(* Title : HOL/RealDef.thy
Author : Jacques D. Fleuriot
Copyright : 1998 University of Cambridge
Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4
Additional contributions by Jeremy Avigad
*)
header{*Defining the Reals from the Positive Reals*}
theory RealDef
imports PReal
begin
definition
realrel :: "((preal * preal) * (preal * preal)) set" where
[code del]: "realrel = {p. \<exists>x1 y1 x2 y2. p = ((x1,y1),(x2,y2)) & x1+y2 = x2+y1}"
typedef (Real) real = "UNIV//realrel"
by (auto simp add: quotient_def)
definition
(** these don't use the overloaded "real" function: users don't see them **)
real_of_preal :: "preal => real" where
[code del]: "real_of_preal m = Abs_Real (realrel `` {(m + 1, 1)})"
instantiation real :: "{zero, one, plus, minus, uminus, times, inverse, ord, abs, sgn}"
begin
definition
real_zero_def [code del]: "0 = Abs_Real(realrel``{(1, 1)})"
definition
real_one_def [code del]: "1 = Abs_Real(realrel``{(1 + 1, 1)})"
definition
real_add_def [code del]: "z + w =
contents (\<Union>(x,y) \<in> Rep_Real(z). \<Union>(u,v) \<in> Rep_Real(w).
{ Abs_Real(realrel``{(x+u, y+v)}) })"
definition
real_minus_def [code del]: "- r = contents (\<Union>(x,y) \<in> Rep_Real(r). { Abs_Real(realrel``{(y,x)}) })"
definition
real_diff_def [code del]: "r - (s::real) = r + - s"
definition
real_mult_def [code del]:
"z * w =
contents (\<Union>(x,y) \<in> Rep_Real(z). \<Union>(u,v) \<in> Rep_Real(w).
{ Abs_Real(realrel``{(x*u + y*v, x*v + y*u)}) })"
definition
real_inverse_def [code del]: "inverse (R::real) = (THE S. (R = 0 & S = 0) | S * R = 1)"
definition
real_divide_def [code del]: "R / (S::real) = R * inverse S"
definition
real_le_def [code del]: "z \<le> (w::real) \<longleftrightarrow>
(\<exists>x y u v. x+v \<le> u+y & (x,y) \<in> Rep_Real z & (u,v) \<in> Rep_Real w)"
definition
real_less_def [code del]: "x < (y\<Colon>real) \<longleftrightarrow> x \<le> y \<and> x \<noteq> y"
definition
real_abs_def: "abs (r::real) = (if r < 0 then - r else r)"
definition
real_sgn_def: "sgn (x::real) = (if x=0 then 0 else if 0<x then 1 else - 1)"
instance ..
end
subsection {* Equivalence relation over positive reals *}
lemma preal_trans_lemma:
assumes "x + y1 = x1 + y"
and "x + y2 = x2 + y"
shows "x1 + y2 = x2 + (y1::preal)"
proof -
have "(x1 + y2) + x = (x + y2) + x1" by (simp add: add_ac)
also have "... = (x2 + y) + x1" by (simp add: prems)
also have "... = x2 + (x1 + y)" by (simp add: add_ac)
also have "... = x2 + (x + y1)" by (simp add: prems)
also have "... = (x2 + y1) + x" by (simp add: add_ac)
finally have "(x1 + y2) + x = (x2 + y1) + x" .
thus ?thesis by (rule add_right_imp_eq)
qed
lemma realrel_iff [simp]: "(((x1,y1),(x2,y2)) \<in> realrel) = (x1 + y2 = x2 + y1)"
by (simp add: realrel_def)
lemma equiv_realrel: "equiv UNIV realrel"
apply (auto simp add: equiv_def refl_on_def sym_def trans_def realrel_def)
apply (blast dest: preal_trans_lemma)
done
text{*Reduces equality of equivalence classes to the @{term realrel} relation:
@{term "(realrel `` {x} = realrel `` {y}) = ((x,y) \<in> realrel)"} *}
lemmas equiv_realrel_iff =
eq_equiv_class_iff [OF equiv_realrel UNIV_I UNIV_I]
declare equiv_realrel_iff [simp]
lemma realrel_in_real [simp]: "realrel``{(x,y)}: Real"
by (simp add: Real_def realrel_def quotient_def, blast)
declare Abs_Real_inject [simp]
declare Abs_Real_inverse [simp]
text{*Case analysis on the representation of a real number as an equivalence
class of pairs of positive reals.*}
lemma eq_Abs_Real [case_names Abs_Real, cases type: real]:
"(!!x y. z = Abs_Real(realrel``{(x,y)}) ==> P) ==> P"
apply (rule Rep_Real [of z, unfolded Real_def, THEN quotientE])
apply (drule arg_cong [where f=Abs_Real])
apply (auto simp add: Rep_Real_inverse)
done
subsection {* Addition and Subtraction *}
lemma real_add_congruent2_lemma:
"[|a + ba = aa + b; ab + bc = ac + bb|]
==> a + ab + (ba + bc) = aa + ac + (b + (bb::preal))"
apply (simp add: add_assoc)
apply (rule add_left_commute [of ab, THEN ssubst])
apply (simp add: add_assoc [symmetric])
apply (simp add: add_ac)
done
lemma real_add:
"Abs_Real (realrel``{(x,y)}) + Abs_Real (realrel``{(u,v)}) =
Abs_Real (realrel``{(x+u, y+v)})"
proof -
have "(\<lambda>z w. (\<lambda>(x,y). (\<lambda>(u,v). {Abs_Real (realrel `` {(x+u, y+v)})}) w) z)
respects2 realrel"
by (simp add: congruent2_def, blast intro: real_add_congruent2_lemma)
thus ?thesis
by (simp add: real_add_def UN_UN_split_split_eq
UN_equiv_class2 [OF equiv_realrel equiv_realrel])
qed
lemma real_minus: "- Abs_Real(realrel``{(x,y)}) = Abs_Real(realrel `` {(y,x)})"
proof -
have "(\<lambda>(x,y). {Abs_Real (realrel``{(y,x)})}) respects realrel"
by (simp add: congruent_def add_commute)
thus ?thesis
by (simp add: real_minus_def UN_equiv_class [OF equiv_realrel])
qed
instance real :: ab_group_add
proof
fix x y z :: real
show "(x + y) + z = x + (y + z)"
by (cases x, cases y, cases z, simp add: real_add add_assoc)
show "x + y = y + x"
by (cases x, cases y, simp add: real_add add_commute)
show "0 + x = x"
by (cases x, simp add: real_add real_zero_def add_ac)
show "- x + x = 0"
by (cases x, simp add: real_minus real_add real_zero_def add_commute)
show "x - y = x + - y"
by (simp add: real_diff_def)
qed
subsection {* Multiplication *}
lemma real_mult_congruent2_lemma:
"!!(x1::preal). [| x1 + y2 = x2 + y1 |] ==>
x * x1 + y * y1 + (x * y2 + y * x2) =
x * x2 + y * y2 + (x * y1 + y * x1)"
apply (simp add: add_left_commute add_assoc [symmetric])
apply (simp add: add_assoc right_distrib [symmetric])
apply (simp add: add_commute)
done
lemma real_mult_congruent2:
"(%p1 p2.
(%(x1,y1). (%(x2,y2).
{ Abs_Real (realrel``{(x1*x2 + y1*y2, x1*y2+y1*x2)}) }) p2) p1)
respects2 realrel"
apply (rule congruent2_commuteI [OF equiv_realrel], clarify)
apply (simp add: mult_commute add_commute)
apply (auto simp add: real_mult_congruent2_lemma)
done
lemma real_mult:
"Abs_Real((realrel``{(x1,y1)})) * Abs_Real((realrel``{(x2,y2)})) =
Abs_Real(realrel `` {(x1*x2+y1*y2,x1*y2+y1*x2)})"
by (simp add: real_mult_def UN_UN_split_split_eq
UN_equiv_class2 [OF equiv_realrel equiv_realrel real_mult_congruent2])
lemma real_mult_commute: "(z::real) * w = w * z"
by (cases z, cases w, simp add: real_mult add_ac mult_ac)
lemma real_mult_assoc: "((z1::real) * z2) * z3 = z1 * (z2 * z3)"
apply (cases z1, cases z2, cases z3)
apply (simp add: real_mult algebra_simps)
done
lemma real_mult_1: "(1::real) * z = z"
apply (cases z)
apply (simp add: real_mult real_one_def algebra_simps)
done
lemma real_add_mult_distrib: "((z1::real) + z2) * w = (z1 * w) + (z2 * w)"
apply (cases z1, cases z2, cases w)
apply (simp add: real_add real_mult algebra_simps)
done
text{*one and zero are distinct*}
lemma real_zero_not_eq_one: "0 \<noteq> (1::real)"
proof -
have "(1::preal) < 1 + 1"
by (simp add: preal_self_less_add_left)
thus ?thesis
by (simp add: real_zero_def real_one_def)
qed
instance real :: comm_ring_1
proof
fix x y z :: real
show "(x * y) * z = x * (y * z)" by (rule real_mult_assoc)
show "x * y = y * x" by (rule real_mult_commute)
show "1 * x = x" by (rule real_mult_1)
show "(x + y) * z = x * z + y * z" by (rule real_add_mult_distrib)
show "0 \<noteq> (1::real)" by (rule real_zero_not_eq_one)
qed
subsection {* Inverse and Division *}
lemma real_zero_iff: "Abs_Real (realrel `` {(x, x)}) = 0"
by (simp add: real_zero_def add_commute)
text{*Instead of using an existential quantifier and constructing the inverse
within the proof, we could define the inverse explicitly.*}
lemma real_mult_inverse_left_ex: "x \<noteq> 0 ==> \<exists>y. y*x = (1::real)"
apply (simp add: real_zero_def real_one_def, cases x)
apply (cut_tac x = xa and y = y in linorder_less_linear)
apply (auto dest!: less_add_left_Ex simp add: real_zero_iff)
apply (rule_tac
x = "Abs_Real (realrel``{(1, inverse (D) + 1)})"
in exI)
apply (rule_tac [2]
x = "Abs_Real (realrel``{(inverse (D) + 1, 1)})"
in exI)
apply (auto simp add: real_mult preal_mult_inverse_right algebra_simps)
done
lemma real_mult_inverse_left: "x \<noteq> 0 ==> inverse(x)*x = (1::real)"
apply (simp add: real_inverse_def)
apply (drule real_mult_inverse_left_ex, safe)
apply (rule theI, assumption, rename_tac z)
apply (subgoal_tac "(z * x) * y = z * (x * y)")
apply (simp add: mult_commute)
apply (rule mult_assoc)
done
subsection{*The Real Numbers form a Field*}
instance real :: field
proof
fix x y z :: real
show "x \<noteq> 0 ==> inverse x * x = 1" by (rule real_mult_inverse_left)
show "x / y = x * inverse y" by (simp add: real_divide_def)
qed
text{*Inverse of zero! Useful to simplify certain equations*}
lemma INVERSE_ZERO: "inverse 0 = (0::real)"
by (simp add: real_inverse_def)
instance real :: division_by_zero
proof
show "inverse 0 = (0::real)" by (rule INVERSE_ZERO)
qed
subsection{*The @{text "\<le>"} Ordering*}
lemma real_le_refl: "w \<le> (w::real)"
by (cases w, force simp add: real_le_def)
text{*The arithmetic decision procedure is not set up for type preal.
This lemma is currently unused, but it could simplify the proofs of the
following two lemmas.*}
lemma preal_eq_le_imp_le:
assumes eq: "a+b = c+d" and le: "c \<le> a"
shows "b \<le> (d::preal)"
proof -
have "c+d \<le> a+d" by (simp add: prems)
hence "a+b \<le> a+d" by (simp add: prems)
thus "b \<le> d" by simp
qed
lemma real_le_lemma:
assumes l: "u1 + v2 \<le> u2 + v1"
and "x1 + v1 = u1 + y1"
and "x2 + v2 = u2 + y2"
shows "x1 + y2 \<le> x2 + (y1::preal)"
proof -
have "(x1+v1) + (u2+y2) = (u1+y1) + (x2+v2)" by (simp add: prems)
hence "(x1+y2) + (u2+v1) = (x2+y1) + (u1+v2)" by (simp add: add_ac)
also have "... \<le> (x2+y1) + (u2+v1)" by (simp add: prems)
finally show ?thesis by simp
qed
lemma real_le:
"(Abs_Real(realrel``{(x1,y1)}) \<le> Abs_Real(realrel``{(x2,y2)})) =
(x1 + y2 \<le> x2 + y1)"
apply (simp add: real_le_def)
apply (auto intro: real_le_lemma)
done
lemma real_le_antisym: "[| z \<le> w; w \<le> z |] ==> z = (w::real)"
by (cases z, cases w, simp add: real_le)
lemma real_trans_lemma:
assumes "x + v \<le> u + y"
and "u + v' \<le> u' + v"
and "x2 + v2 = u2 + y2"
shows "x + v' \<le> u' + (y::preal)"
proof -
have "(x+v') + (u+v) = (x+v) + (u+v')" by (simp add: add_ac)
also have "... \<le> (u+y) + (u+v')" by (simp add: prems)
also have "... \<le> (u+y) + (u'+v)" by (simp add: prems)
also have "... = (u'+y) + (u+v)" by (simp add: add_ac)
finally show ?thesis by simp
qed
lemma real_le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::real)"
apply (cases i, cases j, cases k)
apply (simp add: real_le)
apply (blast intro: real_trans_lemma)
done
instance real :: order
proof
fix u v :: real
show "u < v \<longleftrightarrow> u \<le> v \<and> \<not> v \<le> u"
by (auto simp add: real_less_def intro: real_le_antisym)
qed (assumption | rule real_le_refl real_le_trans real_le_antisym)+
(* Axiom 'linorder_linear' of class 'linorder': *)
lemma real_le_linear: "(z::real) \<le> w | w \<le> z"
apply (cases z, cases w)
apply (auto simp add: real_le real_zero_def add_ac)
done
instance real :: linorder
by (intro_classes, rule real_le_linear)
lemma real_le_eq_diff: "(x \<le> y) = (x-y \<le> (0::real))"
apply (cases x, cases y)
apply (auto simp add: real_le real_zero_def real_diff_def real_add real_minus
add_ac)
apply (simp_all add: add_assoc [symmetric])
done
lemma real_add_left_mono:
assumes le: "x \<le> y" shows "z + x \<le> z + (y::real)"
proof -
have "z + x - (z + y) = (z + -z) + (x - y)"
by (simp add: algebra_simps)
with le show ?thesis
by (simp add: real_le_eq_diff[of x] real_le_eq_diff[of "z+x"] diff_minus)
qed
lemma real_sum_gt_zero_less: "(0 < S + (-W::real)) ==> (W < S)"
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus)
lemma real_less_sum_gt_zero: "(W < S) ==> (0 < S + (-W::real))"
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus)
lemma real_mult_order: "[| 0 < x; 0 < y |] ==> (0::real) < x * y"
apply (cases x, cases y)
apply (simp add: linorder_not_le [where 'a = real, symmetric]
linorder_not_le [where 'a = preal]
real_zero_def real_le real_mult)
--{*Reduce to the (simpler) @{text "\<le>"} relation *}
apply (auto dest!: less_add_left_Ex
simp add: algebra_simps preal_self_less_add_left)
done
lemma real_mult_less_mono2: "[| (0::real) < z; x < y |] ==> z * x < z * y"
apply (rule real_sum_gt_zero_less)
apply (drule real_less_sum_gt_zero [of x y])
apply (drule real_mult_order, assumption)
apply (simp add: right_distrib)
done
instantiation real :: distrib_lattice
begin
definition
"(inf \<Colon> real \<Rightarrow> real \<Rightarrow> real) = min"
definition
"(sup \<Colon> real \<Rightarrow> real \<Rightarrow> real) = max"
instance
by default (auto simp add: inf_real_def sup_real_def min_max.sup_inf_distrib1)
end
subsection{*The Reals Form an Ordered Field*}
instance real :: linordered_field
proof
fix x y z :: real
show "x \<le> y ==> z + x \<le> z + y" by (rule real_add_left_mono)
show "x < y ==> 0 < z ==> z * x < z * y" by (rule real_mult_less_mono2)
show "\<bar>x\<bar> = (if x < 0 then -x else x)" by (simp only: real_abs_def)
show "sgn x = (if x=0 then 0 else if 0<x then 1 else - 1)"
by (simp only: real_sgn_def)
qed
text{*The function @{term real_of_preal} requires many proofs, but it seems
to be essential for proving completeness of the reals from that of the
positive reals.*}
lemma real_of_preal_add:
"real_of_preal ((x::preal) + y) = real_of_preal x + real_of_preal y"
by (simp add: real_of_preal_def real_add algebra_simps)
lemma real_of_preal_mult:
"real_of_preal ((x::preal) * y) = real_of_preal x* real_of_preal y"
by (simp add: real_of_preal_def real_mult algebra_simps)
text{*Gleason prop 9-4.4 p 127*}
lemma real_of_preal_trichotomy:
"\<exists>m. (x::real) = real_of_preal m | x = 0 | x = -(real_of_preal m)"
apply (simp add: real_of_preal_def real_zero_def, cases x)
apply (auto simp add: real_minus add_ac)
apply (cut_tac x = x and y = y in linorder_less_linear)
apply (auto dest!: less_add_left_Ex simp add: add_assoc [symmetric])
done
lemma real_of_preal_leD:
"real_of_preal m1 \<le> real_of_preal m2 ==> m1 \<le> m2"
by (simp add: real_of_preal_def real_le)
lemma real_of_preal_lessI: "m1 < m2 ==> real_of_preal m1 < real_of_preal m2"
by (auto simp add: real_of_preal_leD linorder_not_le [symmetric])
lemma real_of_preal_lessD:
"real_of_preal m1 < real_of_preal m2 ==> m1 < m2"
by (simp add: real_of_preal_def real_le linorder_not_le [symmetric])
lemma real_of_preal_less_iff [simp]:
"(real_of_preal m1 < real_of_preal m2) = (m1 < m2)"
by (blast intro: real_of_preal_lessI real_of_preal_lessD)
lemma real_of_preal_le_iff:
"(real_of_preal m1 \<le> real_of_preal m2) = (m1 \<le> m2)"
by (simp add: linorder_not_less [symmetric])
lemma real_of_preal_zero_less: "0 < real_of_preal m"
apply (insert preal_self_less_add_left [of 1 m])
apply (auto simp add: real_zero_def real_of_preal_def
real_less_def real_le_def add_ac)
apply (rule_tac x="m + 1" in exI, rule_tac x="1" in exI)
apply (simp add: add_ac)
done
lemma real_of_preal_minus_less_zero: "- real_of_preal m < 0"
by (simp add: real_of_preal_zero_less)
lemma real_of_preal_not_minus_gt_zero: "~ 0 < - real_of_preal m"
proof -
from real_of_preal_minus_less_zero
show ?thesis by (blast dest: order_less_trans)
qed
subsection{*Theorems About the Ordering*}
lemma real_gt_zero_preal_Ex: "(0 < x) = (\<exists>y. x = real_of_preal y)"
apply (auto simp add: real_of_preal_zero_less)
apply (cut_tac x = x in real_of_preal_trichotomy)
apply (blast elim!: real_of_preal_not_minus_gt_zero [THEN notE])
done
lemma real_gt_preal_preal_Ex:
"real_of_preal z < x ==> \<exists>y. x = real_of_preal y"
by (blast dest!: real_of_preal_zero_less [THEN order_less_trans]
intro: real_gt_zero_preal_Ex [THEN iffD1])
lemma real_ge_preal_preal_Ex:
"real_of_preal z \<le> x ==> \<exists>y. x = real_of_preal y"
by (blast dest: order_le_imp_less_or_eq real_gt_preal_preal_Ex)
lemma real_less_all_preal: "y \<le> 0 ==> \<forall>x. y < real_of_preal x"
by (auto elim: order_le_imp_less_or_eq [THEN disjE]
intro: real_of_preal_zero_less [THEN [2] order_less_trans]
simp add: real_of_preal_zero_less)
lemma real_less_all_real2: "~ 0 < y ==> \<forall>x. y < real_of_preal x"
by (blast intro!: real_less_all_preal linorder_not_less [THEN iffD1])
subsection{*More Lemmas*}
lemma real_mult_left_cancel: "(c::real) \<noteq> 0 ==> (c*a=c*b) = (a=b)"
by auto
lemma real_mult_right_cancel: "(c::real) \<noteq> 0 ==> (a*c=b*c) = (a=b)"
by auto
lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)"
by (force elim: order_less_asym
simp add: mult_less_cancel_right)
lemma real_mult_le_cancel_iff1 [simp]: "(0::real) < z ==> (x*z \<le> y*z) = (x\<le>y)"
apply (simp add: mult_le_cancel_right)
apply (blast intro: elim: order_less_asym)
done
lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x \<le> z*y) = (x\<le>y)"
by(simp add:mult_commute)
lemma real_inverse_gt_one: "[| (0::real) < x; x < 1 |] ==> 1 < inverse x"
by (simp add: one_less_inverse_iff) (* TODO: generalize/move *)
subsection {* Embedding numbers into the Reals *}
abbreviation
real_of_nat :: "nat \<Rightarrow> real"
where
"real_of_nat \<equiv> of_nat"
abbreviation
real_of_int :: "int \<Rightarrow> real"
where
"real_of_int \<equiv> of_int"
abbreviation
real_of_rat :: "rat \<Rightarrow> real"
where
"real_of_rat \<equiv> of_rat"
consts
(*overloaded constant for injecting other types into "real"*)
real :: "'a => real"
defs (overloaded)
real_of_nat_def [code_unfold]: "real == real_of_nat"
real_of_int_def [code_unfold]: "real == real_of_int"
lemma real_eq_of_nat: "real = of_nat"
unfolding real_of_nat_def ..
lemma real_eq_of_int: "real = of_int"
unfolding real_of_int_def ..
lemma real_of_int_zero [simp]: "real (0::int) = 0"
by (simp add: real_of_int_def)
lemma real_of_one [simp]: "real (1::int) = (1::real)"
by (simp add: real_of_int_def)
lemma real_of_int_add [simp]: "real(x + y) = real (x::int) + real y"
by (simp add: real_of_int_def)
lemma real_of_int_minus [simp]: "real(-x) = -real (x::int)"
by (simp add: real_of_int_def)
lemma real_of_int_diff [simp]: "real(x - y) = real (x::int) - real y"
by (simp add: real_of_int_def)
lemma real_of_int_mult [simp]: "real(x * y) = real (x::int) * real y"
by (simp add: real_of_int_def)
lemma real_of_int_power [simp]: "real (x ^ n) = real (x::int) ^ n"
by (simp add: real_of_int_def of_int_power)
lemmas power_real_of_int = real_of_int_power [symmetric]
lemma real_of_int_setsum [simp]: "real ((SUM x:A. f x)::int) = (SUM x:A. real(f x))"
apply (subst real_eq_of_int)+
apply (rule of_int_setsum)
done
lemma real_of_int_setprod [simp]: "real ((PROD x:A. f x)::int) =
(PROD x:A. real(f x))"
apply (subst real_eq_of_int)+
apply (rule of_int_setprod)
done
lemma real_of_int_zero_cancel [simp, algebra, presburger]: "(real x = 0) = (x = (0::int))"
by (simp add: real_of_int_def)
lemma real_of_int_inject [iff, algebra, presburger]: "(real (x::int) = real y) = (x = y)"
by (simp add: real_of_int_def)
lemma real_of_int_less_iff [iff, presburger]: "(real (x::int) < real y) = (x < y)"
by (simp add: real_of_int_def)
lemma real_of_int_le_iff [simp, presburger]: "(real (x::int) \<le> real y) = (x \<le> y)"
by (simp add: real_of_int_def)
lemma real_of_int_gt_zero_cancel_iff [simp, presburger]: "(0 < real (n::int)) = (0 < n)"
by (simp add: real_of_int_def)
lemma real_of_int_ge_zero_cancel_iff [simp, presburger]: "(0 <= real (n::int)) = (0 <= n)"
by (simp add: real_of_int_def)
lemma real_of_int_lt_zero_cancel_iff [simp, presburger]: "(real (n::int) < 0) = (n < 0)"
by (simp add: real_of_int_def)
lemma real_of_int_le_zero_cancel_iff [simp, presburger]: "(real (n::int) <= 0) = (n <= 0)"
by (simp add: real_of_int_def)
lemma real_of_int_abs [simp]: "real (abs x) = abs(real (x::int))"
by (auto simp add: abs_if)
lemma int_less_real_le: "((n::int) < m) = (real n + 1 <= real m)"
apply (subgoal_tac "real n + 1 = real (n + 1)")
apply (simp del: real_of_int_add)
apply auto
done
lemma int_le_real_less: "((n::int) <= m) = (real n < real m + 1)"
apply (subgoal_tac "real m + 1 = real (m + 1)")
apply (simp del: real_of_int_add)
apply simp
done
lemma real_of_int_div_aux: "d ~= 0 ==> (real (x::int)) / (real d) =
real (x div d) + (real (x mod d)) / (real d)"
proof -
assume "d ~= 0"
have "x = (x div d) * d + x mod d"
by auto
then have "real x = real (x div d) * real d + real(x mod d)"
by (simp only: real_of_int_mult [THEN sym] real_of_int_add [THEN sym])
then have "real x / real d = ... / real d"
by simp
then show ?thesis
by (auto simp add: add_divide_distrib algebra_simps prems)
qed
lemma real_of_int_div: "(d::int) ~= 0 ==> d dvd n ==>
real(n div d) = real n / real d"
apply (frule real_of_int_div_aux [of d n])
apply simp
apply (simp add: dvd_eq_mod_eq_0)
done
lemma real_of_int_div2:
"0 <= real (n::int) / real (x) - real (n div x)"
apply (case_tac "x = 0")
apply simp
apply (case_tac "0 < x")
apply (simp add: algebra_simps)
apply (subst real_of_int_div_aux)
apply simp
apply simp
apply (subst zero_le_divide_iff)
apply auto
apply (simp add: algebra_simps)
apply (subst real_of_int_div_aux)
apply simp
apply simp
apply (subst zero_le_divide_iff)
apply auto
done
lemma real_of_int_div3:
"real (n::int) / real (x) - real (n div x) <= 1"
apply(case_tac "x = 0")
apply simp
apply (simp add: algebra_simps)
apply (subst real_of_int_div_aux)
apply assumption
apply simp
apply (subst divide_le_eq)
apply clarsimp
apply (rule conjI)
apply (rule impI)
apply (rule order_less_imp_le)
apply simp
apply (rule impI)
apply (rule order_less_imp_le)
apply simp
done
lemma real_of_int_div4: "real (n div x) <= real (n::int) / real x"
by (insert real_of_int_div2 [of n x], simp)
lemma Ints_real_of_int [simp]: "real (x::int) \<in> Ints"
unfolding real_of_int_def by (rule Ints_of_int)
subsection{*Embedding the Naturals into the Reals*}
lemma real_of_nat_zero [simp]: "real (0::nat) = 0"
by (simp add: real_of_nat_def)
lemma real_of_nat_1 [simp]: "real (1::nat) = 1"
by (simp add: real_of_nat_def)
lemma real_of_nat_one [simp]: "real (Suc 0) = (1::real)"
by (simp add: real_of_nat_def)
lemma real_of_nat_add [simp]: "real (m + n) = real (m::nat) + real n"
by (simp add: real_of_nat_def)
(*Not for addsimps: often the LHS is used to represent a positive natural*)
lemma real_of_nat_Suc: "real (Suc n) = real n + (1::real)"
by (simp add: real_of_nat_def)
lemma real_of_nat_less_iff [iff]:
"(real (n::nat) < real m) = (n < m)"
by (simp add: real_of_nat_def)
lemma real_of_nat_le_iff [iff]: "(real (n::nat) \<le> real m) = (n \<le> m)"
by (simp add: real_of_nat_def)
lemma real_of_nat_ge_zero [iff]: "0 \<le> real (n::nat)"
by (simp add: real_of_nat_def zero_le_imp_of_nat)
lemma real_of_nat_Suc_gt_zero: "0 < real (Suc n)"
by (simp add: real_of_nat_def del: of_nat_Suc)
lemma real_of_nat_mult [simp]: "real (m * n) = real (m::nat) * real n"
by (simp add: real_of_nat_def of_nat_mult)
lemma real_of_nat_power [simp]: "real (m ^ n) = real (m::nat) ^ n"
by (simp add: real_of_nat_def of_nat_power)
lemmas power_real_of_nat = real_of_nat_power [symmetric]
lemma real_of_nat_setsum [simp]: "real ((SUM x:A. f x)::nat) =
(SUM x:A. real(f x))"
apply (subst real_eq_of_nat)+
apply (rule of_nat_setsum)
done
lemma real_of_nat_setprod [simp]: "real ((PROD x:A. f x)::nat) =
(PROD x:A. real(f x))"
apply (subst real_eq_of_nat)+
apply (rule of_nat_setprod)
done
lemma real_of_card: "real (card A) = setsum (%x.1) A"
apply (subst card_eq_setsum)
apply (subst real_of_nat_setsum)
apply simp
done
lemma real_of_nat_inject [iff]: "(real (n::nat) = real m) = (n = m)"
by (simp add: real_of_nat_def)
lemma real_of_nat_zero_iff [iff]: "(real (n::nat) = 0) = (n = 0)"
by (simp add: real_of_nat_def)
lemma real_of_nat_diff: "n \<le> m ==> real (m - n) = real (m::nat) - real n"
by (simp add: add: real_of_nat_def of_nat_diff)
lemma real_of_nat_gt_zero_cancel_iff [simp]: "(0 < real (n::nat)) = (0 < n)"
by (auto simp: real_of_nat_def)
lemma real_of_nat_le_zero_cancel_iff [simp]: "(real (n::nat) \<le> 0) = (n = 0)"
by (simp add: add: real_of_nat_def)
lemma not_real_of_nat_less_zero [simp]: "~ real (n::nat) < 0"
by (simp add: add: real_of_nat_def)
(* FIXME: duplicates real_of_nat_ge_zero *)
lemma real_of_nat_ge_zero_cancel_iff: "(0 \<le> real (n::nat))"
by (simp add: add: real_of_nat_def)
lemma nat_less_real_le: "((n::nat) < m) = (real n + 1 <= real m)"
apply (subgoal_tac "real n + 1 = real (Suc n)")
apply simp
apply (auto simp add: real_of_nat_Suc)
done
lemma nat_le_real_less: "((n::nat) <= m) = (real n < real m + 1)"
apply (subgoal_tac "real m + 1 = real (Suc m)")
apply (simp add: less_Suc_eq_le)
apply (simp add: real_of_nat_Suc)
done
lemma real_of_nat_div_aux: "0 < d ==> (real (x::nat)) / (real d) =
real (x div d) + (real (x mod d)) / (real d)"
proof -
assume "0 < d"
have "x = (x div d) * d + x mod d"
by auto
then have "real x = real (x div d) * real d + real(x mod d)"
by (simp only: real_of_nat_mult [THEN sym] real_of_nat_add [THEN sym])
then have "real x / real d = \<dots> / real d"
by simp
then show ?thesis
by (auto simp add: add_divide_distrib algebra_simps prems)
qed
lemma real_of_nat_div: "0 < (d::nat) ==> d dvd n ==>
real(n div d) = real n / real d"
apply (frule real_of_nat_div_aux [of d n])
apply simp
apply (subst dvd_eq_mod_eq_0 [THEN sym])
apply assumption
done
lemma real_of_nat_div2:
"0 <= real (n::nat) / real (x) - real (n div x)"
apply(case_tac "x = 0")
apply (simp)
apply (simp add: algebra_simps)
apply (subst real_of_nat_div_aux)
apply simp
apply simp
apply (subst zero_le_divide_iff)
apply simp
done
lemma real_of_nat_div3:
"real (n::nat) / real (x) - real (n div x) <= 1"
apply(case_tac "x = 0")
apply (simp)
apply (simp add: algebra_simps)
apply (subst real_of_nat_div_aux)
apply simp
apply simp
done
lemma real_of_nat_div4: "real (n div x) <= real (n::nat) / real x"
by (insert real_of_nat_div2 [of n x], simp)
lemma real_of_int_real_of_nat: "real (int n) = real n"
by (simp add: real_of_nat_def real_of_int_def int_eq_of_nat)
lemma real_of_int_of_nat_eq [simp]: "real (of_nat n :: int) = real n"
by (simp add: real_of_int_def real_of_nat_def)
lemma real_nat_eq_real [simp]: "0 <= x ==> real(nat x) = real x"
apply (subgoal_tac "real(int(nat x)) = real(nat x)")
apply force
apply (simp only: real_of_int_real_of_nat)
done
lemma Nats_real_of_nat [simp]: "real (n::nat) \<in> Nats"
unfolding real_of_nat_def by (rule of_nat_in_Nats)
lemma Ints_real_of_nat [simp]: "real (n::nat) \<in> Ints"
unfolding real_of_nat_def by (rule Ints_of_nat)
subsection{* Rationals *}
lemma Rats_real_nat[simp]: "real(n::nat) \<in> \<rat>"
by (simp add: real_eq_of_nat)
lemma Rats_eq_int_div_int:
"\<rat> = { real(i::int)/real(j::int) |i j. j \<noteq> 0}" (is "_ = ?S")
proof
show "\<rat> \<subseteq> ?S"
proof
fix x::real assume "x : \<rat>"
then obtain r where "x = of_rat r" unfolding Rats_def ..
have "of_rat r : ?S"
by (cases r)(auto simp add:of_rat_rat real_eq_of_int)
thus "x : ?S" using `x = of_rat r` by simp
qed
next
show "?S \<subseteq> \<rat>"
proof(auto simp:Rats_def)
fix i j :: int assume "j \<noteq> 0"
hence "real i / real j = of_rat(Fract i j)"
by (simp add:of_rat_rat real_eq_of_int)
thus "real i / real j \<in> range of_rat" by blast
qed
qed
lemma Rats_eq_int_div_nat:
"\<rat> = { real(i::int)/real(n::nat) |i n. n \<noteq> 0}"
proof(auto simp:Rats_eq_int_div_int)
fix i j::int assume "j \<noteq> 0"
show "EX (i'::int) (n::nat). real i/real j = real i'/real n \<and> 0<n"
proof cases
assume "j>0"
hence "real i/real j = real i/real(nat j) \<and> 0<nat j"
by (simp add: real_eq_of_int real_eq_of_nat of_nat_nat)
thus ?thesis by blast
next
assume "~ j>0"
hence "real i/real j = real(-i)/real(nat(-j)) \<and> 0<nat(-j)" using `j\<noteq>0`
by (simp add: real_eq_of_int real_eq_of_nat of_nat_nat)
thus ?thesis by blast
qed
next
fix i::int and n::nat assume "0 < n"
hence "real i/real n = real i/real(int n) \<and> int n \<noteq> 0" by simp
thus "\<exists>(i'::int) j::int. real i/real n = real i'/real j \<and> j \<noteq> 0" by blast
qed
lemma Rats_abs_nat_div_natE:
assumes "x \<in> \<rat>"
obtains m n :: nat
where "n \<noteq> 0" and "\<bar>x\<bar> = real m / real n" and "gcd m n = 1"
proof -
from `x \<in> \<rat>` obtain i::int and n::nat where "n \<noteq> 0" and "x = real i / real n"
by(auto simp add: Rats_eq_int_div_nat)
hence "\<bar>x\<bar> = real(nat(abs i)) / real n" by simp
then obtain m :: nat where x_rat: "\<bar>x\<bar> = real m / real n" by blast
let ?gcd = "gcd m n"
from `n\<noteq>0` have gcd: "?gcd \<noteq> 0" by simp
let ?k = "m div ?gcd"
let ?l = "n div ?gcd"
let ?gcd' = "gcd ?k ?l"
have "?gcd dvd m" .. then have gcd_k: "?gcd * ?k = m"
by (rule dvd_mult_div_cancel)
have "?gcd dvd n" .. then have gcd_l: "?gcd * ?l = n"
by (rule dvd_mult_div_cancel)
from `n\<noteq>0` and gcd_l have "?l \<noteq> 0" by (auto iff del: neq0_conv)
moreover
have "\<bar>x\<bar> = real ?k / real ?l"
proof -
from gcd have "real ?k / real ?l =
real (?gcd * ?k) / real (?gcd * ?l)" by simp
also from gcd_k and gcd_l have "\<dots> = real m / real n" by simp
also from x_rat have "\<dots> = \<bar>x\<bar>" ..
finally show ?thesis ..
qed
moreover
have "?gcd' = 1"
proof -
have "?gcd * ?gcd' = gcd (?gcd * ?k) (?gcd * ?l)"
by (rule gcd_mult_distrib_nat)
with gcd_k gcd_l have "?gcd * ?gcd' = ?gcd" by simp
with gcd show ?thesis by auto
qed
ultimately show ?thesis ..
qed
subsection{*Numerals and Arithmetic*}
instantiation real :: number_ring
begin
definition
real_number_of_def [code del]: "number_of w = real_of_int w"
instance
by intro_classes (simp add: real_number_of_def)
end
lemma [code_unfold_post]:
"number_of k = real_of_int (number_of k)"
unfolding number_of_is_id real_number_of_def ..
text{*Collapse applications of @{term real} to @{term number_of}*}
lemma real_number_of [simp]: "real (number_of v :: int) = number_of v"
by (simp add: real_of_int_def)
lemma real_of_nat_number_of [simp]:
"real (number_of v :: nat) =
(if neg (number_of v :: int) then 0
else (number_of v :: real))"
by (simp add: real_of_int_real_of_nat [symmetric])
declaration {*
K (Lin_Arith.add_inj_thms [@{thm real_of_nat_le_iff} RS iffD2, @{thm real_of_nat_inject} RS iffD2]
(* not needed because x < (y::nat) can be rewritten as Suc x <= y: real_of_nat_less_iff RS iffD2 *)
#> Lin_Arith.add_inj_thms [@{thm real_of_int_le_iff} RS iffD2, @{thm real_of_int_inject} RS iffD2]
(* not needed because x < (y::int) can be rewritten as x + 1 <= y: real_of_int_less_iff RS iffD2 *)
#> Lin_Arith.add_simps [@{thm real_of_nat_zero}, @{thm real_of_nat_Suc}, @{thm real_of_nat_add},
@{thm real_of_nat_mult}, @{thm real_of_int_zero}, @{thm real_of_one},
@{thm real_of_int_add}, @{thm real_of_int_minus}, @{thm real_of_int_diff},
@{thm real_of_int_mult}, @{thm real_of_int_of_nat_eq},
@{thm real_of_nat_number_of}, @{thm real_number_of}]
#> Lin_Arith.add_inj_const (@{const_name real}, HOLogic.natT --> HOLogic.realT)
#> Lin_Arith.add_inj_const (@{const_name real}, HOLogic.intT --> HOLogic.realT))
*}
subsection{* Simprules combining x+y and 0: ARE THEY NEEDED?*}
text{*Needed in this non-standard form by Hyperreal/Transcendental*}
lemma real_0_le_divide_iff:
"((0::real) \<le> x/y) = ((x \<le> 0 | 0 \<le> y) & (0 \<le> x | y \<le> 0))"
by (simp add: real_divide_def zero_le_mult_iff, auto)
lemma real_add_minus_iff [simp]: "(x + - a = (0::real)) = (x=a)"
by arith
lemma real_add_eq_0_iff: "(x+y = (0::real)) = (y = -x)"
by auto
lemma real_add_less_0_iff: "(x+y < (0::real)) = (y < -x)"
by auto
lemma real_0_less_add_iff: "((0::real) < x+y) = (-x < y)"
by auto
lemma real_add_le_0_iff: "(x+y \<le> (0::real)) = (y \<le> -x)"
by auto
lemma real_0_le_add_iff: "((0::real) \<le> x+y) = (-x \<le> y)"
by auto
(*
FIXME: we should have this, as for type int, but many proofs would break.
It replaces x+-y by x-y.
declare real_diff_def [symmetric, simp]
*)
subsubsection{*Density of the Reals*}
lemma real_lbound_gt_zero:
"[| (0::real) < d1; 0 < d2 |] ==> \<exists>e. 0 < e & e < d1 & e < d2"
apply (rule_tac x = " (min d1 d2) /2" in exI)
apply (simp add: min_def)
done
text{*Similar results are proved in @{text Fields}*}
lemma real_less_half_sum: "x < y ==> x < (x+y) / (2::real)"
by auto
lemma real_gt_half_sum: "x < y ==> (x+y)/(2::real) < y"
by auto
subsection{*Absolute Value Function for the Reals*}
lemma abs_minus_add_cancel: "abs(x + (-y)) = abs (y + (-(x::real)))"
by (simp add: abs_if)
(* FIXME: redundant, but used by Integration/RealRandVar.thy in AFP *)
lemma abs_le_interval_iff: "(abs x \<le> r) = (-r\<le>x & x\<le>(r::real))"
by (force simp add: abs_le_iff)
lemma abs_add_one_gt_zero [simp]: "(0::real) < 1 + abs(x)"
by (simp add: abs_if)
lemma abs_real_of_nat_cancel [simp]: "abs (real x) = real (x::nat)"
by (rule abs_of_nonneg [OF real_of_nat_ge_zero])
lemma abs_add_one_not_less_self [simp]: "~ abs(x) + (1::real) < x"
by simp
lemma abs_sum_triangle_ineq: "abs ((x::real) + y + (-l + -m)) \<le> abs(x + -l) + abs(y + -m)"
by simp
subsection {* Implementation of rational real numbers *}
definition Ratreal :: "rat \<Rightarrow> real" where
[simp]: "Ratreal = of_rat"
code_datatype Ratreal
lemma Ratreal_number_collapse [code_post]:
"Ratreal 0 = 0"
"Ratreal 1 = 1"
"Ratreal (number_of k) = number_of k"
by simp_all
lemma zero_real_code [code, code_unfold]:
"0 = Ratreal 0"
by simp
lemma one_real_code [code, code_unfold]:
"1 = Ratreal 1"
by simp
lemma number_of_real_code [code_unfold]:
"number_of k = Ratreal (number_of k)"
by simp
lemma Ratreal_number_of_quotient [code_post]:
"Ratreal (number_of r) / Ratreal (number_of s) = number_of r / number_of s"
by simp
lemma Ratreal_number_of_quotient2 [code_post]:
"Ratreal (number_of r / number_of s) = number_of r / number_of s"
unfolding Ratreal_number_of_quotient [symmetric] Ratreal_def of_rat_divide ..
instantiation real :: eq
begin
definition "eq_class.eq (x\<Colon>real) y \<longleftrightarrow> x - y = 0"
instance by default (simp add: eq_real_def)
lemma real_eq_code [code]: "eq_class.eq (Ratreal x) (Ratreal y) \<longleftrightarrow> eq_class.eq x y"
by (simp add: eq_real_def eq)
lemma real_eq_refl [code nbe]:
"eq_class.eq (x::real) x \<longleftrightarrow> True"
by (rule HOL.eq_refl)
end
lemma real_less_eq_code [code]: "Ratreal x \<le> Ratreal y \<longleftrightarrow> x \<le> y"
by (simp add: of_rat_less_eq)
lemma real_less_code [code]: "Ratreal x < Ratreal y \<longleftrightarrow> x < y"
by (simp add: of_rat_less)
lemma real_plus_code [code]: "Ratreal x + Ratreal y = Ratreal (x + y)"
by (simp add: of_rat_add)
lemma real_times_code [code]: "Ratreal x * Ratreal y = Ratreal (x * y)"
by (simp add: of_rat_mult)
lemma real_uminus_code [code]: "- Ratreal x = Ratreal (- x)"
by (simp add: of_rat_minus)
lemma real_minus_code [code]: "Ratreal x - Ratreal y = Ratreal (x - y)"
by (simp add: of_rat_diff)
lemma real_inverse_code [code]: "inverse (Ratreal x) = Ratreal (inverse x)"
by (simp add: of_rat_inverse)
lemma real_divide_code [code]: "Ratreal x / Ratreal y = Ratreal (x / y)"
by (simp add: of_rat_divide)
definition (in term_syntax)
valterm_ratreal :: "rat \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow> real \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
[code_unfold]: "valterm_ratreal k = Code_Evaluation.valtermify Ratreal {\<cdot>} k"
notation fcomp (infixl "o>" 60)
notation scomp (infixl "o\<rightarrow>" 60)
instantiation real :: random
begin
definition
"Quickcheck.random i = Quickcheck.random i o\<rightarrow> (\<lambda>r. Pair (valterm_ratreal r))"
instance ..
end
no_notation fcomp (infixl "o>" 60)
no_notation scomp (infixl "o\<rightarrow>" 60)
text {* Setup for SML code generator *}
types_code
real ("(int */ int)")
attach (term_of) {*
fun term_of_real (p, q) =
let
val rT = HOLogic.realT
in
if q = 1 orelse p = 0 then HOLogic.mk_number rT p
else @{term "op / \<Colon> real \<Rightarrow> real \<Rightarrow> real"} $
HOLogic.mk_number rT p $ HOLogic.mk_number rT q
end;
*}
attach (test) {*
fun gen_real i =
let
val p = random_range 0 i;
val q = random_range 1 (i + 1);
val g = Integer.gcd p q;
val p' = p div g;
val q' = q div g;
val r = (if one_of [true, false] then p' else ~ p',
if p' = 0 then 1 else q')
in
(r, fn () => term_of_real r)
end;
*}
consts_code
Ratreal ("(_)")
consts_code
"of_int :: int \<Rightarrow> real" ("\<module>real'_of'_int")
attach {*
fun real_of_int i = (i, 1);
*}
setup {*
Nitpick.register_frac_type @{type_name real}
[(@{const_name zero_real_inst.zero_real}, @{const_name Nitpick.zero_frac}),
(@{const_name one_real_inst.one_real}, @{const_name Nitpick.one_frac}),
(@{const_name plus_real_inst.plus_real}, @{const_name Nitpick.plus_frac}),
(@{const_name times_real_inst.times_real}, @{const_name Nitpick.times_frac}),
(@{const_name uminus_real_inst.uminus_real}, @{const_name Nitpick.uminus_frac}),
(@{const_name number_real_inst.number_of_real}, @{const_name Nitpick.number_of_frac}),
(@{const_name inverse_real_inst.inverse_real}, @{const_name Nitpick.inverse_frac}),
(@{const_name ord_real_inst.less_eq_real}, @{const_name Nitpick.less_eq_frac})]
*}
lemmas [nitpick_def] = inverse_real_inst.inverse_real
number_real_inst.number_of_real one_real_inst.one_real
ord_real_inst.less_eq_real plus_real_inst.plus_real
times_real_inst.times_real uminus_real_inst.uminus_real
zero_real_inst.zero_real
end