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theory RealDef(* Title : Real/RealDef.thy ID : $Id: RealDef.thy,v 1.58 2005/08/01 17:20:26 wenzelm Exp $ 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 uses ("real_arith.ML") begin constdefs realrel :: "((preal * preal) * (preal * preal)) set" "realrel == {p. ∃x1 y1 x2 y2. p = ((x1,y1),(x2,y2)) & x1+y2 = x2+y1}" typedef (Real) real = "UNIV//realrel" by (auto simp add: quotient_def) instance real :: "{ord, zero, one, plus, times, minus, inverse}" .. constdefs (** these don't use the overloaded "real" function: users don't see them **) real_of_preal :: "preal => real" "real_of_preal m == Abs_Real(realrel``{(m + preal_of_rat 1, preal_of_rat 1)})" consts (*Overloaded constant denoting the Real subset of enclosing types such as hypreal and complex*) Reals :: "'a set" (*overloaded constant for injecting other types into "real"*) real :: "'a => real" syntax (xsymbols) Reals :: "'a set" ("\<real>") defs (overloaded) real_zero_def: "0 == Abs_Real(realrel``{(preal_of_rat 1, preal_of_rat 1)})" real_one_def: "1 == Abs_Real(realrel`` {(preal_of_rat 1 + preal_of_rat 1, preal_of_rat 1)})" real_minus_def: "- r == contents (\<Union>(x,y) ∈ Rep_Real(r). { Abs_Real(realrel``{(y,x)}) })" real_add_def: "z + w == contents (\<Union>(x,y) ∈ Rep_Real(z). \<Union>(u,v) ∈ Rep_Real(w). { Abs_Real(realrel``{(x+u, y+v)}) })" real_diff_def: "r - (s::real) == r + - s" real_mult_def: "z * w == contents (\<Union>(x,y) ∈ Rep_Real(z). \<Union>(u,v) ∈ Rep_Real(w). { Abs_Real(realrel``{(x*u + y*v, x*v + y*u)}) })" real_inverse_def: "inverse (R::real) == (SOME S. (R = 0 & S = 0) | S * R = 1)" real_divide_def: "R / (S::real) == R * inverse S" real_le_def: "z ≤ (w::real) == ∃x y u v. x+v ≤ u+y & (x,y) ∈ Rep_Real z & (u,v) ∈ Rep_Real w" real_less_def: "(x < (y::real)) == (x ≤ y & x ≠ y)" real_abs_def: "abs (r::real) == (if 0 ≤ r then r else -r)" subsection{*Proving that realrel is an equivalence relation*} 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: preal_add_ac) also have "... = (x2 + y) + x1" by (simp add: prems) also have "... = x2 + (x1 + y)" by (simp add: preal_add_ac) also have "... = x2 + (x + y1)" by (simp add: prems) also have "... = (x2 + y1) + x" by (simp add: preal_add_ac) finally have "(x1 + y2) + x = (x2 + y1) + x" . thus ?thesis by (simp add: preal_add_right_cancel_iff) qed lemma realrel_iff [simp]: "(((x1,y1),(x2,y2)) ∈ realrel) = (x1 + y2 = x2 + y1)" by (simp add: realrel_def) lemma equiv_realrel: "equiv UNIV realrel" apply (auto simp add: equiv_def refl_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) ∈ 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) lemma inj_on_Abs_Real: "inj_on Abs_Real Real" apply (rule inj_on_inverseI) apply (erule Abs_Real_inverse) done declare inj_on_Abs_Real [THEN inj_on_iff, 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{*Congruence property for addition*} 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: preal_add_assoc) apply (rule preal_add_left_commute [of ab, THEN ssubst]) apply (simp add: preal_add_assoc [symmetric]) apply (simp add: preal_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 "(λz w. (λ(x,y). (λ(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_add_commute: "(z::real) + w = w + z" by (cases z, cases w, simp add: real_add preal_add_ac) lemma real_add_assoc: "((z1::real) + z2) + z3 = z1 + (z2 + z3)" by (cases z1, cases z2, cases z3, simp add: real_add preal_add_assoc) lemma real_add_zero_left: "(0::real) + z = z" by (cases z, simp add: real_add real_zero_def preal_add_ac) instance real :: comm_monoid_add by (intro_classes, (assumption | rule real_add_commute real_add_assoc real_add_zero_left)+) subsection{*Additive Inverse on real*} lemma real_minus: "- Abs_Real(realrel``{(x,y)}) = Abs_Real(realrel `` {(y,x)})" proof - have "(λ(x,y). {Abs_Real (realrel``{(y,x)})}) respects realrel" by (simp add: congruent_def preal_add_commute) thus ?thesis by (simp add: real_minus_def UN_equiv_class [OF equiv_realrel]) qed lemma real_add_minus_left: "(-z) + z = (0::real)" by (cases z, simp add: real_minus real_add real_zero_def preal_add_commute) subsection{*Congruence property for 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: preal_add_left_commute preal_add_assoc [symmetric]) apply (simp add: preal_add_assoc preal_add_mult_distrib2 [symmetric]) apply (simp add: preal_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: preal_mult_commute preal_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 preal_add_ac preal_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 preal_add_mult_distrib2 preal_add_ac preal_mult_ac) done lemma real_mult_1: "(1::real) * z = z" apply (cases z) apply (simp add: real_mult real_one_def preal_add_mult_distrib2 preal_mult_1_right preal_mult_ac preal_add_ac) 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 preal_add_mult_distrib2 preal_add_ac preal_mult_ac) done text{*one and zero are distinct*} lemma real_zero_not_eq_one: "0 ≠ (1::real)" proof - have "preal_of_rat 1 < preal_of_rat 1 + preal_of_rat 1" by (simp add: preal_self_less_add_left) thus ?thesis by (simp add: real_zero_def real_one_def preal_add_right_cancel_iff) qed subsection{*existence of inverse*} lemma real_zero_iff: "Abs_Real (realrel `` {(x, x)}) = 0" by (simp add: real_zero_def preal_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 ≠ 0 ==> ∃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 `` { (preal_of_rat 1, inverse (D) + preal_of_rat 1)}) " in exI) apply (rule_tac [2] x = "Abs_Real (realrel `` { (inverse (D) + preal_of_rat 1, preal_of_rat 1)})" in exI) apply (auto simp add: real_mult preal_mult_1_right preal_add_mult_distrib2 preal_add_mult_distrib preal_mult_1 preal_mult_inverse_right preal_add_ac preal_mult_ac) done lemma real_mult_inverse_left: "x ≠ 0 ==> inverse(x)*x = (1::real)" apply (simp add: real_inverse_def) apply (frule real_mult_inverse_left_ex, safe) apply (rule someI2, auto) done subsection{*The Real Numbers form a Field*} instance real :: field proof fix x y z :: real show "- x + x = 0" by (rule real_add_minus_left) show "x - y = x + (-y)" by (simp add: real_diff_def) 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 (simp add: real_add_mult_distrib) show "0 ≠ (1::real)" by (rule real_zero_not_eq_one) show "x ≠ 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 (*Pull negations out*) declare minus_mult_right [symmetric, simp] minus_mult_left [symmetric, simp] lemma real_mult_1_right: "z * (1::real) = z" by (rule OrderedGroup.mult_1_right) subsection{*The @{text "≤"} Ordering*} lemma real_le_refl: "w ≤ (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 ≤ a" shows "b ≤ (d::preal)" proof - have "c+d ≤ a+d" by (simp add: prems preal_cancels) hence "a+b ≤ a+d" by (simp add: prems) thus "b ≤ d" by (simp add: preal_cancels) qed lemma real_le_lemma: assumes l: "u1 + v2 ≤ u2 + v1" and "x1 + v1 = u1 + y1" and "x2 + v2 = u2 + y2" shows "x1 + y2 ≤ 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: preal_add_ac) also have "... ≤ (x2+y1) + (u2+v1)" by (simp add: prems preal_add_le_cancel_left) finally show ?thesis by (simp add: preal_add_le_cancel_right) qed lemma real_le: "(Abs_Real(realrel``{(x1,y1)}) ≤ Abs_Real(realrel``{(x2,y2)})) = (x1 + y2 ≤ x2 + y1)" apply (simp add: real_le_def) apply (auto intro: real_le_lemma) done lemma real_le_anti_sym: "[| z ≤ w; w ≤ z |] ==> z = (w::real)" by (cases z, cases w, simp add: real_le) lemma real_trans_lemma: assumes "x + v ≤ u + y" and "u + v' ≤ u' + v" and "x2 + v2 = u2 + y2" shows "x + v' ≤ u' + (y::preal)" proof - have "(x+v') + (u+v) = (x+v) + (u+v')" by (simp add: preal_add_ac) also have "... ≤ (u+y) + (u+v')" by (simp add: preal_add_le_cancel_right prems) also have "... ≤ (u+y) + (u'+v)" by (simp add: preal_add_le_cancel_left prems) also have "... = (u'+y) + (u+v)" by (simp add: preal_add_ac) finally show ?thesis by (simp add: preal_add_le_cancel_right) qed lemma real_le_trans: "[| i ≤ j; j ≤ k |] ==> i ≤ (k::real)" apply (cases i, cases j, cases k) apply (simp add: real_le) apply (blast intro: real_trans_lemma) done (* Axiom 'order_less_le' of class 'order': *) lemma real_less_le: "((w::real) < z) = (w ≤ z & w ≠ z)" by (simp add: real_less_def) instance real :: order proof qed (assumption | rule real_le_refl real_le_trans real_le_anti_sym real_less_le)+ (* Axiom 'linorder_linear' of class 'linorder': *) lemma real_le_linear: "(z::real) ≤ w | w ≤ z" apply (cases z, cases w) apply (auto simp add: real_le real_zero_def preal_add_ac preal_cancels) done instance real :: linorder by (intro_classes, rule real_le_linear) lemma real_le_eq_diff: "(x ≤ y) = (x-y ≤ (0::real))" apply (cases x, cases y) apply (auto simp add: real_le real_zero_def real_diff_def real_add real_minus preal_add_ac) apply (simp_all add: preal_add_assoc [symmetric] preal_cancels) done lemma real_add_left_mono: assumes le: "x ≤ y" shows "z + x ≤ z + (y::real)" proof - have "z + x - (z + y) = (z + -z) + (x - y)" by (simp add: diff_minus add_ac) 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 "≤"} relation *} apply (auto dest!: less_add_left_Ex simp add: preal_add_ac preal_mult_ac preal_add_mult_distrib2 preal_cancels 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 text{*lemma for proving @{term "0<(1::real)"}*} lemma real_zero_le_one: "0 ≤ (1::real)" by (simp add: real_zero_def real_one_def real_le preal_self_less_add_left order_less_imp_le) subsection{*The Reals Form an Ordered Field*} instance real :: ordered_field proof fix x y z :: real show "x ≤ y ==> z + x ≤ z + y" by (rule real_add_left_mono) show "x < y ==> 0 < z ==> z * x < z * y" by (simp add: real_mult_less_mono2) show "¦x¦ = (if x < 0 then -x else x)" by (auto dest: order_le_less_trans simp add: real_abs_def linorder_not_le) 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 preal_add_mult_distrib preal_mult_1 preal_add_ac) 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 preal_add_mult_distrib2 preal_mult_1 preal_mult_1_right preal_add_ac preal_mult_ac) text{*Gleason prop 9-4.4 p 127*} lemma real_of_preal_trichotomy: "∃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 preal_add_ac) apply (cut_tac x = x and y = y in linorder_less_linear) apply (auto dest!: less_add_left_Ex simp add: preal_add_assoc [symmetric]) done lemma real_of_preal_leD: "real_of_preal m1 ≤ real_of_preal m2 ==> m1 ≤ m2" by (simp add: real_of_preal_def real_le preal_cancels) 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] preal_cancels) 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 ≤ real_of_preal m2) = (m1 ≤ m2)" by (simp add: linorder_not_less [symmetric]) lemma real_of_preal_zero_less: "0 < real_of_preal m" apply (auto simp add: real_zero_def real_of_preal_def real_less_def real_le_def preal_add_ac preal_cancels) apply (simp_all add: preal_add_assoc [symmetric] preal_cancels) apply (blast intro: preal_self_less_add_left order_less_imp_le) apply (insert preal_not_eq_self [of "preal_of_rat 1" m]) apply (simp add: preal_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*} text{*obsolete but used a lot*} lemma real_not_refl2: "x < y ==> x ≠ (y::real)" by blast lemma real_le_imp_less_or_eq: "!!(x::real). x ≤ y ==> x < y | x = y" by (simp add: order_le_less) lemma real_gt_zero_preal_Ex: "(0 < x) = (∃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 ==> ∃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 ≤ x ==> ∃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 ≤ 0 ==> ∀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 ==> ∀x. y < real_of_preal x" by (blast intro!: real_less_all_preal linorder_not_less [THEN iffD1]) lemma real_add_less_le_mono: "[| w'<w; z'≤z |] ==> w' + z' < w + (z::real)" by (rule OrderedGroup.add_less_le_mono) lemma real_add_le_less_mono: "!!z z'::real. [| w'≤w; z'<z |] ==> w' + z' < w + z" by (rule OrderedGroup.add_le_less_mono) lemma real_le_square [simp]: "(0::real) ≤ x*x" by (rule Ring_and_Field.zero_le_square) subsection{*More Lemmas*} lemma real_mult_left_cancel: "(c::real) ≠ 0 ==> (c*a=c*b) = (a=b)" by auto lemma real_mult_right_cancel: "(c::real) ≠ 0 ==> (a*c=b*c) = (a=b)" by auto text{*The precondition could be weakened to @{term "0≤x"}*} lemma real_mult_less_mono: "[| u<v; x<y; (0::real) < v; 0 < x |] ==> u*x < v* y" by (simp add: Ring_and_Field.mult_strict_mono order_less_imp_le) lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)" by (force elim: order_less_asym simp add: Ring_and_Field.mult_less_cancel_right) lemma real_mult_le_cancel_iff1 [simp]: "(0::real) < z ==> (x*z ≤ y*z) = (x≤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 ≤ z*y) = (x≤y)" by(simp add:mult_commute) text{*Only two uses?*} lemma real_mult_less_mono': "[| x < y; r1 < r2; (0::real) ≤ r1; 0 ≤ x|] ==> r1 * x < r2 * y" by (rule Ring_and_Field.mult_strict_mono') text{*FIXME: delete or at least combine the next two lemmas*} lemma real_sum_squares_cancel: "x * x + y * y = 0 ==> x = (0::real)" apply (drule OrderedGroup.equals_zero_I [THEN sym]) apply (cut_tac x = y in real_le_square) apply (auto, drule order_antisym, auto) done lemma real_sum_squares_cancel2: "x * x + y * y = 0 ==> y = (0::real)" apply (rule_tac y = x in real_sum_squares_cancel) apply (simp add: add_commute) done lemma real_add_order: "[| 0 < x; 0 < y |] ==> (0::real) < x + y" by (drule add_strict_mono [of concl: 0 0], assumption, simp) lemma real_le_add_order: "[| 0 ≤ x; 0 ≤ y |] ==> (0::real) ≤ x + y" apply (drule order_le_imp_less_or_eq)+ apply (auto intro: real_add_order order_less_imp_le) done lemma real_inverse_unique: "x*y = (1::real) ==> y = inverse x" apply (case_tac "x ≠ 0") apply (rule_tac c1 = x in real_mult_left_cancel [THEN iffD1], auto) done lemma real_inverse_gt_one: "[| (0::real) < x; x < 1 |] ==> 1 < inverse x" by (auto dest: less_imp_inverse_less) lemma real_mult_self_sum_ge_zero: "(0::real) ≤ x*x + y*y" proof - have "0 + 0 ≤ x*x + y*y" by (blast intro: add_mono zero_le_square) thus ?thesis by simp qed subsection{*Embedding the Integers into the Reals*} defs (overloaded) real_of_nat_def: "real z == of_nat z" real_of_int_def: "real z == of_int z" lemma real_eq_of_nat: "real = of_nat" apply (rule ext) apply (unfold real_of_nat_def) apply (rule refl) done lemma real_eq_of_int: "real = of_int" apply (rule ext) apply (unfold real_of_int_def) apply (rule refl) done 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_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]: "(real x = 0) = (x = (0::int))" by (simp add: real_of_int_def) lemma real_of_int_inject [iff]: "(real (x::int) = real y) = (x = y)" by (simp add: real_of_int_def) lemma real_of_int_less_iff [iff]: "(real (x::int) < real y) = (x < y)" by (simp add: real_of_int_def) lemma real_of_int_le_iff [simp]: "(real (x::int) ≤ real y) = (x ≤ y)" by (simp add: real_of_int_def) lemma real_of_int_gt_zero_cancel_iff [simp]: "(0 < real (n::int)) = (0 < n)" by (simp add: real_of_int_def) lemma real_of_int_ge_zero_cancel_iff [simp]: "(0 <= real (n::int)) = (0 <= n)" by (simp add: real_of_int_def) lemma real_of_int_lt_zero_cancel_iff [simp]: "(real (n::int) < 0) = (n < 0)" by (simp add: real_of_int_def) lemma real_of_int_le_zero_cancel_iff [simp]: "(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 ring_eq_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: zdvd_iff_zmod_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: compare_rls) apply (subst real_of_int_div_aux) apply simp apply simp apply (subst zero_le_divide_iff) apply auto apply (simp add: compare_rls) 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: compare_rls) 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) 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_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) ≤ real m) = (n ≤ m)" by (simp add: real_of_nat_def) lemma real_of_nat_ge_zero [iff]: "0 ≤ 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) 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 ≤ m ==> real (m - n) = real (m::nat) - real n" by (simp add: add: real_of_nat_def) lemma real_of_nat_gt_zero_cancel_iff [simp]: "(0 < real (n::nat)) = (0 < n)" by (simp add: add: real_of_nat_def) lemma real_of_nat_le_zero_cancel_iff [simp]: "(real (n::nat) ≤ 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) lemma real_of_nat_ge_zero_cancel_iff [simp]: "(0 ≤ real (n::nat)) = (0 ≤ n)" 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 = … / real d" by simp then show ?thesis by (auto simp add: add_divide_distrib ring_eq_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: compare_rls) apply (subst real_of_nat_div_aux) apply assumption 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: compare_rls) apply (subst real_of_nat_div_aux) apply assumption 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 subsection{*Numerals and Arithmetic*} instance real :: number .. defs (overloaded) real_number_of_def: "(number_of w :: real) == of_int (Rep_Bin w)" --{*the type constraint is essential!*} instance real :: number_ring by (intro_classes, simp add: 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 of_int_number_of_eq) 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] int_nat_number_of) use "real_arith.ML" setup real_arith_setup 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) ≤ x/y) = ((x ≤ 0 | 0 ≤ y) & (0 ≤ x | y ≤ 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 ≤ (0::real)) = (y ≤ -x)" by auto lemma real_0_le_add_iff: "((0::real) ≤ x+y) = (-x ≤ 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 |] ==> ∃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 Ring_and_Field}*} 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) lemma abs_interval_iff: "(abs x < r) = (-r < x & x < (r::real))" by (force simp add: Ring_and_Field.abs_less_iff) lemma abs_le_interval_iff: "(abs x ≤ r) = (-r≤x & x≤(r::real))" by (force simp add: OrderedGroup.abs_le_iff) (*FIXME: used only once, in SEQ.ML*) 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 (simp add: real_of_nat_ge_zero) lemma abs_add_one_not_less_self [simp]: "~ abs(x) + (1::real) < x" apply (simp add: linorder_not_less) apply (auto intro: abs_ge_self [THEN order_trans]) done text{*Used only in Hyperreal/Lim.ML*} lemma abs_sum_triangle_ineq: "abs ((x::real) + y + (-l + -m)) ≤ abs(x + -l) + abs(y + -m)" apply (simp add: real_add_assoc) apply (rule_tac a1 = y in add_left_commute [THEN ssubst]) apply (rule real_add_assoc [THEN subst]) apply (rule abs_triangle_ineq) done ML {* val real_lbound_gt_zero = thm"real_lbound_gt_zero"; val real_less_half_sum = thm"real_less_half_sum"; val real_gt_half_sum = thm"real_gt_half_sum"; val abs_interval_iff = thm"abs_interval_iff"; val abs_le_interval_iff = thm"abs_le_interval_iff"; val abs_add_one_gt_zero = thm"abs_add_one_gt_zero"; val abs_add_one_not_less_self = thm"abs_add_one_not_less_self"; val abs_sum_triangle_ineq = thm"abs_sum_triangle_ineq"; *} end
lemma preal_trans_lemma:
[| x + y1.0 = x1.0 + y; x + y2.0 = x2.0 + y |] ==> x1.0 + y2.0 = x2.0 + y1.0
lemma realrel_iff:
(((x1.0, y1.0), x2.0, y2.0) ∈ realrel) = (x1.0 + y2.0 = x2.0 + y1.0)
lemma equiv_realrel:
equiv UNIV realrel
lemmas equiv_realrel_iff:
(realrel `` {x} = realrel `` {y}) = ((x, y) ∈ realrel)
lemmas equiv_realrel_iff:
(realrel `` {x} = realrel `` {y}) = ((x, y) ∈ realrel)
lemma realrel_in_real:
realrel `` {(x, y)} ∈ Real
lemma inj_on_Abs_Real:
inj_on Abs_Real Real
lemma eq_Abs_Real:
(!!x y. z = Abs_Real (realrel `` {(x, y)}) ==> P) ==> P
lemma real_add_congruent2_lemma:
[| a + ba = aa + b; ab + bc = ac + bb |] ==> a + ab + (ba + bc) = aa + ac + (b + bb)
lemma real_add:
Abs_Real (realrel `` {(x, y)}) + Abs_Real (realrel `` {(u, v)}) = Abs_Real (realrel `` {(x + u, y + v)})
lemma real_add_commute:
z + w = w + z
lemma real_add_assoc:
z1.0 + z2.0 + z3.0 = z1.0 + (z2.0 + z3.0)
lemma real_add_zero_left:
0 + z = z
lemma real_minus:
- Abs_Real (realrel `` {(x, y)}) = Abs_Real (realrel `` {(y, x)})
lemma real_add_minus_left:
- z + z = 0
lemma real_mult_congruent2_lemma:
x1.0 + y2.0 = x2.0 + y1.0 ==> x * x1.0 + y * y1.0 + (x * y2.0 + y * x2.0) = x * x2.0 + y * y2.0 + (x * y1.0 + y * x1.0)
lemma real_mult_congruent2:
congruent2 realrel realrel (%p1 p2. (%(x1, y1). (%(x2, y2). {Abs_Real (realrel `` {(x1 * x2 + y1 * y2, x1 * y2 + y1 * x2)})}) p2) p1)
lemma real_mult:
Abs_Real (realrel `` {(x1.0, y1.0)}) * Abs_Real (realrel `` {(x2.0, y2.0)}) = Abs_Real (realrel `` {(x1.0 * x2.0 + y1.0 * y2.0, x1.0 * y2.0 + y1.0 * x2.0)})
lemma real_mult_commute:
z * w = w * z
lemma real_mult_assoc:
z1.0 * z2.0 * z3.0 = z1.0 * (z2.0 * z3.0)
lemma real_mult_1:
1 * z = z
lemma real_add_mult_distrib:
(z1.0 + z2.0) * w = z1.0 * w + z2.0 * w
lemma real_zero_not_eq_one:
0 ≠ 1
lemma real_zero_iff:
Abs_Real (realrel `` {(x, x)}) = 0
lemma real_mult_inverse_left_ex:
x ≠ 0 ==> ∃y. y * x = 1
lemma real_mult_inverse_left:
x ≠ 0 ==> inverse x * x = 1
lemma INVERSE_ZERO:
inverse 0 = 0
lemma real_mult_1_right:
z * 1 = z
lemma real_le_refl:
w ≤ w
lemma preal_eq_le_imp_le:
[| a + b = c + d; c ≤ a |] ==> b ≤ d
lemma real_le_lemma:
[| u1.0 + v2.0 ≤ u2.0 + v1.0; x1.0 + v1.0 = u1.0 + y1.0; x2.0 + v2.0 = u2.0 + y2.0 |] ==> x1.0 + y2.0 ≤ x2.0 + y1.0
lemma real_le:
(Abs_Real (realrel `` {(x1.0, y1.0)}) ≤ Abs_Real (realrel `` {(x2.0, y2.0)})) = (x1.0 + y2.0 ≤ x2.0 + y1.0)
lemma real_le_anti_sym:
[| z ≤ w; w ≤ z |] ==> z = w
lemma real_trans_lemma:
[| x + v ≤ u + y; u + v' ≤ u' + v; x2.0 + v2.0 = u2.0 + y2.0 |] ==> x + v' ≤ u' + y
lemma real_le_trans:
[| i ≤ j; j ≤ k |] ==> i ≤ k
lemma real_less_le:
(w < z) = (w ≤ z ∧ w ≠ z)
lemma real_le_linear:
z ≤ w ∨ w ≤ z
lemma real_le_eq_diff:
(x ≤ y) = (x - y ≤ 0)
lemma real_add_left_mono:
x ≤ y ==> z + x ≤ z + y
lemma real_sum_gt_zero_less:
0 < S + - W ==> W < S
lemma real_less_sum_gt_zero:
W < S ==> 0 < S + - W
lemma real_mult_order:
[| 0 < x; 0 < y |] ==> 0 < x * y
lemma real_mult_less_mono2:
[| 0 < z; x < y |] ==> z * x < z * y
lemma real_zero_le_one:
0 ≤ 1
lemma real_of_preal_add:
real_of_preal (x + y) = real_of_preal x + real_of_preal y
lemma real_of_preal_mult:
real_of_preal (x * y) = real_of_preal x * real_of_preal y
lemma real_of_preal_trichotomy:
∃m. x = real_of_preal m ∨ x = 0 ∨ x = - real_of_preal m
lemma real_of_preal_leD:
real_of_preal m1.0 ≤ real_of_preal m2.0 ==> m1.0 ≤ m2.0
lemma real_of_preal_lessI:
m1.0 < m2.0 ==> real_of_preal m1.0 < real_of_preal m2.0
lemma real_of_preal_lessD:
real_of_preal m1.0 < real_of_preal m2.0 ==> m1.0 < m2.0
lemma real_of_preal_less_iff:
(real_of_preal m1.0 < real_of_preal m2.0) = (m1.0 < m2.0)
lemma real_of_preal_le_iff:
(real_of_preal m1.0 ≤ real_of_preal m2.0) = (m1.0 ≤ m2.0)
lemma real_of_preal_zero_less:
0 < real_of_preal m
lemma real_of_preal_minus_less_zero:
- real_of_preal m < 0
lemma real_of_preal_not_minus_gt_zero:
¬ 0 < - real_of_preal m
lemma real_not_refl2:
x < y ==> x ≠ y
lemma real_le_imp_less_or_eq:
x ≤ y ==> x < y ∨ x = y
lemma real_gt_zero_preal_Ex:
(0 < x) = (∃y. x = real_of_preal y)
lemma real_gt_preal_preal_Ex:
real_of_preal z < x ==> ∃y. x = real_of_preal y
lemma real_ge_preal_preal_Ex:
real_of_preal z ≤ x ==> ∃y. x = real_of_preal y
lemma real_less_all_preal:
y ≤ 0 ==> ∀x. y < real_of_preal x
lemma real_less_all_real2:
¬ 0 < y ==> ∀x. y < real_of_preal x
lemma real_add_less_le_mono:
[| w' < w; z' ≤ z |] ==> w' + z' < w + z
lemma real_add_le_less_mono:
[| w' ≤ w; z' < z |] ==> w' + z' < w + z
lemma real_le_square:
0 ≤ x * x
lemma real_mult_left_cancel:
c ≠ 0 ==> (c * a = c * b) = (a = b)
lemma real_mult_right_cancel:
c ≠ 0 ==> (a * c = b * c) = (a = b)
lemma real_mult_less_mono:
[| u < v; x < y; 0 < v; 0 < x |] ==> u * x < v * y
lemma real_mult_less_iff1:
0 < z ==> (x * z < y * z) = (x < y)
lemma real_mult_le_cancel_iff1:
0 < z ==> (x * z ≤ y * z) = (x ≤ y)
lemma real_mult_le_cancel_iff2:
0 < z ==> (z * x ≤ z * y) = (x ≤ y)
lemma real_mult_less_mono':
[| x < y; r1.0 < r2.0; 0 ≤ r1.0; 0 ≤ x |] ==> r1.0 * x < r2.0 * y
lemma real_sum_squares_cancel:
x * x + y * y = 0 ==> x = 0
lemma real_sum_squares_cancel2:
x * x + y * y = 0 ==> y = 0
lemma real_add_order:
[| 0 < x; 0 < y |] ==> 0 < x + y
lemma real_le_add_order:
[| 0 ≤ x; 0 ≤ y |] ==> 0 ≤ x + y
lemma real_inverse_unique:
x * y = 1 ==> y = inverse x
lemma real_inverse_gt_one:
[| 0 < x; x < 1 |] ==> 1 < inverse x
lemma real_mult_self_sum_ge_zero:
0 ≤ x * x + y * y
lemma real_eq_of_nat:
real = of_nat
lemma real_eq_of_int:
real = of_int
lemma real_of_int_zero:
real 0 = 0
lemma real_of_one:
real 1 = 1
lemma real_of_int_add:
real (x + y) = real x + real y
lemma real_of_int_minus:
real (- x) = - real x
lemma real_of_int_diff:
real (x - y) = real x - real y
lemma real_of_int_mult:
real (x * y) = real x * real y
lemma real_of_int_setsum:
real (setsum f A) = (∑x∈A. real (f x))
lemma real_of_int_setprod:
real (setprod f A) = (∏x∈A. real (f x))
lemma real_of_int_zero_cancel:
(real x = 0) = (x = 0)
lemma real_of_int_inject:
(real x = real y) = (x = y)
lemma real_of_int_less_iff:
(real x < real y) = (x < y)
lemma real_of_int_le_iff:
(real x ≤ real y) = (x ≤ y)
lemma real_of_int_gt_zero_cancel_iff:
(0 < real n) = (0 < n)
lemma real_of_int_ge_zero_cancel_iff:
(0 ≤ real n) = (0 ≤ n)
lemma real_of_int_lt_zero_cancel_iff:
(real n < 0) = (n < 0)
lemma real_of_int_le_zero_cancel_iff:
(real n ≤ 0) = (n ≤ 0)
lemma real_of_int_abs:
real ¦x¦ = ¦real x¦
lemma int_less_real_le:
(n < m) = (real n + 1 ≤ real m)
lemma int_le_real_less:
(n ≤ m) = (real n < real m + 1)
lemma real_of_int_div_aux:
d ≠ 0 ==> real x / real d = real (x div d) + real (x mod d) / real d
lemma real_of_int_div:
[| d ≠ 0; d dvd n |] ==> real (n div d) = real n / real d
lemma real_of_int_div2:
0 ≤ real n / real x - real (n div x)
lemma real_of_int_div3:
real n / real x - real (n div x) ≤ 1
lemma real_of_int_div4:
real (n div x) ≤ real n / real x
lemma real_of_nat_zero:
real 0 = 0
lemma real_of_nat_one:
real (Suc 0) = 1
lemma real_of_nat_add:
real (m + n) = real m + real n
lemma real_of_nat_Suc:
real (Suc n) = real n + 1
lemma real_of_nat_less_iff:
(real n < real m) = (n < m)
lemma real_of_nat_le_iff:
(real n ≤ real m) = (n ≤ m)
lemma real_of_nat_ge_zero:
0 ≤ real n
lemma real_of_nat_Suc_gt_zero:
0 < real (Suc n)
lemma real_of_nat_mult:
real (m * n) = real m * real n
lemma real_of_nat_setsum:
real (setsum f A) = (∑x∈A. real (f x))
lemma real_of_nat_setprod:
real (setprod f A) = (∏x∈A. real (f x))
lemma real_of_card:
real (card A) = (∑x∈A. 1)
lemma real_of_nat_inject:
(real n = real m) = (n = m)
lemma real_of_nat_zero_iff:
(real n = 0) = (n = 0)
lemma real_of_nat_diff:
n ≤ m ==> real (m - n) = real m - real n
lemma real_of_nat_gt_zero_cancel_iff:
(0 < real n) = (0 < n)
lemma real_of_nat_le_zero_cancel_iff:
(real n ≤ 0) = (n = 0)
lemma not_real_of_nat_less_zero:
¬ real n < 0
lemma real_of_nat_ge_zero_cancel_iff:
(0 ≤ real n) = (0 ≤ n)
lemma nat_less_real_le:
(n < m) = (real n + 1 ≤ real m)
lemma nat_le_real_less:
(n ≤ m) = (real n < real m + 1)
lemma real_of_nat_div_aux:
0 < d ==> real x / real d = real (x div d) + real (x mod d) / real d
lemma real_of_nat_div:
[| 0 < d; d dvd n |] ==> real (n div d) = real n / real d
lemma real_of_nat_div2:
0 ≤ real n / real x - real (n div x)
lemma real_of_nat_div3:
real n / real x - real (n div x) ≤ 1
lemma real_of_nat_div4:
real (n div x) ≤ real n / real x
lemma real_of_int_real_of_nat:
real (int n) = real n
lemma real_of_int_of_nat_eq:
real (of_nat n) = real n
lemma real_nat_eq_real:
0 ≤ x ==> real (nat x) = real x
lemma real_number_of:
real (number_of v) = number_of v
lemma real_of_nat_number_of:
real (number_of v) = (if neg (number_of v) then 0 else number_of v)
lemma real_0_le_divide_iff:
(0 ≤ x / y) = ((x ≤ 0 ∨ 0 ≤ y) ∧ (0 ≤ x ∨ y ≤ 0))
lemma real_add_minus_iff:
(x + - a = 0) = (x = a)
lemma real_add_eq_0_iff:
(x + y = 0) = (y = - x)
lemma real_add_less_0_iff:
(x + y < 0) = (y < - x)
lemma real_0_less_add_iff:
(0 < x + y) = (- x < y)
lemma real_add_le_0_iff:
(x + y ≤ 0) = (y ≤ - x)
lemma real_0_le_add_iff:
(0 ≤ x + y) = (- x ≤ y)
lemma real_lbound_gt_zero:
[| 0 < d1.0; 0 < d2.0 |] ==> ∃e>0. e < d1.0 ∧ e < d2.0
lemma real_less_half_sum:
x < y ==> x < (x + y) / 2
lemma real_gt_half_sum:
x < y ==> (x + y) / 2 < y
lemma abs_minus_add_cancel:
¦x + - y¦ = ¦y + - x¦
lemma abs_interval_iff:
(¦x¦ < r) = (- r < x ∧ x < r)
lemma abs_le_interval_iff:
(¦x¦ ≤ r) = (- r ≤ x ∧ x ≤ r)
lemma abs_add_one_gt_zero:
0 < 1 + ¦x¦
lemma abs_real_of_nat_cancel:
¦real x¦ = real x
lemma abs_add_one_not_less_self:
¬ ¦x¦ + 1 < x
lemma abs_sum_triangle_ineq:
¦x + y + (- l + - m)¦ ≤ ¦x + - l¦ + ¦y + - m¦