfinite; 0 < x; y \<in> Reals |] ==> y < x"
+by (auto dest!: bspec simp add: HInfinite_def hrabs_def order_less_imp_le)
+
+lemma HInfinite_gt_zero_gt_one: "[| x \<in> HInfinite; 0 < x |] ==> 1 < x"
+by (auto intro: HInfinite_gt_SReal)
+
+
+lemma not_HInfinite_one: "1 \<notin> HInfinite"
+apply (simp (no_asm) add: HInfinite_HFinite_iff)
+done
+declare not_HInfinite_one [simp]
+
+(*------------------------------------------------------------------
+ more about absolute value (hrabs)
+ ------------------------------------------------------------------*)
+
+lemma approx_hrabs_disj: "abs x @= x | abs x @= -x"
+by (cut_tac x = x in hrabs_disj, auto)
+
+(*------------------------------------------------------------------
+ Theorems about monads
+ ------------------------------------------------------------------*)
+
+lemma monad_hrabs_Un_subset: "monad (abs x) <= monad(x) Un monad(-x)"
+by (rule_tac x1 = x in hrabs_disj [THEN disjE], auto)
+
+lemma Infinitesimal_monad_eq: "e \<in> Infinitesimal ==> monad (x+e) = monad x"
+by (fast intro!: Infinitesimal_add_approx_self [THEN approx_sym] approx_monad_iff [THEN iffD1])
+
+lemma mem_monad_iff: "(u \<in> monad x) = (-u \<in> monad (-x))"
+by (simp add: monad_def)
+
+lemma Infinitesimal_monad_zero_iff: "(x \<in> Infinitesimal) = (x \<in> monad 0)"
+by (auto intro: approx_sym simp add: monad_def mem_infmal_iff)
+
+lemma monad_zero_minus_iff: "(x \<in> monad 0) = (-x \<in> monad 0)"
+apply (simp (no_asm) add: Infinitesimal_monad_zero_iff [symmetric])
+done
+
+lemma monad_zero_hrabs_iff: "(x \<in> monad 0) = (abs x \<in> monad 0)"
+apply (rule_tac x1 = x in hrabs_disj [THEN disjE])
+apply (auto simp add: monad_zero_minus_iff [symmetric])
+done
+
+lemma mem_monad_self: "x \<in> monad x"
+by (simp add: monad_def)
+declare mem_monad_self [simp]
+
+(*------------------------------------------------------------------
+ Proof that x @= y ==> abs x @= abs y
+ ------------------------------------------------------------------*)
+lemma approx_subset_monad: "x @= y ==> {x,y}<=monad x"
+apply (simp (no_asm))
+apply (simp add: approx_monad_iff)
+done
+
+lemma approx_subset_monad2: "x @= y ==> {x,y}<=monad y"
+apply (drule approx_sym)
+apply (fast dest: approx_subset_monad)
+done
+
+lemma mem_monad_approx: "u \<in> monad x ==> x @= u"
+by (simp add: monad_def)
+
+lemma approx_mem_monad: "x @= u ==> u \<in> monad x"
+by (simp add: monad_def)
+
+lemma approx_mem_monad2: "x @= u ==> x \<in> monad u"
+apply (simp add: monad_def)
+apply (blast intro!: approx_sym)
+done
+
+lemma approx_mem_monad_zero: "[| x @= y;x \<in> monad 0 |] ==> y \<in> monad 0"
+apply (drule mem_monad_approx)
+apply (fast intro: approx_mem_monad approx_trans)
+done
+
+lemma Infinitesimal_approx_hrabs: "[| x @= y; x \<in> Infinitesimal |] ==> abs x @= abs y"
+apply (drule Infinitesimal_monad_zero_iff [THEN iffD1])
+apply (blast intro: approx_mem_monad_zero monad_zero_hrabs_iff [THEN iffD1] mem_monad_approx approx_trans3)
+done
+
+lemma less_Infinitesimal_less: "[| 0 < x; x \<notin>Infinitesimal; e :Infinitesimal |] ==> e < x"
+apply (rule ccontr)
+apply (auto intro: Infinitesimal_zero [THEN [2] Infinitesimal_interval]
+ dest!: order_le_imp_less_or_eq simp add: linorder_not_less)
+done
+
+lemma Ball_mem_monad_gt_zero: "[| 0 < x; x \<notin> Infinitesimal; u \<in> monad x |] ==> 0 < u"
+apply (drule mem_monad_approx [THEN approx_sym])
+apply (erule bex_Infinitesimal_iff2 [THEN iffD2, THEN bexE])
+apply (drule_tac e = "-xa" in less_Infinitesimal_less, auto)
+done
+
+lemma Ball_mem_monad_less_zero: "[| x < 0; x \<notin> Infinitesimal; u \<in> monad x |] ==> u < 0"
+apply (drule mem_monad_approx [THEN approx_sym])
+apply (erule bex_Infinitesimal_iff [THEN iffD2, THEN bexE])
+apply (cut_tac x = "-x" and e = xa in less_Infinitesimal_less, auto)
+done
+
+lemma lemma_approx_gt_zero: "[|0 < x; x \<notin> Infinitesimal; x @= y|] ==> 0 < y"
+by (blast dest: Ball_mem_monad_gt_zero approx_subset_monad)
+
+lemma lemma_approx_less_zero: "[|x < 0; x \<notin> Infinitesimal; x @= y|] ==> y < 0"
+by (blast dest: Ball_mem_monad_less_zero approx_subset_monad)
+
+lemma approx_hrabs1: "[| x @= y; x < 0; x \<notin> Infinitesimal |] ==> abs x @= abs y"
+apply (frule lemma_approx_less_zero)
+apply (assumption+)
+apply (simp add: abs_if)
+done
+
+lemma approx_hrabs2: "[| x @= y; 0 < x; x \<notin> Infinitesimal |] ==> abs x @= abs y"
+apply (frule lemma_approx_gt_zero)
+apply (assumption+)
+apply (simp add: abs_if)
+done
+
+lemma approx_hrabs: "x @= y ==> abs x @= abs y"
+apply (rule_tac Q = "x \<in> Infinitesimal" in excluded_middle [THEN disjE])
+apply (rule_tac x1 = x and y1 = 0 in linorder_less_linear [THEN disjE])
+apply (auto intro: approx_hrabs1 approx_hrabs2 Infinitesimal_approx_hrabs)
+done
+
+lemma approx_hrabs_zero_cancel: "abs(x) @= 0 ==> x @= 0"
+apply (cut_tac x = x in hrabs_disj)
+apply (auto dest: approx_minus)
+done
+
+lemma approx_hrabs_add_Infinitesimal: "e \<in> Infinitesimal ==> abs x @= abs(x+e)"
+by (fast intro: approx_hrabs Infinitesimal_add_approx_self)
+
+lemma approx_hrabs_add_minus_Infinitesimal: "e \<in> Infinitesimal ==> abs x @= abs(x + -e)"
+by (fast intro: approx_hrabs Infinitesimal_add_minus_approx_self)
+
+lemma hrabs_add_Infinitesimal_cancel: "[| e \<in> Infinitesimal; e' \<in> Infinitesimal;
+ abs(x+e) = abs(y+e')|] ==> abs x @= abs y"
+apply (drule_tac x = x in approx_hrabs_add_Infinitesimal)
+apply (drule_tac x = y in approx_hrabs_add_Infinitesimal)
+apply (auto intro: approx_trans2)
+done
+
+lemma hrabs_add_minus_Infinitesimal_cancel: "[| e \<in> Infinitesimal; e' \<in> Infinitesimal;
+ abs(x + -e) = abs(y + -e')|] ==> abs x @= abs y"
+apply (drule_tac x = x in approx_hrabs_add_minus_Infinitesimal)
+apply (drule_tac x = y in approx_hrabs_add_minus_Infinitesimal)
+apply (auto intro: approx_trans2)
+done
+
+lemma hypreal_less_minus_iff: "((x::hypreal) < y) = (0 < y + -x)"
+by arith
- (*standard real numbers as a subset of the hyperreals*)
- SReal_def "Reals == {x. EX r. x = hypreal_of_real r}"
+(* interesting slightly counterintuitive theorem: necessary
+ for proving that an open interval is an NS open set
+*)
+lemma Infinitesimal_add_hypreal_of_real_less:
+ "[| x < y; u \<in> Infinitesimal |]
+ ==> hypreal_of_real x + u < hypreal_of_real y"
+apply (simp add: Infinitesimal_def)
+apply (drule hypreal_of_real_less_iff [THEN iffD2])
+apply (drule_tac x = "hypreal_of_real y + -hypreal_of_real x" in bspec)
+apply (auto simp add: hypreal_add_commute abs_less_iff SReal_add SReal_minus)
+done
+
+lemma Infinitesimal_add_hrabs_hypreal_of_real_less: "[| x \<in> Infinitesimal; abs(hypreal_of_real r) < hypreal_of_real y |]
+ ==> abs (hypreal_of_real r + x) < hypreal_of_real y"
+apply (drule_tac x = "hypreal_of_real r" in approx_hrabs_add_Infinitesimal)
+apply (drule approx_sym [THEN bex_Infinitesimal_iff2 [THEN iffD2]])
+apply (auto intro!: Infinitesimal_add_hypreal_of_real_less simp add: hypreal_of_real_hrabs)
+done
+
+lemma Infinitesimal_add_hrabs_hypreal_of_real_less2: "[| x \<in> Infinitesimal; abs(hypreal_of_real r) < hypreal_of_real y |]
+ ==> abs (x + hypreal_of_real r) < hypreal_of_real y"
+apply (rule hypreal_add_commute [THEN subst])
+apply (erule Infinitesimal_add_hrabs_hypreal_of_real_less, assumption)
+done
+
+lemma hypreal_of_real_le_add_Infininitesimal_cancel: "[| u \<in> Infinitesimal; v \<in> Infinitesimal;
+ hypreal_of_real x + u <= hypreal_of_real y + v |]
+ ==> hypreal_of_real x <= hypreal_of_real y"
+apply (simp add: linorder_not_less [symmetric], auto)
+apply (drule_tac u = "v-u" in Infinitesimal_add_hypreal_of_real_less)
+apply (auto simp add: Infinitesimal_diff)
+done
+
+lemma hypreal_of_real_le_add_Infininitesimal_cancel2: "[| u \<in> Infinitesimal; v \<in> Infinitesimal;
+ hypreal_of_real x + u <= hypreal_of_real y + v |]
+ ==> x <= y"
+apply (blast intro!: hypreal_of_real_le_iff [THEN iffD1] hypreal_of_real_le_add_Infininitesimal_cancel)
+done
+
+lemma hypreal_of_real_less_Infinitesimal_le_zero: "[| hypreal_of_real x < e; e \<in> Infinitesimal |] ==> hypreal_of_real x <= 0"
+apply (rule linorder_not_less [THEN iffD1], safe)
+apply (drule Infinitesimal_interval)
+apply (drule_tac [4] SReal_hypreal_of_real [THEN SReal_Infinitesimal_zero], auto)
+done
+
+(*used once, in Lim/NSDERIV_inverse*)
+lemma Infinitesimal_add_not_zero: "[| h \<in> Infinitesimal; x \<noteq> 0 |] ==> hypreal_of_real x + h \<noteq> 0"
+apply auto
+apply (subgoal_tac "h = - hypreal_of_real x", auto)
+done
+
+lemma Infinitesimal_square_cancel: "x*x + y*y \<in> Infinitesimal ==> x*x \<in> Infinitesimal"
+apply (rule Infinitesimal_interval2)
+apply (rule_tac [3] zero_le_square, assumption)
+apply (auto simp add: zero_le_square)
+done
+declare Infinitesimal_square_cancel [simp]
+
+lemma HFinite_square_cancel: "x*x + y*y \<in> HFinite ==> x*x \<in> HFinite"
+apply (rule HFinite_bounded, assumption)
+apply (auto simp add: zero_le_square)
+done
+declare HFinite_square_cancel [simp]
+
+lemma Infinitesimal_square_cancel2: "x*x + y*y \<in> Infinitesimal ==> y*y \<in> Infinitesimal"
+apply (rule Infinitesimal_square_cancel)
+apply (rule hypreal_add_commute [THEN subst])
+apply (simp (no_asm))
+done
+declare Infinitesimal_square_cancel2 [simp]
+
+lemma HFinite_square_cancel2: "x*x + y*y \<in> HFinite ==> y*y \<in> HFinite"
+apply (rule HFinite_square_cancel)
+apply (rule hypreal_add_commute [THEN subst])
+apply (simp (no_asm))
+done
+declare HFinite_square_cancel2 [simp]
+
+lemma Infinitesimal_sum_square_cancel: "x*x + y*y + z*z \<in> Infinitesimal ==> x*x \<in> Infinitesimal"
+apply (rule Infinitesimal_interval2, assumption)
+apply (rule_tac [2] zero_le_square, simp)
+apply (insert zero_le_square [of y])
+apply (insert zero_le_square [of z], simp)
+done
+declare Infinitesimal_sum_square_cancel [simp]
+
+lemma HFinite_sum_square_cancel: "x*x + y*y + z*z \<in> HFinite ==> x*x \<in> HFinite"
+apply (rule HFinite_bounded, assumption)
+apply (rule_tac [2] zero_le_square)
+apply (insert zero_le_square [of y])
+apply (insert zero_le_square [of z], simp)
+done
+declare HFinite_sum_square_cancel [simp]
+
+lemma Infinitesimal_sum_square_cancel2: "y*y + x*x + z*z \<in> Infinitesimal ==> x*x \<in> Infinitesimal"
+apply (rule Infinitesimal_sum_square_cancel)
+apply (simp add: add_ac)
+done
+declare Infinitesimal_sum_square_cancel2 [simp]
+
+lemma HFinite_sum_square_cancel2: "y*y + x*x + z*z \<in> HFinite ==> x*x \<in> HFinite"
+apply (rule HFinite_sum_square_cancel)
+apply (simp add: add_ac)
+done
+declare HFinite_sum_square_cancel2 [simp]
+
+lemma Infinitesimal_sum_square_cancel3: "z*z + y*y + x*x \<in> Infinitesimal ==> x*x \<in> Infinitesimal"
+apply (rule Infinitesimal_sum_square_cancel)
+apply (simp add: add_ac)
+done
+declare Infinitesimal_sum_square_cancel3 [simp]
+
+lemma HFinite_sum_square_cancel3: "z*z + y*y + x*x \<in> HFinite ==> x*x \<in> HFinite"
+apply (rule HFinite_sum_square_cancel)
+apply (simp add: add_ac)
+done
+declare HFinite_sum_square_cancel3 [simp]
+
+lemma monad_hrabs_less: "[| y \<in> monad x; 0 < hypreal_of_real e |]
+ ==> abs (y + -x) < hypreal_of_real e"
+apply (drule mem_monad_approx [THEN approx_sym])
+apply (drule bex_Infinitesimal_iff [THEN iffD2])
+apply (auto dest!: InfinitesimalD)
+done
+
+lemma mem_monad_SReal_HFinite: "x \<in> monad (hypreal_of_real a) ==> x \<in> HFinite"
+apply (drule mem_monad_approx [THEN approx_sym])
+apply (drule bex_Infinitesimal_iff2 [THEN iffD2])
+apply (safe dest!: Infinitesimal_subset_HFinite [THEN subsetD])
+apply (erule SReal_hypreal_of_real [THEN SReal_subset_HFinite [THEN subsetD], THEN HFinite_add])
+done
+
+(*------------------------------------------------------------------
+ Theorems about standard part
+ ------------------------------------------------------------------*)
+
+lemma st_approx_self: "x \<in> HFinite ==> st x @= x"
+apply (simp add: st_def)
+apply (frule st_part_Ex, safe)
+apply (rule someI2)
+apply (auto intro: approx_sym)
+done
+
+lemma st_SReal: "x \<in> HFinite ==> st x \<in> Reals"
+apply (simp add: st_def)
+apply (frule st_part_Ex, safe)
+apply (rule someI2)
+apply (auto intro: approx_sym)
+done
+
+lemma st_HFinite: "x \<in> HFinite ==> st x \<in> HFinite"
+by (erule st_SReal [THEN SReal_subset_HFinite [THEN subsetD]])
+
+lemma st_SReal_eq: "x \<in> Reals ==> st x = x"
+apply (simp add: st_def)
+apply (rule some_equality)
+apply (fast intro: SReal_subset_HFinite [THEN subsetD])
+apply (blast dest: SReal_approx_iff [THEN iffD1])
+done
+
+(* ???should be added to simpset *)
+lemma st_hypreal_of_real: "st (hypreal_of_real x) = hypreal_of_real x"
+by (rule SReal_hypreal_of_real [THEN st_SReal_eq])
+
+lemma st_eq_approx: "[| x \<in> HFinite; y \<in> HFinite; st x = st y |] ==> x @= y"
+by (auto dest!: st_approx_self elim!: approx_trans3)
+
+lemma approx_st_eq:
+ assumes "x \<in> HFinite" and "y \<in> HFinite" and "x @= y"
+ shows "st x = st y"
+proof -
+ have "st x @= x" "st y @= y" "st x \<in> Reals" "st y \<in> Reals"
+ by (simp_all add: st_approx_self st_SReal prems)
+ with prems show ?thesis
+ by (fast elim: approx_trans approx_trans2 SReal_approx_iff [THEN iffD1])
+qed
+
+lemma st_eq_approx_iff: "[| x \<in> HFinite; y \<in> HFinite|]
+ ==> (x @= y) = (st x = st y)"
+by (blast intro: approx_st_eq st_eq_approx)
+
+lemma st_Infinitesimal_add_SReal: "[| x \<in> Reals; e \<in> Infinitesimal |] ==> st(x + e) = x"
+apply (frule st_SReal_eq [THEN subst])
+prefer 2 apply assumption
+apply (frule SReal_subset_HFinite [THEN subsetD])
+apply (frule Infinitesimal_subset_HFinite [THEN subsetD])
+apply (drule st_SReal_eq)
+apply (rule approx_st_eq)
+apply (auto intro: HFinite_add simp add: Infinitesimal_add_approx_self [THEN approx_sym])
+done
+
+lemma st_Infinitesimal_add_SReal2: "[| x \<in> Reals; e \<in> Infinitesimal |]
+ ==> st(e + x) = x"
+apply (rule hypreal_add_commute [THEN subst])
+apply (blast intro!: st_Infinitesimal_add_SReal)
+done
+
+lemma HFinite_st_Infinitesimal_add: "x \<in> HFinite ==>
+ \<exists>e \<in> Infinitesimal. x = st(x) + e"
+apply (blast dest!: st_approx_self [THEN approx_sym] bex_Infinitesimal_iff2 [THEN iffD2])
+done
+
+lemma st_add:
+ assumes x: "x \<in> HFinite" and y: "y \<in> HFinite"
+ shows "st (x + y) = st(x) + st(y)"
+proof -
+ from HFinite_st_Infinitesimal_add [OF x]
+ obtain ex where ex: "ex \<in> Infinitesimal" "st x + ex = x"
+ by (blast intro: sym)
+ from HFinite_st_Infinitesimal_add [OF y]
+ obtain ey where ey: "ey \<in> Infinitesimal" "st y + ey = y"
+ by (blast intro: sym)
+ have "st (x + y) = st ((st x + ex) + (st y + ey))"
+ by (simp add: ex ey)
+ also have "... = st ((ex + ey) + (st x + st y))" by (simp add: add_ac)
+ also have "... = st x + st y"
+ by (simp add: prems st_SReal SReal_add Infinitesimal_add
+ st_Infinitesimal_add_SReal2)
+ finally show ?thesis .
+qed
+
+lemma st_number_of: "st (number_of w) = number_of w"
+by (rule SReal_number_of [THEN st_SReal_eq])
+declare st_number_of [simp]
+
+(*the theorem above for the special cases of zero and one*)
+lemma [simp]: "st 0 = 0" "st 1 = 1"
+by (simp_all add: st_SReal_eq)
+
+lemma st_minus: assumes "y \<in> HFinite" shows "st(-y) = -st(y)"
+proof -
+ have "st (- y) + st y = 0"
+ by (simp add: prems st_add [symmetric] HFinite_minus_iff)
+ thus ?thesis by arith
+qed
+
+lemma st_diff:
+ "[| x \<in> HFinite; y \<in> HFinite |] ==> st (x-y) = st(x) - st(y)"
+apply (simp add: hypreal_diff_def)
+apply (frule_tac y1 = y in st_minus [symmetric])
+apply (drule_tac x1 = y in HFinite_minus_iff [THEN iffD2])
+apply (simp (no_asm_simp) add: st_add)
+done
+
+(* lemma *)
+lemma lemma_st_mult: "[| x \<in> HFinite; y \<in> HFinite;
+ e \<in> Infinitesimal;
+ ea \<in> Infinitesimal |]
+ ==> e*y + x*ea + e*ea \<in> Infinitesimal"
+apply (frule_tac x = e and y = y in Infinitesimal_HFinite_mult)
+apply (frule_tac [2] x = ea and y = x in Infinitesimal_HFinite_mult)
+apply (drule_tac [3] Infinitesimal_mult)
+apply (auto intro: Infinitesimal_add simp add: add_ac mult_ac)
+done
+
+lemma st_mult: "[| x \<in> HFinite; y \<in> HFinite |]
+ ==> st (x * y) = st(x) * st(y)"
+apply (frule HFinite_st_Infinitesimal_add)
+apply (frule_tac x = y in HFinite_st_Infinitesimal_add, safe)
+apply (subgoal_tac "st (x * y) = st ((st x + e) * (st y + ea))")
+apply (drule_tac [2] sym, drule_tac [2] sym)
+ prefer 2 apply simp
+apply (erule_tac V = "x = st x + e" in thin_rl)
+apply (erule_tac V = "y = st y + ea" in thin_rl)
+apply (simp add: left_distrib right_distrib)
+apply (drule st_SReal)+
+apply (simp (no_asm_use) add: hypreal_add_assoc)
+apply (rule st_Infinitesimal_add_SReal)
+apply (blast intro!: SReal_mult)
+apply (drule SReal_subset_HFinite [THEN subsetD])+
+apply (rule hypreal_add_assoc [THEN subst])
+apply (blast intro!: lemma_st_mult)
+done
+
+lemma st_Infinitesimal: "x \<in> Infinitesimal ==> st x = 0"
+apply (subst hypreal_numeral_0_eq_0 [symmetric])
+apply (rule st_number_of [THEN subst])
+apply (rule approx_st_eq)
+apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD] simp add: mem_infmal_iff [symmetric])
+done
+
+lemma st_not_Infinitesimal: "st(x) \<noteq> 0 ==> x \<notin> Infinitesimal"
+by (fast intro: st_Infinitesimal)
+
+lemma st_inverse: "[| x \<in> HFinite; st x \<noteq> 0 |]
+ ==> st(inverse x) = inverse (st x)"
+apply (rule_tac c1 = "st x" in hypreal_mult_left_cancel [THEN iffD1])
+apply (auto simp add: st_mult [symmetric] st_not_Infinitesimal HFinite_inverse)
+apply (subst hypreal_mult_inverse, auto)
+done
+
+lemma st_divide: "[| x \<in> HFinite; y \<in> HFinite; st y \<noteq> 0 |]
+ ==> st(x/y) = (st x) / (st y)"
+apply (auto simp add: hypreal_divide_def st_mult st_not_Infinitesimal HFinite_inverse st_inverse)
+done
+declare st_divide [simp]
+
+lemma st_idempotent: "x \<in> HFinite ==> st(st(x)) = st(x)"
+by (blast intro: st_HFinite st_approx_self approx_st_eq)
+declare st_idempotent [simp]
+
+(*** lemmas ***)
+lemma Infinitesimal_add_st_less: "[| x \<in> HFinite; y \<in> HFinite;
+ u \<in> Infinitesimal; st x < st y
+ |] ==> st x + u < st y"
+apply (drule st_SReal)+
+apply (auto intro!: Infinitesimal_add_hypreal_of_real_less simp add: SReal_iff)
+done
+
+lemma Infinitesimal_add_st_le_cancel: "[| x \<in> HFinite; y \<in> HFinite;
+ u \<in> Infinitesimal; st x <= st y + u
+ |] ==> st x <= st y"
+apply (simp add: linorder_not_less [symmetric])
+apply (auto dest: Infinitesimal_add_st_less)
+done
+
+lemma st_le: "[| x \<in> HFinite; y \<in> HFinite; x <= y |] ==> st(x) <= st(y)"
+apply (frule HFinite_st_Infinitesimal_add)
+apply (rotate_tac 1)
+apply (frule HFinite_st_Infinitesimal_add, safe)
+apply (rule Infinitesimal_add_st_le_cancel)
+apply (rule_tac [3] x = ea and y = e in Infinitesimal_diff)
+apply (auto simp add: hypreal_add_assoc [symmetric])
+done
+
+lemma st_zero_le: "[| 0 <= x; x \<in> HFinite |] ==> 0 <= st x"
+apply (subst hypreal_numeral_0_eq_0 [symmetric])
+apply (rule st_number_of [THEN subst])
+apply (rule st_le, auto)
+done
+
+lemma st_zero_ge: "[| x <= 0; x \<in> HFinite |] ==> st x <= 0"
+apply (subst hypreal_numeral_0_eq_0 [symmetric])
+apply (rule st_number_of [THEN subst])
+apply (rule st_le, auto)
+done
+
+lemma st_hrabs: "x \<in> HFinite ==> abs(st x) = st(abs x)"
+apply (simp add: linorder_not_le st_zero_le abs_if st_minus
+ linorder_not_less)
+apply (auto dest!: st_zero_ge [OF order_less_imp_le])
+done
+
+
+
+(*--------------------------------------------------------------------
+ Alternative definitions for HFinite using Free ultrafilter
+ --------------------------------------------------------------------*)
+
+lemma FreeUltrafilterNat_Rep_hypreal: "[| X \<in> Rep_hypreal x; Y \<in> Rep_hypreal x |]
+ ==> {n. X n = Y n} \<in> FreeUltrafilterNat"
+apply (rule_tac z = x in eq_Abs_hypreal, auto, ultra)
+done
+
+lemma HFinite_FreeUltrafilterNat:
+ "x \<in> HFinite
+ ==> \<exists>X \<in> Rep_hypreal x. \<exists>u. {n. abs (X n) < u} \<in> FreeUltrafilterNat"
+apply (rule eq_Abs_hypreal [of x])
+apply (auto simp add: HFinite_def abs_less_iff minus_less_iff [of x]
+ hypreal_less SReal_iff hypreal_minus hypreal_of_real_def)
+apply (rule_tac x=x in bexI)
+apply (rule_tac x=y in exI, auto, ultra)
+done
+
+lemma FreeUltrafilterNat_HFinite:
+ "\<exists>X \<in> Rep_hypreal x.
+ \<exists>u. {n. abs (X n) < u} \<in> FreeUltrafilterNat
+ ==> x \<in> HFinite"
+apply (rule eq_Abs_hypreal [of x])
+apply (auto simp add: HFinite_def abs_less_iff minus_less_iff [of x])
+apply (rule_tac x = "hypreal_of_real u" in bexI)
+apply (auto simp add: hypreal_less SReal_iff hypreal_minus hypreal_of_real_def, ultra+)
+done
+
+lemma HFinite_FreeUltrafilterNat_iff: "(x \<in> HFinite) = (\<exists>X \<in> Rep_hypreal x.
+ \<exists>u. {n. abs (X n) < u} \<in> FreeUltrafilterNat)"
+apply (blast intro!: HFinite_FreeUltrafilterNat FreeUltrafilterNat_HFinite)
+done
+
+(*--------------------------------------------------------------------
+ Alternative definitions for HInfinite using Free ultrafilter
+ --------------------------------------------------------------------*)
+lemma lemma_Compl_eq: "- {n. (u::real) < abs (xa n)} = {n. abs (xa n) <= u}"
+by auto
+
+lemma lemma_Compl_eq2: "- {n. abs (xa n) < (u::real)} = {n. u <= abs (xa n)}"
+by auto
+
+lemma lemma_Int_eq1: "{n. abs (xa n) <= (u::real)} Int {n. u <= abs (xa n)}
+ = {n. abs(xa n) = u}"
+apply auto
+done
+
+lemma lemma_FreeUltrafilterNat_one: "{n. abs (xa n) = u} <= {n. abs (xa n) < u + (1::real)}"
+by auto
+
+(*-------------------------------------
+ Exclude this type of sets from free
+ ultrafilter for Infinite numbers!
+ -------------------------------------*)
+lemma FreeUltrafilterNat_const_Finite: "[| xa: Rep_hypreal x;
+ {n. abs (xa n) = u} \<in> FreeUltrafilterNat
+ |] ==> x \<in> HFinite"
+apply (rule FreeUltrafilterNat_HFinite)
+apply (rule_tac x = xa in bexI)
+apply (rule_tac x = "u + 1" in exI)
+apply (ultra, assumption)
+done
+
+lemma HInfinite_FreeUltrafilterNat:
+ "x \<in> HInfinite ==> \<exists>X \<in> Rep_hypreal x.
+ \<forall>u. {n. u < abs (X n)} \<in> FreeUltrafilterNat"
+apply (frule HInfinite_HFinite_iff [THEN iffD1])
+apply (cut_tac x = x in Rep_hypreal_nonempty)
+apply (auto simp del: Rep_hypreal_nonempty simp add: HFinite_FreeUltrafilterNat_iff Bex_def)
+apply (drule spec)+
+apply auto
+apply (drule_tac x = u in spec)
+apply (drule FreeUltrafilterNat_Compl_mem)+
+apply (drule FreeUltrafilterNat_Int, assumption)
+apply (simp add: lemma_Compl_eq lemma_Compl_eq2 lemma_Int_eq1)
+apply (auto dest: FreeUltrafilterNat_const_Finite simp
+ add: HInfinite_HFinite_iff [THEN iffD1])
+done
+
+(* yet more lemmas! *)
+lemma lemma_Int_HI: "{n. abs (Xa n) < u} Int {n. X n = Xa n}
+ <= {n. abs (X n) < (u::real)}"
+apply auto
+done
+
+lemma lemma_Int_HIa: "{n. u < abs (X n)} Int {n. abs (X n) < (u::real)} = {}"
+by (auto intro: order_less_asym)
+
+lemma FreeUltrafilterNat_HInfinite: "\<exists>X \<in> Rep_hypreal x. \<forall>u.
+ {n. u < abs (X n)} \<in> FreeUltrafilterNat
+ ==> x \<in> HInfinite"
+apply (rule HInfinite_HFinite_iff [THEN iffD2])
+apply (safe, drule HFinite_FreeUltrafilterNat, auto)
+apply (drule_tac x = u in spec)
+apply (drule FreeUltrafilterNat_Rep_hypreal, assumption)
+apply (drule_tac Y = "{n. X n = Xa n}" in FreeUltrafilterNat_Int, simp)
+apply (drule lemma_Int_HI [THEN [2] FreeUltrafilterNat_subset])
+apply (drule_tac Y = "{n. abs (X n) < u}" in FreeUltrafilterNat_Int)
+apply (auto simp add: lemma_Int_HIa FreeUltrafilterNat_empty)
+done
+
+lemma HInfinite_FreeUltrafilterNat_iff: "(x \<in> HInfinite) = (\<exists>X \<in> Rep_hypreal x.
+ \<forall>u. {n. u < abs (X n)} \<in> FreeUltrafilterNat)"
+apply (blast intro!: HInfinite_FreeUltrafilterNat FreeUltrafilterNat_HInfinite)
+done
+
+(*--------------------------------------------------------------------
+ Alternative definitions for Infinitesimal using Free ultrafilter
+ --------------------------------------------------------------------*)
+
-syntax (xsymbols)
- approx :: "[hypreal, hypreal] => bool" (infixl "\\<approx>" 50)
-
+lemma Infinitesimal_FreeUltrafilterNat:
+ "x \<in> Infinitesimal ==> \<exists>X \<in> Rep_hypreal x.
+ \<forall>u. 0 < u --> {n. abs (X n) < u} \<in> FreeUltrafilterNat"
+apply (simp add: Infinitesimal_def)
+apply (auto simp add: abs_less_iff minus_less_iff [of x])
+apply (rule eq_Abs_hypreal [of x])
+apply (auto, rule bexI [OF _ lemma_hyprel_refl], safe)
+apply (drule hypreal_of_real_less_iff [THEN iffD2])
+apply (drule_tac x = "hypreal_of_real u" in bspec, auto)
+apply (auto simp add: hypreal_less hypreal_minus hypreal_of_real_def, ultra)
+done
+
+lemma FreeUltrafilterNat_Infinitesimal:
+ "\<exists>X \<in> Rep_hypreal x.
+ \<forall>u. 0 < u --> {n. abs (X n) < u} \<in> FreeUltrafilterNat
+ ==> x \<in> Infinitesimal"
+apply (simp add: Infinitesimal_def)
+apply (rule eq_Abs_hypreal [of x])
+apply (auto simp add: abs_less_iff abs_interval_iff minus_less_iff [of x])
+apply (auto simp add: SReal_iff)
+apply (drule_tac [!] x=y in spec)
+apply (auto simp add: hypreal_less hypreal_minus hypreal_of_real_def, ultra+)
+done
+
+lemma Infinitesimal_FreeUltrafilterNat_iff: "(x \<in> Infinitesimal) = (\<exists>X \<in> Rep_hypreal x.
+ \<forall>u. 0 < u --> {n. abs (X n) < u} \<in> FreeUltrafilterNat)"
+apply (blast intro!: Infinitesimal_FreeUltrafilterNat FreeUltrafilterNat_Infinitesimal)
+done
+
+(*------------------------------------------------------------------------
+ Infinitesimals as smaller than 1/n for all n::nat (> 0)
+ ------------------------------------------------------------------------*)
+
+lemma lemma_Infinitesimal: "(\<forall>r. 0 < r --> x < r) = (\<forall>n. x < inverse(real (Suc n)))"
+apply (auto simp add: real_of_nat_Suc_gt_zero)
+apply (blast dest!: reals_Archimedean intro: order_less_trans)
+done
+
+lemma lemma_Infinitesimal2: "(\<forall>r \<in> Reals. 0 < r --> x < r) =
+ (\<forall>n. x < inverse(hypreal_of_nat (Suc n)))"
+apply safe
+apply (drule_tac x = "inverse (hypreal_of_real (real (Suc n))) " in bspec)
+apply (simp (no_asm_use) add: SReal_inverse)
+apply (rule real_of_nat_Suc_gt_zero [THEN positive_imp_inverse_positive, THEN hypreal_of_real_less_iff [THEN iffD2], THEN [2] impE])
+prefer 2 apply assumption
+apply (simp add: real_of_nat_Suc_gt_zero hypreal_of_nat_def)
+apply (auto dest!: reals_Archimedean simp add: SReal_iff)
+apply (drule hypreal_of_real_less_iff [THEN iffD2])
+apply (simp add: real_of_nat_Suc_gt_zero hypreal_of_nat_def)
+apply (blast intro: order_less_trans)
+done
+
+lemma Infinitesimal_hypreal_of_nat_iff:
+ "Infinitesimal = {x. \<forall>n. abs x < inverse (hypreal_of_nat (Suc n))}"
+apply (simp add: Infinitesimal_def)
+apply (auto simp add: lemma_Infinitesimal2)
+done
+
+
+(*-------------------------------------------------------------------------
+ Proof that omega is an infinite number and
+ hence that epsilon is an infinitesimal number.
+ -------------------------------------------------------------------------*)
+lemma Suc_Un_eq: "{n. n < Suc m} = {n. n < m} Un {n. n = m}"
+by (auto simp add: less_Suc_eq)
+
+(*-------------------------------------------
+ Prove that any segment is finite and
+ hence cannot belong to FreeUltrafilterNat
+ -------------------------------------------*)
+lemma finite_nat_segment: "finite {n::nat. n < m}"
+apply (induct_tac "m")
+apply (auto simp add: Suc_Un_eq)
+done
+
+lemma finite_real_of_nat_segment: "finite {n::nat. real n < real (m::nat)}"
+by (auto intro: finite_nat_segment)
+
+lemma finite_real_of_nat_less_real: "finite {n::nat. real n < u}"
+apply (cut_tac x = u in reals_Archimedean2, safe)
+apply (rule finite_real_of_nat_segment [THEN [2] finite_subset])
+apply (auto dest: order_less_trans)
+done
+
+lemma lemma_real_le_Un_eq: "{n. f n <= u} = {n. f n < u} Un {n. u = (f n :: real)}"
+by (auto dest: order_le_imp_less_or_eq simp add: order_less_imp_le)
+
+lemma finite_real_of_nat_le_real: "finite {n::nat. real n <= u}"
+by (auto simp add: lemma_real_le_Un_eq lemma_finite_omega_set finite_real_of_nat_less_real)
+
+lemma finite_rabs_real_of_nat_le_real: "finite {n::nat. abs(real n) <= u}"
+apply (simp (no_asm) add: real_of_nat_Suc_gt_zero finite_real_of_nat_le_real)
+done
+
+lemma rabs_real_of_nat_le_real_FreeUltrafilterNat: "{n. abs(real n) <= u} \<notin> FreeUltrafilterNat"
+by (blast intro!: FreeUltrafilterNat_finite finite_rabs_real_of_nat_le_real)
+
+lemma FreeUltrafilterNat_nat_gt_real: "{n. u < real n} \<in> FreeUltrafilterNat"
+apply (rule ccontr, drule FreeUltrafilterNat_Compl_mem)
+apply (subgoal_tac "- {n::nat. u < real n} = {n. real n <= u}")
+prefer 2 apply force
+apply (simp add: finite_real_of_nat_le_real [THEN FreeUltrafilterNat_finite])
+done
+
+(*--------------------------------------------------------------
+ The complement of {n. abs(real n) <= u} =
+ {n. u < abs (real n)} is in FreeUltrafilterNat
+ by property of (free) ultrafilters
+ --------------------------------------------------------------*)
+
+lemma Compl_real_le_eq: "- {n::nat. real n <= u} = {n. u < real n}"
+by (auto dest!: order_le_less_trans simp add: linorder_not_le)
+
+(*-----------------------------------------------
+ Omega is a member of HInfinite
+ -----------------------------------------------*)
+
+lemma hypreal_omega: "hyprel``{%n::nat. real (Suc n)} \<in> hypreal"
+by auto
+
+lemma FreeUltrafilterNat_omega: "{n. u < real n} \<in> FreeUltrafilterNat"
+apply (cut_tac u = u in rabs_real_of_nat_le_real_FreeUltrafilterNat)
+apply (auto dest: FreeUltrafilterNat_Compl_mem simp add: Compl_real_le_eq)
+done
+
+lemma HInfinite_omega: "omega: HInfinite"
+apply (simp add: omega_def)
+apply (auto intro!: FreeUltrafilterNat_HInfinite)
+apply (rule bexI)
+apply (rule_tac [2] lemma_hyprel_refl, auto)
+apply (simp (no_asm) add: real_of_nat_Suc diff_less_eq [symmetric] FreeUltrafilterNat_omega)
+done
+declare HInfinite_omega [simp]
+
+(*-----------------------------------------------
+ Epsilon is a member of Infinitesimal
+ -----------------------------------------------*)
+
+lemma Infinitesimal_epsilon: "epsilon \<in> Infinitesimal"
+by (auto intro!: HInfinite_inverse_Infinitesimal HInfinite_omega simp add: hypreal_epsilon_inverse_omega)
+declare Infinitesimal_epsilon [simp]
+
+lemma HFinite_epsilon: "epsilon \<in> HFinite"
+by (auto intro: Infinitesimal_subset_HFinite [THEN subsetD])
+declare HFinite_epsilon [simp]
+
+lemma epsilon_approx_zero: "epsilon @= 0"
+apply (simp (no_asm) add: mem_infmal_iff [symmetric])
+done
+declare epsilon_approx_zero [simp]
+
+(*------------------------------------------------------------------------
+ Needed for proof that we define a hyperreal [<X(n)] @= hypreal_of_real a given
+ that \<forall>n. |X n - a| < 1/n. Used in proof of NSLIM => LIM.
+ -----------------------------------------------------------------------*)
+
+lemma real_of_nat_less_inverse_iff: "0 < u ==>
+ (u < inverse (real(Suc n))) = (real(Suc n) < inverse u)"
+apply (simp add: inverse_eq_divide)
+apply (subst pos_less_divide_eq, assumption)
+apply (subst pos_less_divide_eq)
+ apply (simp add: real_of_nat_Suc_gt_zero)
+apply (simp add: real_mult_commute)
+done
+
+lemma finite_inverse_real_of_posnat_gt_real: "0 < u ==> finite {n. u < inverse(real(Suc n))}"
+apply (simp (no_asm_simp) add: real_of_nat_less_inverse_iff)
+apply (simp (no_asm_simp) add: real_of_nat_Suc less_diff_eq [symmetric])
+apply (rule finite_real_of_nat_less_real)
+done
+
+lemma lemma_real_le_Un_eq2: "{n. u <= inverse(real(Suc n))} =
+ {n. u < inverse(real(Suc n))} Un {n. u = inverse(real(Suc n))}"
+apply (auto dest: order_le_imp_less_or_eq simp add: order_less_imp_le)
+done
+
+lemma real_of_nat_inverse_le_iff: "(inverse (real(Suc n)) <= r) = (1 <= r * real(Suc n))"
+apply (simp (no_asm) add: linorder_not_less [symmetric])
+apply (simp (no_asm) add: inverse_eq_divide)
+apply (subst pos_less_divide_eq)
+apply (simp (no_asm) add: real_of_nat_Suc_gt_zero)
+apply (simp (no_asm) add: real_mult_commute)
+done
+
+lemma real_of_nat_inverse_eq_iff: "(u = inverse (real(Suc n))) = (real(Suc n) = inverse u)"
+by (auto simp add: inverse_inverse_eq real_of_nat_Suc_gt_zero real_not_refl2 [THEN not_sym])
+
+lemma lemma_finite_omega_set2: "finite {n::nat. u = inverse(real(Suc n))}"
+apply (simp (no_asm_simp) add: real_of_nat_inverse_eq_iff)
+apply (cut_tac x = "inverse u - 1" in lemma_finite_omega_set)
+apply (simp add: real_of_nat_Suc diff_eq_eq [symmetric] eq_commute)
+done
+
+lemma finite_inverse_real_of_posnat_ge_real: "0 < u ==> finite {n. u <= inverse(real(Suc n))}"
+by (auto simp add: lemma_real_le_Un_eq2 lemma_finite_omega_set2 finite_inverse_real_of_posnat_gt_real)
+
+lemma inverse_real_of_posnat_ge_real_FreeUltrafilterNat: "0 < u ==>
+ {n. u <= inverse(real(Suc n))} \<notin> FreeUltrafilterNat"
+apply (blast intro!: FreeUltrafilterNat_finite finite_inverse_real_of_posnat_ge_real)
+done
+
+(*--------------------------------------------------------------
+ The complement of {n. u <= inverse(real(Suc n))} =
+ {n. inverse(real(Suc n)) < u} is in FreeUltrafilterNat
+ by property of (free) ultrafilters
+ --------------------------------------------------------------*)
+lemma Compl_le_inverse_eq: "- {n. u <= inverse(real(Suc n))} =
+ {n. inverse(real(Suc n)) < u}"
+apply (auto dest!: order_le_less_trans simp add: linorder_not_le)
+done
+
+lemma FreeUltrafilterNat_inverse_real_of_posnat: "0 < u ==>
+ {n. inverse(real(Suc n)) < u} \<in> FreeUltrafilterNat"
+apply (cut_tac u = u in inverse_real_of_posnat_ge_real_FreeUltrafilterNat)
+apply (auto dest: FreeUltrafilterNat_Compl_mem simp add: Compl_le_inverse_eq)
+done
+
+(*--------------------------------------------------------------
+ Example where we get a hyperreal from a real sequence
+ for which a particular property holds. The theorem is
+ used in proofs about equivalence of nonstandard and
+ standard neighbourhoods. Also used for equivalence of
+ nonstandard ans standard definitions of pointwise
+ limit (the theorem was previously in REALTOPOS.thy).
+ -------------------------------------------------------------*)
+(*-----------------------------------------------------
+ |X(n) - x| < 1/n ==> [<X n>] - hypreal_of_real x| \<in> Infinitesimal
+ -----------------------------------------------------*)
+lemma real_seq_to_hypreal_Infinitesimal: "\<forall>n. abs(X n + -x) < inverse(real(Suc n))
+ ==> Abs_hypreal(hyprel``{X}) + -hypreal_of_real x \<in> Infinitesimal"
+apply (auto intro!: bexI dest: FreeUltrafilterNat_inverse_real_of_posnat FreeUltrafilterNat_all FreeUltrafilterNat_Int intro: order_less_trans FreeUltrafilterNat_subset simp add: hypreal_minus hypreal_of_real_def hypreal_add Infinitesimal_FreeUltrafilterNat_iff hypreal_inverse)
+done
+
+lemma real_seq_to_hypreal_approx: "\<forall>n. abs(X n + -x) < inverse(real(Suc n))
+ ==> Abs_hypreal(hyprel``{X}) @= hypreal_of_real x"
+apply (subst approx_minus_iff)
+apply (rule mem_infmal_iff [THEN subst])
+apply (erule real_seq_to_hypreal_Infinitesimal)
+done
+
+lemma real_seq_to_hypreal_approx2: "\<forall>n. abs(x + -X n) < inverse(real(Suc n))
+ ==> Abs_hypreal(hyprel``{X}) @= hypreal_of_real x"
+apply (simp add: abs_minus_add_cancel real_seq_to_hypreal_approx)
+done
+
+lemma real_seq_to_hypreal_Infinitesimal2: "\<forall>n. abs(X n + -Y n) < inverse(real(Suc n))
+ ==> Abs_hypreal(hyprel``{X}) +
+ -Abs_hypreal(hyprel``{Y}) \<in> Infinitesimal"
+by (auto intro!: bexI
+ dest: FreeUltrafilterNat_inverse_real_of_posnat
+ FreeUltrafilterNat_all FreeUltrafilterNat_Int
+ intro: order_less_trans FreeUltrafilterNat_subset
+ simp add: Infinitesimal_FreeUltrafilterNat_iff hypreal_minus
+ hypreal_add hypreal_inverse)
+
+
+ML
+{*
+val Infinitesimal_def = thm"Infinitesimal_def";
+val HFinite_def = thm"HFinite_def";
+val HInfinite_def = thm"HInfinite_def";
+val st_def = thm"st_def";
+val monad_def = thm"monad_def";
+val galaxy_def = thm"galaxy_def";
+val approx_def = thm"approx_def";
+val SReal_def = thm"SReal_def";
+
+val Infinitesimal_approx_minus = thm "Infinitesimal_approx_minus";
+val approx_monad_iff = thm "approx_monad_iff";
+val Infinitesimal_approx = thm "Infinitesimal_approx";
+val approx_add = thm "approx_add";
+val approx_minus = thm "approx_minus";
+val approx_minus2 = thm "approx_minus2";
+val approx_minus_cancel = thm "approx_minus_cancel";
+val approx_add_minus = thm "approx_add_minus";
+val approx_mult1 = thm "approx_mult1";
+val approx_mult2 = thm "approx_mult2";
+val approx_mult_subst = thm "approx_mult_subst";
+val approx_mult_subst2 = thm "approx_mult_subst2";
+val approx_mult_subst_SReal = thm "approx_mult_subst_SReal";
+val approx_eq_imp = thm "approx_eq_imp";
+val Infinitesimal_minus_approx = thm "Infinitesimal_minus_approx";
+val bex_Infinitesimal_iff = thm "bex_Infinitesimal_iff";
+val bex_Infinitesimal_iff2 = thm "bex_Infinitesimal_iff2";
+val Infinitesimal_add_approx = thm "Infinitesimal_add_approx";
+val Infinitesimal_add_approx_self = thm "Infinitesimal_add_approx_self";
+val Infinitesimal_add_approx_self2 = thm "Infinitesimal_add_approx_self2";
+val Infinitesimal_add_minus_approx_self = thm "Infinitesimal_add_minus_approx_self";
+val Infinitesimal_add_cancel = thm "Infinitesimal_add_cancel";
+val Infinitesimal_add_right_cancel = thm "Infinitesimal_add_right_cancel";
+val approx_add_left_cancel = thm "approx_add_left_cancel";
+val approx_add_right_cancel = thm "approx_add_right_cancel";
+val approx_add_mono1 = thm "approx_add_mono1";
+val approx_add_mono2 = thm "approx_add_mono2";
+val approx_add_left_iff = thm "approx_add_left_iff";
+val approx_add_right_iff = thm "approx_add_right_iff";
+val approx_HFinite = thm "approx_HFinite";
+val approx_hypreal_of_real_HFinite = thm "approx_hypreal_of_real_HFinite";
+val approx_mult_HFinite = thm "approx_mult_HFinite";
+val approx_mult_hypreal_of_real = thm "approx_mult_hypreal_of_real";
+val approx_SReal_mult_cancel_zero = thm "approx_SReal_mult_cancel_zero";
+val approx_mult_SReal1 = thm "approx_mult_SReal1";
+val approx_mult_SReal2 = thm "approx_mult_SReal2";
+val approx_mult_SReal_zero_cancel_iff = thm "approx_mult_SReal_zero_cancel_iff";
+val approx_SReal_mult_cancel = thm "approx_SReal_mult_cancel";
+val approx_SReal_mult_cancel_iff1 = thm "approx_SReal_mult_cancel_iff1";
+val approx_le_bound = thm "approx_le_bound";
+val Infinitesimal_less_SReal = thm "Infinitesimal_less_SReal";
+val Infinitesimal_less_SReal2 = thm "Infinitesimal_less_SReal2";
+val SReal_not_Infinitesimal = thm "SReal_not_Infinitesimal";
+val SReal_minus_not_Infinitesimal = thm "SReal_minus_not_Infinitesimal";
+val SReal_Int_Infinitesimal_zero = thm "SReal_Int_Infinitesimal_zero";
+val SReal_Infinitesimal_zero = thm "SReal_Infinitesimal_zero";
+val SReal_HFinite_diff_Infinitesimal = thm "SReal_HFinite_diff_Infinitesimal";
+val hypreal_of_real_HFinite_diff_Infinitesimal = thm "hypreal_of_real_HFinite_diff_Infinitesimal";
+val hypreal_of_real_Infinitesimal_iff_0 = thm "hypreal_of_real_Infinitesimal_iff_0";
+val number_of_not_Infinitesimal = thm "number_of_not_Infinitesimal";
+val one_not_Infinitesimal = thm "one_not_Infinitesimal";
+val approx_SReal_not_zero = thm "approx_SReal_not_zero";
+val HFinite_diff_Infinitesimal_approx = thm "HFinite_diff_Infinitesimal_approx";
+val Infinitesimal_ratio = thm "Infinitesimal_ratio";
+val SReal_approx_iff = thm "SReal_approx_iff";
+val number_of_approx_iff = thm "number_of_approx_iff";
+val hypreal_of_real_approx_iff = thm "hypreal_of_real_approx_iff";
+val hypreal_of_real_approx_number_of_iff = thm "hypreal_of_real_approx_number_of_iff";
+val approx_unique_real = thm "approx_unique_real";
+val hypreal_isLub_unique = thm "hypreal_isLub_unique";
+val hypreal_setle_less_trans = thm "hypreal_setle_less_trans";
+val hypreal_gt_isUb = thm "hypreal_gt_isUb";
+val st_part_Ex = thm "st_part_Ex";
+val st_part_Ex1 = thm "st_part_Ex1";
+val HFinite_Int_HInfinite_empty = thm "HFinite_Int_HInfinite_empty";
+val HFinite_not_HInfinite = thm "HFinite_not_HInfinite";
+val not_HFinite_HInfinite = thm "not_HFinite_HInfinite";
+val HInfinite_HFinite_disj = thm "HInfinite_HFinite_disj";
+val HInfinite_HFinite_iff = thm "HInfinite_HFinite_iff";
+val HFinite_HInfinite_iff = thm "HFinite_HInfinite_iff";
+val HInfinite_diff_HFinite_Infinitesimal_disj = thm "HInfinite_diff_HFinite_Infinitesimal_disj";
+val HFinite_inverse = thm "HFinite_inverse";
+val HFinite_inverse2 = thm "HFinite_inverse2";
+val Infinitesimal_inverse_HFinite = thm "Infinitesimal_inverse_HFinite";
+val HFinite_not_Infinitesimal_inverse = thm "HFinite_not_Infinitesimal_inverse";
+val approx_inverse = thm "approx_inverse";
+val hypreal_of_real_approx_inverse = thm "hypreal_of_real_approx_inverse";
+val inverse_add_Infinitesimal_approx = thm "inverse_add_Infinitesimal_approx";
+val inverse_add_Infinitesimal_approx2 = thm "inverse_add_Infinitesimal_approx2";
+val inverse_add_Infinitesimal_approx_Infinitesimal = thm "inverse_add_Infinitesimal_approx_Infinitesimal";
+val Infinitesimal_square_iff = thm "Infinitesimal_square_iff";
+val HFinite_square_iff = thm "HFinite_square_iff";
+val HInfinite_square_iff = thm "HInfinite_square_iff";
+val approx_HFinite_mult_cancel = thm "approx_HFinite_mult_cancel";
+val approx_HFinite_mult_cancel_iff1 = thm "approx_HFinite_mult_cancel_iff1";
+val approx_hrabs_disj = thm "approx_hrabs_disj";
+val monad_hrabs_Un_subset = thm "monad_hrabs_Un_subset";
+val Infinitesimal_monad_eq = thm "Infinitesimal_monad_eq";
+val mem_monad_iff = thm "mem_monad_iff";
+val Infinitesimal_monad_zero_iff = thm "Infinitesimal_monad_zero_iff";
+val monad_zero_minus_iff = thm "monad_zero_minus_iff";
+val monad_zero_hrabs_iff = thm "monad_zero_hrabs_iff";
+val mem_monad_self = thm "mem_monad_self";
+val approx_subset_monad = thm "approx_subset_monad";
+val approx_subset_monad2 = thm "approx_subset_monad2";
+val mem_monad_approx = thm "mem_monad_approx";
+val approx_mem_monad = thm "approx_mem_monad";
+val approx_mem_monad2 = thm "approx_mem_monad2";
+val approx_mem_monad_zero = thm "approx_mem_monad_zero";
+val Infinitesimal_approx_hrabs = thm "Infinitesimal_approx_hrabs";
+val less_Infinitesimal_less = thm "less_Infinitesimal_less";
+val Ball_mem_monad_gt_zero = thm "Ball_mem_monad_gt_zero";
+val Ball_mem_monad_less_zero = thm "Ball_mem_monad_less_zero";
+val approx_hrabs1 = thm "approx_hrabs1";
+val approx_hrabs2 = thm "approx_hrabs2";
+val approx_hrabs = thm "approx_hrabs";
+val approx_hrabs_zero_cancel = thm "approx_hrabs_zero_cancel";
+val approx_hrabs_add_Infinitesimal = thm "approx_hrabs_add_Infinitesimal";
+val approx_hrabs_add_minus_Infinitesimal = thm "approx_hrabs_add_minus_Infinitesimal";
+val hrabs_add_Infinitesimal_cancel = thm "hrabs_add_Infinitesimal_cancel";
+val hrabs_add_minus_Infinitesimal_cancel = thm "hrabs_add_minus_Infinitesimal_cancel";
+val hypreal_less_minus_iff = thm "hypreal_less_minus_iff";
+val Infinitesimal_add_hypreal_of_real_less = thm "Infinitesimal_add_hypreal_of_real_less";
+val Infinitesimal_add_hrabs_hypreal_of_real_less = thm "Infinitesimal_add_hrabs_hypreal_of_real_less";
+val Infinitesimal_add_hrabs_hypreal_of_real_less2 = thm "Infinitesimal_add_hrabs_hypreal_of_real_less2";
+val hypreal_of_real_le_add_Infininitesimal_cancel2 = thm"hypreal_of_real_le_add_Infininitesimal_cancel2";
+val hypreal_of_real_less_Infinitesimal_le_zero = thm "hypreal_of_real_less_Infinitesimal_le_zero";
+val Infinitesimal_add_not_zero = thm "Infinitesimal_add_not_zero";
+val Infinitesimal_square_cancel = thm "Infinitesimal_square_cancel";
+val HFinite_square_cancel = thm "HFinite_square_cancel";
+val Infinitesimal_square_cancel2 = thm "Infinitesimal_square_cancel2";
+val HFinite_square_cancel2 = thm "HFinite_square_cancel2";
+val Infinitesimal_sum_square_cancel = thm "Infinitesimal_sum_square_cancel";
+val HFinite_sum_square_cancel = thm "HFinite_sum_square_cancel";
+val Infinitesimal_sum_square_cancel2 = thm "Infinitesimal_sum_square_cancel2";
+val HFinite_sum_square_cancel2 = thm "HFinite_sum_square_cancel2";
+val Infinitesimal_sum_square_cancel3 = thm "Infinitesimal_sum_square_cancel3";
+val HFinite_sum_square_cancel3 = thm "HFinite_sum_square_cancel3";
+val monad_hrabs_less = thm "monad_hrabs_less";
+val mem_monad_SReal_HFinite = thm "mem_monad_SReal_HFinite";
+val st_approx_self = thm "st_approx_self";
+val st_SReal = thm "st_SReal";
+val st_HFinite = thm "st_HFinite";
+val st_SReal_eq = thm "st_SReal_eq";
+val st_hypreal_of_real = thm "st_hypreal_of_real";
+val st_eq_approx = thm "st_eq_approx";
+val approx_st_eq = thm "approx_st_eq";
+val st_eq_approx_iff = thm "st_eq_approx_iff";
+val st_Infinitesimal_add_SReal = thm "st_Infinitesimal_add_SReal";
+val st_Infinitesimal_add_SReal2 = thm "st_Infinitesimal_add_SReal2";
+val HFinite_st_Infinitesimal_add = thm "HFinite_st_Infinitesimal_add";
+val st_add = thm "st_add";
+val st_number_of = thm "st_number_of";
+val st_minus = thm "st_minus";
+val st_diff = thm "st_diff";
+val st_mult = thm "st_mult";
+val st_Infinitesimal = thm "st_Infinitesimal";
+val st_not_Infinitesimal = thm "st_not_Infinitesimal";
+val st_inverse = thm "st_inverse";
+val st_divide = thm "st_divide";
+val st_idempotent = thm "st_idempotent";
+val Infinitesimal_add_st_less = thm "Infinitesimal_add_st_less";
+val Infinitesimal_add_st_le_cancel = thm "Infinitesimal_add_st_le_cancel";
+val st_le = thm "st_le";
+val st_zero_le = thm "st_zero_le";
+val st_zero_ge = thm "st_zero_ge";
+val st_hrabs = thm "st_hrabs";
+val FreeUltrafilterNat_HFinite = thm "FreeUltrafilterNat_HFinite";
+val HFinite_FreeUltrafilterNat_iff = thm "HFinite_FreeUltrafilterNat_iff";
+val FreeUltrafilterNat_const_Finite = thm "FreeUltrafilterNat_const_Finite";
+val FreeUltrafilterNat_HInfinite = thm "FreeUltrafilterNat_HInfinite";
+val HInfinite_FreeUltrafilterNat_iff = thm "HInfinite_FreeUltrafilterNat_iff";
+val Infinitesimal_FreeUltrafilterNat = thm "Infinitesimal_FreeUltrafilterNat";
+val FreeUltrafilterNat_Infinitesimal = thm "FreeUltrafilterNat_Infinitesimal";
+val Infinitesimal_FreeUltrafilterNat_iff = thm "Infinitesimal_FreeUltrafilterNat_iff";
+val Infinitesimal_hypreal_of_nat_iff = thm "Infinitesimal_hypreal_of_nat_iff";
+val Suc_Un_eq = thm "Suc_Un_eq";
+val finite_nat_segment = thm "finite_nat_segment";
+val finite_real_of_nat_segment = thm "finite_real_of_nat_segment";
+val finite_real_of_nat_less_real = thm "finite_real_of_nat_less_real";
+val finite_real_of_nat_le_real = thm "finite_real_of_nat_le_real";
+val finite_rabs_real_of_nat_le_real = thm "finite_rabs_real_of_nat_le_real";
+val rabs_real_of_nat_le_real_FreeUltrafilterNat = thm "rabs_real_of_nat_le_real_FreeUltrafilterNat";
+val FreeUltrafilterNat_nat_gt_real = thm "FreeUltrafilterNat_nat_gt_real";
+val hypreal_omega = thm "hypreal_omega";
+val FreeUltrafilterNat_omega = thm "FreeUltrafilterNat_omega";
+val HInfinite_omega = thm "HInfinite_omega";
+val Infinitesimal_epsilon = thm "Infinitesimal_epsilon";
+val HFinite_epsilon = thm "HFinite_epsilon";
+val epsilon_approx_zero = thm "epsilon_approx_zero";
+val real_of_nat_less_inverse_iff = thm "real_of_nat_less_inverse_iff";
+val finite_inverse_real_of_posnat_gt_real = thm "finite_inverse_real_of_posnat_gt_real";
+val real_of_nat_inverse_le_iff = thm "real_of_nat_inverse_le_iff";
+val real_of_nat_inverse_eq_iff = thm "real_of_nat_inverse_eq_iff";
+val finite_inverse_real_of_posnat_ge_real = thm "finite_inverse_real_of_posnat_ge_real";
+val inverse_real_of_posnat_ge_real_FreeUltrafilterNat = thm "inverse_real_of_posnat_ge_real_FreeUltrafilterNat";
+val FreeUltrafilterNat_inverse_real_of_posnat = thm "FreeUltrafilterNat_inverse_real_of_posnat";
+val real_seq_to_hypreal_Infinitesimal = thm "real_seq_to_hypreal_Infinitesimal";
+val real_seq_to_hypreal_approx = thm "real_seq_to_hypreal_approx";
+val real_seq_to_hypreal_approx2 = thm "real_seq_to_hypreal_approx2";
+val real_seq_to_hypreal_Infinitesimal2 = thm "real_seq_to_hypreal_Infinitesimal2";
+val HInfinite_HFinite_add = thm "HInfinite_HFinite_add";
+val HInfinite_ge_HInfinite = thm "HInfinite_ge_HInfinite";
+val Infinitesimal_inverse_HInfinite = thm "Infinitesimal_inverse_HInfinite";
+val HInfinite_HFinite_not_Infinitesimal_mult = thm "HInfinite_HFinite_not_Infinitesimal_mult";
+val HInfinite_HFinite_not_Infinitesimal_mult2 = thm "HInfinite_HFinite_not_Infinitesimal_mult2";
+val HInfinite_gt_SReal = thm "HInfinite_gt_SReal";
+val HInfinite_gt_zero_gt_one = thm "HInfinite_gt_zero_gt_one";
+val not_HInfinite_one = thm "not_HInfinite_one";
+*}
+
end
--- a/src/HOL/Hyperreal/Star.ML Wed Jan 28 17:01:01 2004 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,490 +0,0 @@
-(* Title : STAR.ML
- Author : Jacques D. Fleuriot
- Copyright : 1998 University of Cambridge
- Description : *-transforms
-*)
-
-(*--------------------------------------------------------
- Preamble - Pulling "EX" over "ALL"
- ---------------------------------------------------------*)
-
-(* This proof does not need AC and was suggested by the
- referee for the JCM Paper: let f(x) be least y such
- that Q(x,y)
-*)
-Goal "ALL x. EX y. Q x y ==> EX (f :: nat => nat). ALL x. Q x (f x)";
-by (res_inst_tac [("x","%x. LEAST y. Q x y")] exI 1);
-by (blast_tac (claset() addIs [LeastI]) 1);
-qed "no_choice";
-
-(*------------------------------------------------------------
- Properties of the *-transform applied to sets of reals
- ------------------------------------------------------------*)
-
-Goalw [starset_def] "*s*(UNIV::real set) = (UNIV::hypreal set)";
-by (Auto_tac);
-qed "STAR_real_set";
-Addsimps [STAR_real_set];
-
-Goalw [starset_def] "*s* {} = {}";
-by (Step_tac 1);
-by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
-by (dres_inst_tac [("x","%n. xa n")] bspec 1);
-by (Auto_tac);
-qed "STAR_empty_set";
-Addsimps [STAR_empty_set];
-
-Goalw [starset_def] "*s* (A Un B) = *s* A Un *s* B";
-by (Auto_tac);
-by (REPEAT(blast_tac (claset() addIs [FreeUltrafilterNat_subset]) 2));
-by (dtac FreeUltrafilterNat_Compl_mem 1);
-by (dtac bspec 1 THEN assume_tac 1);
-by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
-by (Auto_tac);
-by (Fuf_tac 1);
-qed "STAR_Un";
-
-Goalw [starset_def] "*s* (A Int B) = *s* A Int *s* B";
-by (Auto_tac);
-by (blast_tac (claset() addIs [FreeUltrafilterNat_Int,
- FreeUltrafilterNat_subset]) 3);
-by (REPEAT(blast_tac (claset() addIs [FreeUltrafilterNat_subset]) 1));
-qed "STAR_Int";
-
-Goalw [starset_def] "*s* -A = -( *s* A)";
-by (Auto_tac);
-by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
-by (res_inst_tac [("z","x")] eq_Abs_hypreal 2);
-by (REPEAT(Step_tac 1) THEN Auto_tac);
-by (Fuf_empty_tac 1);
-by (dtac FreeUltrafilterNat_Compl_mem 1);
-by (Fuf_tac 1);
-qed "STAR_Compl";
-
-goal Set.thy "(A - B) = (A Int (- B))";
-by (Step_tac 1);
-qed "set_diff_iff2";
-
-Goal "x ~: *s* F ==> x : *s* (- F)";
-by (auto_tac (claset(),simpset() addsimps [STAR_Compl]));
-qed "STAR_mem_Compl";
-
-Goal "*s* (A - B) = *s* A - *s* B";
-by (auto_tac (claset(),simpset() addsimps
- [set_diff_iff2,STAR_Int,STAR_Compl]));
-qed "STAR_diff";
-
-Goalw [starset_def] "A <= B ==> *s* A <= *s* B";
-by (REPEAT(blast_tac (claset() addIs [FreeUltrafilterNat_subset]) 1));
-qed "STAR_subset";
-
-Goalw [starset_def,hypreal_of_real_def]
- "a : A ==> hypreal_of_real a : *s* A";
-by (auto_tac (claset() addIs [FreeUltrafilterNat_subset],simpset()));
-qed "STAR_mem";
-
-Goalw [starset_def] "hypreal_of_real ` A <= *s* A";
-by (auto_tac (claset(),simpset() addsimps [hypreal_of_real_def]));
-by (blast_tac (claset() addIs [FreeUltrafilterNat_subset]) 1);
-qed "STAR_hypreal_of_real_image_subset";
-
-Goalw [starset_def] "*s* X Int Reals = hypreal_of_real ` X";
-by (auto_tac (claset(),simpset() addsimps [hypreal_of_real_def,SReal_def]));
-by (fold_tac [hypreal_of_real_def]);
-by (rtac imageI 1 THEN rtac ccontr 1);
-by (dtac bspec 1);
-by (rtac lemma_hyprel_refl 1);
-by (blast_tac (claset() addIs [FreeUltrafilterNat_subset]) 2);
-by (Auto_tac);
-qed "STAR_hypreal_of_real_Int";
-
-Goal "x ~: hypreal_of_real ` A ==> ALL y: A. x ~= hypreal_of_real y";
-by (Auto_tac);
-qed "lemma_not_hyprealA";
-
-Goal "- {n. X n = xa} = {n. X n ~= xa}";
-by (Auto_tac);
-qed "lemma_Compl_eq";
-
-Goalw [starset_def]
- "ALL n. (X n) ~: M \
-\ ==> Abs_hypreal(hyprel``{X}) ~: *s* M";
-by (Auto_tac THEN rtac bexI 1 THEN rtac lemma_hyprel_refl 2);
-by (Auto_tac);
-qed "STAR_real_seq_to_hypreal";
-
-Goalw [starset_def] "*s* {x} = {hypreal_of_real x}";
-by (auto_tac (claset(),simpset() addsimps [hypreal_of_real_def]));
-by (res_inst_tac [("z","xa")] eq_Abs_hypreal 1);
-by (auto_tac (claset() addIs [FreeUltrafilterNat_subset],simpset()));
-qed "STAR_singleton";
-Addsimps [STAR_singleton];
-
-Goal "x ~: F ==> hypreal_of_real x ~: *s* F";
-by (auto_tac (claset(),simpset() addsimps
- [starset_def,hypreal_of_real_def]));
-by (rtac bexI 1 THEN rtac lemma_hyprel_refl 2);
-by (Auto_tac);
-qed "STAR_not_mem";
-
-Goal "[| x : *s* A; A <= B |] ==> x : *s* B";
-by (blast_tac (claset() addDs [STAR_subset]) 1);
-qed "STAR_subset_closed";
-
-(*------------------------------------------------------------------
- Nonstandard extension of a set (defined using a constant
- sequence) as a special case of an internal set
- -----------------------------------------------------------------*)
-
-Goalw [starset_n_def,starset_def]
- "ALL n. (As n = A) ==> *sn* As = *s* A";
-by (Auto_tac);
-qed "starset_n_starset";
-
-
-(*----------------------------------------------------------------*)
-(* Theorems about nonstandard extensions of functions *)
-(*----------------------------------------------------------------*)
-
-(*----------------------------------------------------------------*)
-(* Nonstandard extension of a function (defined using a *)
-(* constant sequence) as a special case of an internal function *)
-(*----------------------------------------------------------------*)
-
-Goalw [starfun_n_def,starfun_def]
- "ALL n. (F n = f) ==> *fn* F = *f* f";
-by (Auto_tac);
-qed "starfun_n_starfun";
-
-
-(*
- Prove that hrabs is a nonstandard extension of rabs without
- use of congruence property (proved after this for general
- nonstandard extensions of real valued functions). This makes
- proof much longer- see comments at end of HREALABS.thy where
- we proved a congruence theorem for hrabs.
-
- NEW!!! No need to prove all the lemmas anymore. Use the ultrafilter
- tactic!
-*)
-
-Goalw [is_starext_def] "is_starext abs abs";
-by (Step_tac 1);
-by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
-by (res_inst_tac [("z","y")] eq_Abs_hypreal 1);
-by Auto_tac;
-by (rtac bexI 1 THEN rtac lemma_hyprel_refl 2);
-by (rtac bexI 1 THEN rtac lemma_hyprel_refl 2);
-by (auto_tac (claset() addSDs [spec],
- simpset() addsimps [hypreal_minus,hrabs_def, hypreal_zero_def,
- hypreal_le_def, hypreal_less_def]));
-by (TRYALL(Ultra_tac));
-by (TRYALL(arith_tac));
-qed "hrabs_is_starext_rabs";
-
-Goal "[| X: Rep_hypreal z; Y: Rep_hypreal z |] \
-\ ==> {n. X n = Y n} : FreeUltrafilterNat";
-by (res_inst_tac [("z","z")] eq_Abs_hypreal 1);
-by (Auto_tac THEN Fuf_tac 1);
-qed "Rep_hypreal_FreeUltrafilterNat";
-
-(*-----------------------------------------------------------------------
- Nonstandard extension of functions- congruence
- -----------------------------------------------------------------------*)
-
-Goalw [congruent_def] "congruent hyprel (%X. hyprel``{%n. f (X n)})";
-by Auto_tac;
-by (ALLGOALS(Fuf_tac));
-qed "starfun_congruent";
-
-Goalw [starfun_def]
- "( *f* f) (Abs_hypreal(hyprel``{%n. X n})) = \
-\ Abs_hypreal(hyprel `` {%n. f (X n)})";
-by (res_inst_tac [("f","Abs_hypreal")] arg_cong 1);
-by (simp_tac (simpset() addsimps
- [hyprel_in_hypreal RS Abs_hypreal_inverse,[equiv_hyprel,
- starfun_congruent] MRS UN_equiv_class]) 1);
-qed "starfun";
-
-(*-------------------------------------------
- multiplication: ( *f ) x ( *g ) = *(f x g)
- ------------------------------------------*)
-Goal "( *f* f) xa * ( *f* g) xa = ( *f* (%x. f x * g x)) xa";
-by (res_inst_tac [("z","xa")] eq_Abs_hypreal 1);
-by (auto_tac (claset(),simpset() addsimps [starfun,hypreal_mult]));
-qed "starfun_mult";
-Addsimps [starfun_mult RS sym];
-
-(*---------------------------------------
- addition: ( *f ) + ( *g ) = *(f + g)
- ---------------------------------------*)
-Goal "( *f* f) xa + ( *f* g) xa = ( *f* (%x. f x + g x)) xa";
-by (res_inst_tac [("z","xa")] eq_Abs_hypreal 1);
-by (auto_tac (claset(),simpset() addsimps [starfun,hypreal_add]));
-qed "starfun_add";
-Addsimps [starfun_add RS sym];
-
-(*--------------------------------------------
- subtraction: ( *f ) + -( *g ) = *(f + -g)
- -------------------------------------------*)
-
-Goal "- ( *f* f) x = ( *f* (%x. - f x)) x";
-by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
-by (auto_tac (claset(),simpset() addsimps [starfun, hypreal_minus]));
-qed "starfun_minus";
-Addsimps [starfun_minus RS sym];
-
-(*FIXME: delete*)
-Goal "( *f* f) xa + -( *f* g) xa = ( *f* (%x. f x + -g x)) xa";
-by (Simp_tac 1);
-qed "starfun_add_minus";
-Addsimps [starfun_add_minus RS sym];
-
-Goalw [hypreal_diff_def,real_diff_def]
- "( *f* f) xa - ( *f* g) xa = ( *f* (%x. f x - g x)) xa";
-by (rtac starfun_add_minus 1);
-qed "starfun_diff";
-Addsimps [starfun_diff RS sym];
-
-(*--------------------------------------
- composition: ( *f ) o ( *g ) = *(f o g)
- ---------------------------------------*)
-
-Goal "(%x. ( *f* f) (( *f* g) x)) = *f* (%x. f (g x))";
-by (rtac ext 1);
-by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
-by (auto_tac (claset(),simpset() addsimps [starfun]));
-qed "starfun_o2";
-
-Goalw [o_def] "( *f* f) o ( *f* g) = ( *f* (f o g))";
-by (simp_tac (simpset() addsimps [starfun_o2]) 1);
-qed "starfun_o";
-
-(*--------------------------------------
- NS extension of constant function
- --------------------------------------*)
-Goal "( *f* (%x. k)) xa = hypreal_of_real k";
-by (res_inst_tac [("z","xa")] eq_Abs_hypreal 1);
-by (auto_tac (claset(),simpset() addsimps [starfun,
- hypreal_of_real_def]));
-qed "starfun_const_fun";
-
-Addsimps [starfun_const_fun];
-
-(*----------------------------------------------------
- the NS extension of the identity function
- ----------------------------------------------------*)
-
-Goal "x @= hypreal_of_real a ==> ( *f* (%x. x)) x @= hypreal_of_real a";
-by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
-by (auto_tac (claset(),simpset() addsimps [starfun]));
-qed "starfun_Idfun_approx";
-
-Goal "( *f* (%x. x)) x = x";
-by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
-by (auto_tac (claset(),simpset() addsimps [starfun]));
-qed "starfun_Id";
-Addsimps [starfun_Id];
-
-(*----------------------------------------------------------------------
- the *-function is a (nonstandard) extension of the function
- ----------------------------------------------------------------------*)
-
-Goalw [is_starext_def] "is_starext ( *f* f) f";
-by (Auto_tac);
-by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
-by (res_inst_tac [("z","y")] eq_Abs_hypreal 1);
-by (auto_tac (claset() addSIs [bexI] ,simpset() addsimps [starfun]));
-qed "is_starext_starfun";
-
-(*----------------------------------------------------------------------
- Any nonstandard extension is in fact the *-function
- ----------------------------------------------------------------------*)
-
-Goalw [is_starext_def] "is_starext F f ==> F = *f* f";
-by (rtac ext 1);
-by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
-by (dres_inst_tac [("x","x")] spec 1);
-by (dres_inst_tac [("x","( *f* f) x")] spec 1);
-by (auto_tac (claset() addSDs [FreeUltrafilterNat_Compl_mem],
- simpset() addsimps [starfun]));
-by (Fuf_empty_tac 1);
-qed "is_starfun_starext";
-
-Goal "(is_starext F f) = (F = *f* f)";
-by (blast_tac (claset() addIs [is_starfun_starext,is_starext_starfun]) 1);
-qed "is_starext_starfun_iff";
-
-(*--------------------------------------------------------
- extented function has same solution as its standard
- version for real arguments. i.e they are the same
- for all real arguments
- -------------------------------------------------------*)
-Goal "( *f* f) (hypreal_of_real a) = hypreal_of_real (f a)";
-by (auto_tac (claset(),simpset() addsimps
- [starfun,hypreal_of_real_def]));
-qed "starfun_eq";
-
-Addsimps [starfun_eq];
-
-Goal "( *f* f) (hypreal_of_real a) @= hypreal_of_real (f a)";
-by (Auto_tac);
-qed "starfun_approx";
-
-(* useful for NS definition of derivatives *)
-Goal "( *f* (%h. f (x + h))) xa = ( *f* f) (hypreal_of_real x + xa)";
-by (res_inst_tac [("z","xa")] eq_Abs_hypreal 1);
-by (auto_tac (claset(),simpset() addsimps [starfun,
- hypreal_of_real_def,hypreal_add]));
-qed "starfun_lambda_cancel";
-
-Goal "( *f* (%h. f(g(x + h)))) xa = ( *f* (f o g)) (hypreal_of_real x + xa)";
-by (res_inst_tac [("z","xa")] eq_Abs_hypreal 1);
-by (auto_tac (claset(),simpset() addsimps [starfun,
- hypreal_of_real_def,hypreal_add]));
-qed "starfun_lambda_cancel2";
-
-Goal "[| ( *f* f) xa @= l; ( *f* g) xa @= m; \
-\ l: HFinite; m: HFinite \
-\ |] ==> ( *f* (%x. f x * g x)) xa @= l * m";
-by (dtac approx_mult_HFinite 1);
-by (REPEAT(assume_tac 1));
-by (auto_tac (claset() addIs [approx_sym RSN (2,approx_HFinite)],
- simpset()));
-qed "starfun_mult_HFinite_approx";
-
-Goal "[| ( *f* f) xa @= l; ( *f* g) xa @= m \
-\ |] ==> ( *f* (%x. f x + g x)) xa @= l + m";
-by (auto_tac (claset() addIs [approx_add], simpset()));
-qed "starfun_add_approx";
-
-(*----------------------------------------------------
- Examples: hrabs is nonstandard extension of rabs
- inverse is nonstandard extension of inverse
- ---------------------------------------------------*)
-
-(* can be proved easily using theorem "starfun" and *)
-(* properties of ultrafilter as for inverse below we *)
-(* use the theorem we proved above instead *)
-
-Goal "*f* abs = abs";
-by (rtac (hrabs_is_starext_rabs RS
- (is_starext_starfun_iff RS iffD1) RS sym) 1);
-qed "starfun_rabs_hrabs";
-
-Goal "( *f* inverse) x = inverse(x)";
-by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
-by (auto_tac (claset(),
- simpset() addsimps [starfun, hypreal_inverse, hypreal_zero_def]));
-qed "starfun_inverse_inverse";
-Addsimps [starfun_inverse_inverse];
-
-Goal "inverse (( *f* f) x) = ( *f* (%x. inverse (f x))) x";
-by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
-by (auto_tac (claset(),
- simpset() addsimps [starfun, hypreal_inverse]));
-qed "starfun_inverse";
-Addsimps [starfun_inverse RS sym];
-
-Goalw [hypreal_divide_def,real_divide_def]
- "( *f* f) xa / ( *f* g) xa = ( *f* (%x. f x / g x)) xa";
-by Auto_tac;
-qed "starfun_divide";
-Addsimps [starfun_divide RS sym];
-
-Goal "inverse (( *f* f) x) = ( *f* (%x. inverse (f x))) x";
-by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
-by (auto_tac (claset() addIs [FreeUltrafilterNat_subset]
- addSDs [FreeUltrafilterNat_Compl_mem],
- simpset() addsimps [starfun, hypreal_inverse, hypreal_zero_def]));
-qed "starfun_inverse2";
-
-(*-------------------------------------------------------------
- General lemma/theorem needed for proofs in elementary
- topology of the reals
- ------------------------------------------------------------*)
-Goalw [starset_def]
- "( *f* f) x : *s* A ==> x : *s* {x. f x : A}";
-by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
-by (auto_tac (claset(),simpset() addsimps [starfun]));
-by (rename_tac "X" 1);
-by (dres_inst_tac [("x","%n. f (X n)")] bspec 1);
-by (Auto_tac THEN Fuf_tac 1);
-qed "starfun_mem_starset";
-
-(*------------------------------------------------------------
- Alternative definition for hrabs with rabs function
- applied entrywise to equivalence class representative.
- This is easily proved using starfun and ns extension thm
- ------------------------------------------------------------*)
-Goal "abs (Abs_hypreal (hyprel `` {X})) = \
-\ Abs_hypreal(hyprel `` {%n. abs (X n)})";
-by (simp_tac (simpset() addsimps [starfun_rabs_hrabs RS sym,starfun]) 1);
-qed "hypreal_hrabs";
-
-(*----------------------------------------------------------------
- nonstandard extension of set through nonstandard extension
- of rabs function i.e hrabs. A more general result should be
- where we replace rabs by some arbitrary function f and hrabs
- by its NS extenson ( *f* f). See second NS set extension below.
- ----------------------------------------------------------------*)
-Goalw [starset_def]
- "*s* {x. abs (x + - y) < r} = \
-\ {x. abs(x + -hypreal_of_real y) < hypreal_of_real r}";
-by (Step_tac 1);
-by (ALLGOALS(res_inst_tac [("z","x")] eq_Abs_hypreal));
-by (auto_tac (claset() addSIs [exI] addSDs [bspec],
- simpset() addsimps [hypreal_minus, hypreal_of_real_def,hypreal_add,
- hypreal_hrabs,hypreal_less_def]));
-by (Fuf_tac 1);
-qed "STAR_rabs_add_minus";
-
-Goalw [starset_def]
- "*s* {x. abs (f x + - y) < r} = \
-\ {x. abs(( *f* f) x + -hypreal_of_real y) < hypreal_of_real r}";
-by (Step_tac 1);
-by (ALLGOALS(res_inst_tac [("z","x")] eq_Abs_hypreal));
-by (auto_tac (claset() addSIs [exI] addSDs [bspec],
- simpset() addsimps [hypreal_minus, hypreal_of_real_def,hypreal_add,
- hypreal_hrabs,hypreal_less_def,starfun]));
-by (Fuf_tac 1);
-qed "STAR_starfun_rabs_add_minus";
-
-(*-------------------------------------------------------------------
- Another characterization of Infinitesimal and one of @= relation.
- In this theory since hypreal_hrabs proved here. (To Check:) Maybe
- move both if possible?
- -------------------------------------------------------------------*)
-Goal "(x:Infinitesimal) = \
-\ (EX X:Rep_hypreal(x). \
-\ ALL m. {n. abs(X n) < inverse(real(Suc m))} \
-\ : FreeUltrafilterNat)";
-by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
-by (auto_tac (claset() addSIs [bexI,lemma_hyprel_refl],
- simpset() addsimps [Infinitesimal_hypreal_of_nat_iff,
- hypreal_of_real_def,hypreal_inverse,
- hypreal_hrabs,hypreal_less, hypreal_of_nat_def]));
-by (asm_full_simp_tac (simpset() addsimps [real_of_nat_Suc_gt_zero,
- real_not_refl2 RS not_sym]) 1) ;
-by (dres_inst_tac [("x","n")] spec 1);
-by (Fuf_tac 1);
-qed "Infinitesimal_FreeUltrafilterNat_iff2";
-
-Goal "(Abs_hypreal(hyprel``{X}) @= Abs_hypreal(hyprel``{Y})) = \
-\ (ALL m. {n. abs (X n + - Y n) < \
-\ inverse(real(Suc m))} : FreeUltrafilterNat)";
-by (stac approx_minus_iff 1);
-by (rtac (mem_infmal_iff RS subst) 1);
-by (auto_tac (claset(),
- simpset() addsimps [hypreal_minus, hypreal_add,
- Infinitesimal_FreeUltrafilterNat_iff2]));
-by (dres_inst_tac [("x","m")] spec 1);
-by (Fuf_tac 1);
-qed "approx_FreeUltrafilterNat_iff";
-
-Goal "inj starfun";
-by (rtac injI 1);
-by (rtac ext 1 THEN rtac ccontr 1);
-by (dres_inst_tac [("x","Abs_hypreal(hyprel ``{%n. xa})")] fun_cong 1);
-by (auto_tac (claset(),simpset() addsimps [starfun]));
-qed "inj_starfun";
--- a/src/HOL/Hyperreal/Star.thy Wed Jan 28 17:01:01 2004 +0100
+++ b/src/HOL/Hyperreal/Star.thy Thu Jan 29 16:51:17 2004 +0100
@@ -1,39 +1,589 @@
(* Title : Star.thy
Author : Jacques D. Fleuriot
Copyright : 1998 University of Cambridge
- Description : defining *-transforms in NSA which extends sets of reals,
+ Description : defining *-transforms in NSA which extends sets of reals,
and real=>real functions
-*)
+*)
-Star = NSA +
+header{*Star-Transforms in Non-Standard Analysis*}
+
+theory Star = NSA:
constdefs
(* nonstandard extension of sets *)
- starset :: real set => hypreal set ("*s* _" [80] 80)
+ starset :: "real set => hypreal set" ("*s* _" [80] 80)
"*s* A == {x. ALL X: Rep_hypreal(x). {n::nat. X n : A}: FreeUltrafilterNat}"
(* internal sets *)
- starset_n :: (nat => real set) => hypreal set ("*sn* _" [80] 80)
- "*sn* As == {x. ALL X: Rep_hypreal(x). {n::nat. X n : (As n)}: FreeUltrafilterNat}"
-
+ starset_n :: "(nat => real set) => hypreal set" ("*sn* _" [80] 80)
+ "*sn* As == {x. ALL X: Rep_hypreal(x). {n::nat. X n : (As n)}: FreeUltrafilterNat}"
+
InternalSets :: "hypreal set set"
"InternalSets == {X. EX As. X = *sn* As}"
(* nonstandard extension of function *)
- is_starext :: [hypreal => hypreal, real => real] => bool
+ is_starext :: "[hypreal => hypreal, real => real] => bool"
"is_starext F f == (ALL x y. EX X: Rep_hypreal(x). EX Y: Rep_hypreal(y).
((y = (F x)) = ({n. Y n = f(X n)} : FreeUltrafilterNat)))"
-
- starfun :: (real => real) => hypreal => hypreal ("*f* _" [80] 80)
- "*f* f == (%x. Abs_hypreal(UN X: Rep_hypreal(x). hyprel``{%n. f(X n)}))"
+
+ starfun :: "(real => real) => hypreal => hypreal" ("*f* _" [80] 80)
+ "*f* f == (%x. Abs_hypreal(UN X: Rep_hypreal(x). hyprel``{%n. f(X n)}))"
(* internal functions *)
- starfun_n :: (nat => (real => real)) => hypreal => hypreal ("*fn* _" [80] 80)
- "*fn* F == (%x. Abs_hypreal(UN X: Rep_hypreal(x). hyprel``{%n. (F n)(X n)}))"
+ starfun_n :: "(nat => (real => real)) => hypreal => hypreal"
+ ("*fn* _" [80] 80)
+ "*fn* F == (%x. Abs_hypreal(UN X: Rep_hypreal(x). hyprel``{%n. (F n)(X n)}))"
- InternalFuns :: (hypreal => hypreal) set
+ InternalFuns :: "(hypreal => hypreal) set"
"InternalFuns == {X. EX F. X = *fn* F}"
-end
+(*--------------------------------------------------------
+ Preamble - Pulling "EX" over "ALL"
+ ---------------------------------------------------------*)
+
+(* This proof does not need AC and was suggested by the
+ referee for the JCM Paper: let f(x) be least y such
+ that Q(x,y)
+*)
+lemma no_choice: "ALL x. EX y. Q x y ==> EX (f :: nat => nat). ALL x. Q x (f x)"
+apply (rule_tac x = "%x. LEAST y. Q x y" in exI)
+apply (blast intro: LeastI)
+done
+
+(*------------------------------------------------------------
+ Properties of the *-transform applied to sets of reals
+ ------------------------------------------------------------*)
+
+lemma STAR_real_set: "*s*(UNIV::real set) = (UNIV::hypreal set)"
+
+apply (unfold starset_def)
+apply auto
+done
+declare STAR_real_set [simp]
+
+lemma STAR_empty_set: "*s* {} = {}"
+apply (unfold starset_def)
+apply safe
+apply (rule_tac z = "x" in eq_Abs_hypreal)
+apply (drule_tac x = "%n. xa n" in bspec)
+apply auto
+done
+declare STAR_empty_set [simp]
+
+lemma STAR_Un: "*s* (A Un B) = *s* A Un *s* B"
+apply (unfold starset_def)
+apply auto
+ prefer 3 apply (blast intro: FreeUltrafilterNat_subset)
+ prefer 2 apply (blast intro: FreeUltrafilterNat_subset)
+apply (drule FreeUltrafilterNat_Compl_mem)
+apply (drule bspec , assumption)
+apply (rule_tac z = "x" in eq_Abs_hypreal)
+apply auto
+apply ultra
+done
+
+lemma STAR_Int: "*s* (A Int B) = *s* A Int *s* B"
+apply (unfold starset_def)
+apply auto
+prefer 3 apply (blast intro: FreeUltrafilterNat_Int FreeUltrafilterNat_subset)
+apply (blast intro: FreeUltrafilterNat_subset)+
+done
+
+lemma STAR_Compl: "*s* -A = -( *s* A)"
+apply (auto simp add: starset_def)
+apply (rule_tac [!] z = "x" in eq_Abs_hypreal)
+apply (auto dest!: bspec);
+apply ultra
+apply (drule FreeUltrafilterNat_Compl_mem)
+apply ultra
+done
+
+lemma STAR_mem_Compl: "x \<notin> *s* F ==> x : *s* (- F)"
+apply (auto simp add: STAR_Compl)
+done
+
+lemma STAR_diff: "*s* (A - B) = *s* A - *s* B"
+apply (auto simp add: Diff_eq STAR_Int STAR_Compl)
+done
+
+lemma STAR_subset: "A <= B ==> *s* A <= *s* B"
+apply (unfold starset_def)
+apply (blast intro: FreeUltrafilterNat_subset)+
+done
+
+lemma STAR_mem: "a : A ==> hypreal_of_real a : *s* A"
+apply (unfold starset_def hypreal_of_real_def)
+apply (auto intro: FreeUltrafilterNat_subset)
+done
+
+lemma STAR_hypreal_of_real_image_subset: "hypreal_of_real ` A <= *s* A"
+apply (unfold starset_def)
+apply (auto simp add: hypreal_of_real_def)
+apply (blast intro: FreeUltrafilterNat_subset)
+done
+
+lemma STAR_hypreal_of_real_Int: "*s* X Int Reals = hypreal_of_real ` X"
+apply (unfold starset_def)
+apply (auto simp add: hypreal_of_real_def SReal_def)
+apply (simp add: hypreal_of_real_def [symmetric])
+apply (rule imageI , rule ccontr)
+apply (drule bspec)
+apply (rule lemma_hyprel_refl)
+prefer 2 apply (blast intro: FreeUltrafilterNat_subset)
+apply auto
+done
+
+lemma lemma_not_hyprealA: "x \<notin> hypreal_of_real ` A ==> ALL y: A. x \<noteq> hypreal_of_real y"
+apply auto
+done
+
+lemma lemma_Compl_eq: "- {n. X n = xa} = {n. X n \<noteq> xa}"
+apply auto
+done
+
+lemma STAR_real_seq_to_hypreal:
+ "ALL n. (X n) \<notin> M
+ ==> Abs_hypreal(hyprel``{X}) \<notin> *s* M"
+apply (unfold starset_def)
+apply (auto , rule bexI , rule_tac [2] lemma_hyprel_refl)
+apply auto
+done
+
+lemma STAR_singleton: "*s* {x} = {hypreal_of_real x}"
+apply (unfold starset_def)
+apply (auto simp add: hypreal_of_real_def)
+apply (rule_tac z = "xa" in eq_Abs_hypreal)
+apply (auto intro: FreeUltrafilterNat_subset)
+done
+declare STAR_singleton [simp]
+
+lemma STAR_not_mem: "x \<notin> F ==> hypreal_of_real x \<notin> *s* F"
+apply (auto simp add: starset_def hypreal_of_real_def)
+apply (rule bexI , rule_tac [2] lemma_hyprel_refl)
+apply auto
+done
+
+lemma STAR_subset_closed: "[| x : *s* A; A <= B |] ==> x : *s* B"
+apply (blast dest: STAR_subset)
+done
+
+(*------------------------------------------------------------------
+ Nonstandard extension of a set (defined using a constant
+ sequence) as a special case of an internal set
+ -----------------------------------------------------------------*)
+
+lemma starset_n_starset:
+ "ALL n. (As n = A) ==> *sn* As = *s* A"
+
+apply (unfold starset_n_def starset_def)
+apply auto
+done
+
+
+(*----------------------------------------------------------------*)
+(* Theorems about nonstandard extensions of functions *)
+(*----------------------------------------------------------------*)
+
+(*----------------------------------------------------------------*)
+(* Nonstandard extension of a function (defined using a *)
+(* constant sequence) as a special case of an internal function *)
+(*----------------------------------------------------------------*)
+
+lemma starfun_n_starfun:
+ "ALL n. (F n = f) ==> *fn* F = *f* f"
+
+apply (unfold starfun_n_def starfun_def)
+apply auto
+done
+
+
+(*
+ Prove that hrabs is a nonstandard extension of rabs without
+ use of congruence property (proved after this for general
+ nonstandard extensions of real valued functions). This makes
+ proof much longer- see comments at end of HREALABS.thy where
+ we proved a congruence theorem for hrabs.
+
+ NEW!!! No need to prove all the lemmas anymore. Use the ultrafilter
+ tactic!
+*)
+
+lemma hrabs_is_starext_rabs: "is_starext abs abs"
+
+apply (unfold is_starext_def)
+apply safe
+apply (rule_tac z = "x" in eq_Abs_hypreal)
+apply (rule_tac z = "y" in eq_Abs_hypreal)
+apply auto
+apply (rule bexI , rule_tac [2] lemma_hyprel_refl)
+apply (rule bexI , rule_tac [2] lemma_hyprel_refl)
+apply (auto dest!: spec simp add: hypreal_minus hrabs_def hypreal_zero_def hypreal_le_def hypreal_less_def)
+apply (arith | ultra)+
+done
+
+lemma Rep_hypreal_FreeUltrafilterNat: "[| X: Rep_hypreal z; Y: Rep_hypreal z |]
+ ==> {n. X n = Y n} : FreeUltrafilterNat"
+apply (rule_tac z = "z" in eq_Abs_hypreal)
+apply (auto , ultra)
+done
+
+(*-----------------------------------------------------------------------
+ Nonstandard extension of functions- congruence
+ -----------------------------------------------------------------------*)
+
+lemma starfun_congruent: "congruent hyprel (%X. hyprel``{%n. f (X n)})"
+apply (unfold congruent_def)
+apply auto
+apply ultra
+done
+
+lemma starfun:
+ "( *f* f) (Abs_hypreal(hyprel``{%n. X n})) =
+ Abs_hypreal(hyprel `` {%n. f (X n)})"
+apply (unfold starfun_def)
+apply (rule_tac f = "Abs_hypreal" in arg_cong)
+apply (simp add: hyprel_in_hypreal [THEN Abs_hypreal_inverse]
+ UN_equiv_class [OF equiv_hyprel starfun_congruent])
+done
+
+(*-------------------------------------------
+ multiplication: ( *f ) x ( *g ) = *(f x g)
+ ------------------------------------------*)
+lemma starfun_mult: "( *f* f) xa * ( *f* g) xa = ( *f* (%x. f x * g x)) xa"
+apply (rule_tac z = "xa" in eq_Abs_hypreal)
+apply (auto simp add: starfun hypreal_mult)
+done
+declare starfun_mult [symmetric, simp]
+
+(*---------------------------------------
+ addition: ( *f ) + ( *g ) = *(f + g)
+ ---------------------------------------*)
+lemma starfun_add: "( *f* f) xa + ( *f* g) xa = ( *f* (%x. f x + g x)) xa"
+apply (rule_tac z = "xa" in eq_Abs_hypreal)
+apply (auto simp add: starfun hypreal_add)
+done
+declare starfun_add [symmetric, simp]
+
+(*--------------------------------------------
+ subtraction: ( *f ) + -( *g ) = *(f + -g)
+ -------------------------------------------*)
+
+lemma starfun_minus: "- ( *f* f) x = ( *f* (%x. - f x)) x"
+apply (rule_tac z = "x" in eq_Abs_hypreal)
+apply (auto simp add: starfun hypreal_minus)
+done
+declare starfun_minus [symmetric, simp]
+
+(*FIXME: delete*)
+lemma starfun_add_minus: "( *f* f) xa + -( *f* g) xa = ( *f* (%x. f x + -g x)) xa"
+apply (simp (no_asm))
+done
+declare starfun_add_minus [symmetric, simp]
+
+lemma starfun_diff:
+ "( *f* f) xa - ( *f* g) xa = ( *f* (%x. f x - g x)) xa"
+apply (unfold hypreal_diff_def real_diff_def)
+apply (rule starfun_add_minus)
+done
+declare starfun_diff [symmetric, simp]
+
+(*--------------------------------------
+ composition: ( *f ) o ( *g ) = *(f o g)
+ ---------------------------------------*)
+
+lemma starfun_o2: "(%x. ( *f* f) (( *f* g) x)) = *f* (%x. f (g x))"
+apply (rule ext)
+apply (rule_tac z = "x" in eq_Abs_hypreal)
+apply (auto simp add: starfun)
+done
+
+lemma starfun_o: "( *f* f) o ( *f* g) = ( *f* (f o g))"
+apply (unfold o_def)
+apply (simp (no_asm) add: starfun_o2)
+done
+
+(*--------------------------------------
+ NS extension of constant function
+ --------------------------------------*)
+lemma starfun_const_fun: "( *f* (%x. k)) xa = hypreal_of_real k"
+apply (rule_tac z = "xa" in eq_Abs_hypreal)
+apply (auto simp add: starfun hypreal_of_real_def)
+done
+
+declare starfun_const_fun [simp]
+
+(*----------------------------------------------------
+ the NS extension of the identity function
+ ----------------------------------------------------*)
+
+lemma starfun_Idfun_approx: "x @= hypreal_of_real a ==> ( *f* (%x. x)) x @= hypreal_of_real a"
+apply (rule_tac z = "x" in eq_Abs_hypreal)
+apply (auto simp add: starfun)
+done
+
+lemma starfun_Id: "( *f* (%x. x)) x = x"
+apply (rule_tac z = "x" in eq_Abs_hypreal)
+apply (auto simp add: starfun)
+done
+declare starfun_Id [simp]
+
+(*----------------------------------------------------------------------
+ the *-function is a (nonstandard) extension of the function
+ ----------------------------------------------------------------------*)
+
+lemma is_starext_starfun: "is_starext ( *f* f) f"
+
+apply (unfold is_starext_def)
+apply auto
+apply (rule_tac z = "x" in eq_Abs_hypreal)
+apply (rule_tac z = "y" in eq_Abs_hypreal)
+apply (auto intro!: bexI simp add: starfun)
+done
+
+(*----------------------------------------------------------------------
+ Any nonstandard extension is in fact the *-function
+ ----------------------------------------------------------------------*)
+
+lemma is_starfun_starext: "is_starext F f ==> F = *f* f"
+
+apply (unfold is_starext_def)
+apply (rule ext)
+apply (rule_tac z = "x" in eq_Abs_hypreal)
+apply (drule_tac x = "x" in spec)
+apply (drule_tac x = "( *f* f) x" in spec)
+apply (auto dest!: FreeUltrafilterNat_Compl_mem simp add: starfun)
+apply ultra
+done
+
+lemma is_starext_starfun_iff: "(is_starext F f) = (F = *f* f)"
+apply (blast intro: is_starfun_starext is_starext_starfun)
+done
+
+(*--------------------------------------------------------
+ extented function has same solution as its standard
+ version for real arguments. i.e they are the same
+ for all real arguments
+ -------------------------------------------------------*)
+lemma starfun_eq: "( *f* f) (hypreal_of_real a) = hypreal_of_real (f a)"
+apply (auto simp add: starfun hypreal_of_real_def)
+done
+
+declare starfun_eq [simp]
+
+lemma starfun_approx: "( *f* f) (hypreal_of_real a) @= hypreal_of_real (f a)"
+apply auto
+done
+
+(* useful for NS definition of derivatives *)
+lemma starfun_lambda_cancel: "( *f* (%h. f (x + h))) xa = ( *f* f) (hypreal_of_real x + xa)"
+apply (rule_tac z = "xa" in eq_Abs_hypreal)
+apply (auto simp add: starfun hypreal_of_real_def hypreal_add)
+done
+
+lemma starfun_lambda_cancel2: "( *f* (%h. f(g(x + h)))) xa = ( *f* (f o g)) (hypreal_of_real x + xa)"
+apply (rule_tac z = "xa" in eq_Abs_hypreal)
+apply (auto simp add: starfun hypreal_of_real_def hypreal_add)
+done
+
+lemma starfun_mult_HFinite_approx: "[| ( *f* f) xa @= l; ( *f* g) xa @= m;
+ l: HFinite; m: HFinite
+ |] ==> ( *f* (%x. f x * g x)) xa @= l * m"
+apply (drule approx_mult_HFinite)
+apply (assumption)+
+apply (auto intro: approx_HFinite [OF _ approx_sym])
+done
+
+lemma starfun_add_approx: "[| ( *f* f) xa @= l; ( *f* g) xa @= m
+ |] ==> ( *f* (%x. f x + g x)) xa @= l + m"
+apply (auto intro: approx_add)
+done
+
+(*----------------------------------------------------
+ Examples: hrabs is nonstandard extension of rabs
+ inverse is nonstandard extension of inverse
+ ---------------------------------------------------*)
+
+(* can be proved easily using theorem "starfun" and *)
+(* properties of ultrafilter as for inverse below we *)
+(* use the theorem we proved above instead *)
+
+lemma starfun_rabs_hrabs: "*f* abs = abs"
+apply (rule hrabs_is_starext_rabs [THEN is_starext_starfun_iff [THEN iffD1], symmetric])
+done
+
+lemma starfun_inverse_inverse: "( *f* inverse) x = inverse(x)"
+apply (rule_tac z = "x" in eq_Abs_hypreal)
+apply (auto simp add: starfun hypreal_inverse hypreal_zero_def)
+done
+declare starfun_inverse_inverse [simp]
+
+lemma starfun_inverse: "inverse (( *f* f) x) = ( *f* (%x. inverse (f x))) x"
+apply (rule_tac z = "x" in eq_Abs_hypreal)
+apply (auto simp add: starfun hypreal_inverse)
+done
+declare starfun_inverse [symmetric, simp]
+
+lemma starfun_divide:
+ "( *f* f) xa / ( *f* g) xa = ( *f* (%x. f x / g x)) xa"
+apply (unfold hypreal_divide_def real_divide_def)
+apply auto
+done
+declare starfun_divide [symmetric, simp]
+
+lemma starfun_inverse2: "inverse (( *f* f) x) = ( *f* (%x. inverse (f x))) x"
+apply (rule_tac z = "x" in eq_Abs_hypreal)
+apply (auto intro: FreeUltrafilterNat_subset dest!: FreeUltrafilterNat_Compl_mem simp add: starfun hypreal_inverse hypreal_zero_def)
+done
+
+(*-------------------------------------------------------------
+ General lemma/theorem needed for proofs in elementary
+ topology of the reals
+ ------------------------------------------------------------*)
+lemma starfun_mem_starset:
+ "( *f* f) x : *s* A ==> x : *s* {x. f x : A}"
+apply (unfold starset_def)
+apply (rule_tac z = "x" in eq_Abs_hypreal)
+apply (auto simp add: starfun)
+apply (rename_tac "X")
+apply (drule_tac x = "%n. f (X n) " in bspec)
+apply (auto , ultra)
+done
+
+(*------------------------------------------------------------
+ Alternative definition for hrabs with rabs function
+ applied entrywise to equivalence class representative.
+ This is easily proved using starfun and ns extension thm
+ ------------------------------------------------------------*)
+lemma hypreal_hrabs: "abs (Abs_hypreal (hyprel `` {X})) =
+ Abs_hypreal(hyprel `` {%n. abs (X n)})"
+apply (simp (no_asm) add: starfun_rabs_hrabs [symmetric] starfun)
+done
+
+(*----------------------------------------------------------------
+ nonstandard extension of set through nonstandard extension
+ of rabs function i.e hrabs. A more general result should be
+ where we replace rabs by some arbitrary function f and hrabs
+ by its NS extenson ( *f* f). See second NS set extension below.
+ ----------------------------------------------------------------*)
+lemma STAR_rabs_add_minus:
+ "*s* {x. abs (x + - y) < r} =
+ {x. abs(x + -hypreal_of_real y) < hypreal_of_real r}"
+apply (unfold starset_def)
+apply safe
+apply (rule_tac [!] z = "x" in eq_Abs_hypreal)
+apply (auto intro!: exI dest!: bspec simp add: hypreal_minus hypreal_of_real_def hypreal_add hypreal_hrabs hypreal_less)
+apply ultra
+done
+
+lemma STAR_starfun_rabs_add_minus:
+ "*s* {x. abs (f x + - y) < r} =
+ {x. abs(( *f* f) x + -hypreal_of_real y) < hypreal_of_real r}"
+apply (unfold starset_def)
+apply safe
+apply (rule_tac [!] z = "x" in eq_Abs_hypreal)
+apply (auto intro!: exI dest!: bspec simp add: hypreal_minus hypreal_of_real_def hypreal_add hypreal_hrabs hypreal_less starfun)
+apply ultra
+done
+
+(*-------------------------------------------------------------------
+ Another characterization of Infinitesimal and one of @= relation.
+ In this theory since hypreal_hrabs proved here. (To Check:) Maybe
+ move both if possible?
+ -------------------------------------------------------------------*)
+lemma Infinitesimal_FreeUltrafilterNat_iff2: "(x:Infinitesimal) =
+ (EX X:Rep_hypreal(x).
+ ALL m. {n. abs(X n) < inverse(real(Suc m))}
+ : FreeUltrafilterNat)"
+apply (rule_tac z = "x" in eq_Abs_hypreal)
+apply (auto intro!: bexI lemma_hyprel_refl
+ simp add: Infinitesimal_hypreal_of_nat_iff hypreal_of_real_def
+ hypreal_inverse hypreal_hrabs hypreal_less hypreal_of_nat_def)
+apply (drule_tac x = "n" in spec)
+apply ultra
+done
+
+lemma approx_FreeUltrafilterNat_iff: "(Abs_hypreal(hyprel``{X}) @= Abs_hypreal(hyprel``{Y})) =
+ (ALL m. {n. abs (X n + - Y n) <
+ inverse(real(Suc m))} : FreeUltrafilterNat)"
+apply (subst approx_minus_iff)
+apply (rule mem_infmal_iff [THEN subst])
+apply (auto simp add: hypreal_minus hypreal_add Infinitesimal_FreeUltrafilterNat_iff2)
+apply (drule_tac x = "m" in spec)
+apply ultra
+done
+
+lemma inj_starfun: "inj starfun"
+apply (rule inj_onI)
+apply (rule ext , rule ccontr)
+apply (drule_tac x = "Abs_hypreal (hyprel ``{%n. xa}) " in fun_cong)
+apply (auto simp add: starfun)
+done
+
+ML
+{*
+val starset_def = thm"starset_def";
+val starset_n_def = thm"starset_n_def";
+val InternalSets_def = thm"InternalSets_def";
+val is_starext_def = thm"is_starext_def";
+val starfun_def = thm"starfun_def";
+val starfun_n_def = thm"starfun_n_def";
+val InternalFuns_def = thm"InternalFuns_def";
+
+val no_choice = thm "no_choice";
+val STAR_real_set = thm "STAR_real_set";
+val STAR_empty_set = thm "STAR_empty_set";
+val STAR_Un = thm "STAR_Un";
+val STAR_Int = thm "STAR_Int";
+val STAR_Compl = thm "STAR_Compl";
+val STAR_mem_Compl = thm "STAR_mem_Compl";
+val STAR_diff = thm "STAR_diff";
+val STAR_subset = thm "STAR_subset";
+val STAR_mem = thm "STAR_mem";
+val STAR_hypreal_of_real_image_subset = thm "STAR_hypreal_of_real_image_subset";
+val STAR_hypreal_of_real_Int = thm "STAR_hypreal_of_real_Int";
+val STAR_real_seq_to_hypreal = thm "STAR_real_seq_to_hypreal";
+val STAR_singleton = thm "STAR_singleton";
+val STAR_not_mem = thm "STAR_not_mem";
+val STAR_subset_closed = thm "STAR_subset_closed";
+val starset_n_starset = thm "starset_n_starset";
+val starfun_n_starfun = thm "starfun_n_starfun";
+val hrabs_is_starext_rabs = thm "hrabs_is_starext_rabs";
+val Rep_hypreal_FreeUltrafilterNat = thm "Rep_hypreal_FreeUltrafilterNat";
+val starfun_congruent = thm "starfun_congruent";
+val starfun = thm "starfun";
+val starfun_mult = thm "starfun_mult";
+val starfun_add = thm "starfun_add";
+val starfun_minus = thm "starfun_minus";
+val starfun_add_minus = thm "starfun_add_minus";
+val starfun_diff = thm "starfun_diff";
+val starfun_o2 = thm "starfun_o2";
+val starfun_o = thm "starfun_o";
+val starfun_const_fun = thm "starfun_const_fun";
+val starfun_Idfun_approx = thm "starfun_Idfun_approx";
+val starfun_Id = thm "starfun_Id";
+val is_starext_starfun = thm "is_starext_starfun";
+val is_starfun_starext = thm "is_starfun_starext";
+val is_starext_starfun_iff = thm "is_starext_starfun_iff";
+val starfun_eq = thm "starfun_eq";
+val starfun_approx = thm "starfun_approx";
+val starfun_lambda_cancel = thm "starfun_lambda_cancel";
+val starfun_lambda_cancel2 = thm "starfun_lambda_cancel2";
+val starfun_mult_HFinite_approx = thm "starfun_mult_HFinite_approx";
+val starfun_add_approx = thm "starfun_add_approx";
+val starfun_rabs_hrabs = thm "starfun_rabs_hrabs";
+val starfun_inverse_inverse = thm "starfun_inverse_inverse";
+val starfun_inverse = thm "starfun_inverse";
+val starfun_divide = thm "starfun_divide";
+val starfun_inverse2 = thm "starfun_inverse2";
+val starfun_mem_starset = thm "starfun_mem_starset";
+val hypreal_hrabs = thm "hypreal_hrabs";
+val STAR_rabs_add_minus = thm "STAR_rabs_add_minus";
+val STAR_starfun_rabs_add_minus = thm "STAR_starfun_rabs_add_minus";
+val Infinitesimal_FreeUltrafilterNat_iff2 = thm "Infinitesimal_FreeUltrafilterNat_iff2";
+val approx_FreeUltrafilterNat_iff = thm "approx_FreeUltrafilterNat_iff";
+val inj_starfun = thm "inj_starfun";
+*}
+
+end
--- a/src/HOL/Hyperreal/Transcendental.ML Wed Jan 28 17:01:01 2004 +0100
+++ b/src/HOL/Hyperreal/Transcendental.ML Thu Jan 29 16:51:17 2004 +0100
@@ -1216,7 +1216,6 @@
by Auto_tac;
qed "exp_ln_eq";
-Addsimps [hypreal_less_not_refl];
(* ------------------------------------------------------------------------ *)
(* Basic properties of the trig functions *)
--- a/src/HOL/Hyperreal/hypreal_arith.ML Wed Jan 28 17:01:01 2004 +0100
+++ b/src/HOL/Hyperreal/hypreal_arith.ML Thu Jan 29 16:51:17 2004 +0100
@@ -9,6 +9,8 @@
*)
(*FIXME DELETE*)
+val hypreal_mult_less_mono2 = thm"hypreal_mult_less_mono2";
+
val hypreal_mult_left_mono =
read_instantiate_sg(sign_of (the_context())) [("a","?a::hypreal")] mult_left_mono;
--- a/src/HOL/IsaMakefile Wed Jan 28 17:01:01 2004 +0100
+++ b/src/HOL/IsaMakefile Thu Jan 29 16:51:17 2004 +0100
@@ -154,10 +154,10 @@
Hyperreal/Lim.ML Hyperreal/Lim.thy Hyperreal/Log.ML Hyperreal/Log.thy\
Hyperreal/MacLaurin.ML Hyperreal/MacLaurin.thy\
Hyperreal/NatStar.ML Hyperreal/NatStar.thy\
- Hyperreal/NSA.ML Hyperreal/NSA.thy Hyperreal/NthRoot.thy\
+ Hyperreal/NSA.thy Hyperreal/NthRoot.thy\
Hyperreal/Poly.ML Hyperreal/Poly.thy\
Hyperreal/SEQ.ML Hyperreal/SEQ.thy Hyperreal/Series.ML Hyperreal/Series.thy\
- Hyperreal/Star.ML Hyperreal/Star.thy Hyperreal/Transcendental.ML\
+ Hyperreal/Star.thy Hyperreal/Transcendental.ML\
Hyperreal/Transcendental.thy Hyperreal/fuf.ML Hyperreal/hypreal_arith.ML \
Complex/Complex_Main.thy\
Complex/CLim.ML Complex/CLim.thy\
--- a/src/HOL/Ring_and_Field.thy Wed Jan 28 17:01:01 2004 +0100
+++ b/src/HOL/Ring_and_Field.thy Thu Jan 29 16:51:17 2004 +0100
@@ -249,7 +249,8 @@
apply (erule add_strict_left_mono)
done
-lemma add_less_le_mono: "[| a<b; c\<le>d |] ==> a + c < b + (d::'a::ordered_semiring)"
+lemma add_less_le_mono:
+ "[| a<b; c\<le>d |] ==> a + c < b + (d::'a::ordered_semiring)"
apply (erule add_strict_right_mono [THEN order_less_le_trans])
apply (erule add_left_mono)
done