author | paulson |
Tue, 10 Feb 2004 12:02:11 +0100 | |
changeset 14378 | 69c4d5997669 |
parent 14369 | c50188fe6366 |
child 14387 | e96d5c42c4b0 |
permissions | -rw-r--r-- |
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(* Title : Real/RealDef.thy |
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ID : $Id$ |
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Author : Jacques D. Fleuriot |
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Copyright : 1998 University of Cambridge |
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Description : The reals |
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*) |
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theory RealDef = PReal: |
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constdefs |
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realrel :: "((preal * preal) * (preal * preal)) set" |
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"realrel == {p. \<exists>x1 y1 x2 y2. p = ((x1,y1),(x2,y2)) & x1+y2 = x2+y1}" |
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typedef (REAL) real = "UNIV//realrel" |
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by (auto simp add: quotient_def) |
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instance real :: ord .. |
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instance real :: zero .. |
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instance real :: one .. |
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instance real :: plus .. |
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instance real :: times .. |
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instance real :: minus .. |
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instance real :: inverse .. |
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consts |
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(*Overloaded constant denoting the Real subset of enclosing |
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types such as hypreal and complex*) |
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Reals :: "'a set" |
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(*overloaded constant for injecting other types into "real"*) |
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real :: "'a => real" |
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defs (overloaded) |
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real_zero_def: |
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"0 == Abs_REAL(realrel``{(preal_of_rat 1, preal_of_rat 1)})" |
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real_one_def: |
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"1 == Abs_REAL(realrel`` |
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{(preal_of_rat 1 + preal_of_rat 1, |
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preal_of_rat 1)})" |
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real_minus_def: |
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"- R == Abs_REAL(UN (x,y):Rep_REAL(R). realrel``{(y,x)})" |
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real_diff_def: |
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"R - (S::real) == R + - S" |
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real_inverse_def: |
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"inverse (R::real) == (SOME S. (R = 0 & S = 0) | S * R = 1)" |
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real_divide_def: |
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"R / (S::real) == R * inverse S" |
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constdefs |
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(** these don't use the overloaded "real" function: users don't see them **) |
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real_of_preal :: "preal => real" |
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"real_of_preal m == |
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Abs_REAL(realrel``{(m + preal_of_rat 1, preal_of_rat 1)})" |
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defs (overloaded) |
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real_add_def: |
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"P+Q == Abs_REAL(\<Union>p1\<in>Rep_REAL(P). \<Union>p2\<in>Rep_REAL(Q). |
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(%(x1,y1). (%(x2,y2). realrel``{(x1+x2, y1+y2)}) p2) p1)" |
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real_mult_def: |
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"P*Q == Abs_REAL(\<Union>p1\<in>Rep_REAL(P). \<Union>p2\<in>Rep_REAL(Q). |
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(%(x1,y1). (%(x2,y2). realrel``{(x1*x2+y1*y2,x1*y2+x2*y1)}) |
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p2) p1)" |
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real_less_def: "(x < (y::real)) == (x \<le> y & x \<noteq> y)" |
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real_le_def: |
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"P \<le> (Q::real) == \<exists>x1 y1 x2 y2. x1 + y2 \<le> x2 + y1 & |
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(x1,y1) \<in> Rep_REAL(P) & (x2,y2) \<in> Rep_REAL(Q)" |
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real_abs_def: "abs (r::real) == (if 0 \<le> r then r else -r)" |
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syntax (xsymbols) |
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Reals :: "'a set" ("\<real>") |
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subsection{*Proving that realrel is an equivalence relation*} |
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lemma preal_trans_lemma: |
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assumes "x + y1 = x1 + y" |
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and "x + y2 = x2 + y" |
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shows "x1 + y2 = x2 + (y1::preal)" |
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proof - |
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have "(x1 + y2) + x = (x + y2) + x1" by (simp add: preal_add_ac) |
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also have "... = (x2 + y) + x1" by (simp add: prems) |
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also have "... = x2 + (x1 + y)" by (simp add: preal_add_ac) |
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also have "... = x2 + (x + y1)" by (simp add: prems) |
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also have "... = (x2 + y1) + x" by (simp add: preal_add_ac) |
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finally have "(x1 + y2) + x = (x2 + y1) + x" . |
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thus ?thesis by (simp add: preal_add_right_cancel_iff) |
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qed |
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lemma realrel_iff [simp]: "(((x1,y1),(x2,y2)): realrel) = (x1 + y2 = x2 + y1)" |
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by (unfold realrel_def, blast) |
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lemma realrel_refl: "(x,x): realrel" |
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apply (case_tac "x") |
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apply (simp add: realrel_def) |
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done |
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lemma equiv_realrel: "equiv UNIV realrel" |
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apply (auto simp add: equiv_def refl_def sym_def trans_def realrel_def) |
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apply (blast dest: preal_trans_lemma) |
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done |
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(* (realrel `` {x} = realrel `` {y}) = ((x,y) : realrel) *) |
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lemmas equiv_realrel_iff = |
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eq_equiv_class_iff [OF equiv_realrel UNIV_I UNIV_I] |
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declare equiv_realrel_iff [simp] |
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lemma realrel_in_real [simp]: "realrel``{(x,y)}: REAL" |
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by (unfold REAL_def realrel_def quotient_def, blast) |
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lemma inj_on_Abs_REAL: "inj_on Abs_REAL REAL" |
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apply (rule inj_on_inverseI) |
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apply (erule Abs_REAL_inverse) |
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done |
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declare inj_on_Abs_REAL [THEN inj_on_iff, simp] |
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declare Abs_REAL_inverse [simp] |
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lemmas eq_realrelD = equiv_realrel [THEN [2] eq_equiv_class] |
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lemma inj_Rep_REAL: "inj Rep_REAL" |
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apply (rule inj_on_inverseI) |
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apply (rule Rep_REAL_inverse) |
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done |
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(** real_of_preal: the injection from preal to real **) |
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lemma inj_real_of_preal: "inj(real_of_preal)" |
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apply (rule inj_onI) |
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apply (unfold real_of_preal_def) |
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apply (drule inj_on_Abs_REAL [THEN inj_onD]) |
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apply (rule realrel_in_real)+ |
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apply (drule eq_equiv_class) |
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apply (rule equiv_realrel, blast) |
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apply (simp add: realrel_def preal_add_right_cancel_iff) |
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done |
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lemma eq_Abs_REAL: |
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"(!!x y. z = Abs_REAL(realrel``{(x,y)}) ==> P) ==> P" |
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apply (rule_tac x1 = z in Rep_REAL [unfolded REAL_def, THEN quotientE]) |
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apply (drule_tac f = Abs_REAL in arg_cong) |
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apply (case_tac "x") |
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apply (simp add: Rep_REAL_inverse) |
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done |
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subsection{*Congruence property for addition*} |
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lemma real_add_congruent2_lemma: |
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"[|a + ba = aa + b; ab + bc = ac + bb|] |
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==> a + ab + (ba + bc) = aa + ac + (b + (bb::preal))" |
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apply (simp add: preal_add_assoc) |
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apply (rule preal_add_left_commute [of ab, THEN ssubst]) |
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apply (simp add: preal_add_assoc [symmetric]) |
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apply (simp add: preal_add_ac) |
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done |
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lemma real_add: |
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"Abs_REAL(realrel``{(x1,y1)}) + Abs_REAL(realrel``{(x2,y2)}) = |
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Abs_REAL(realrel``{(x1+x2, y1+y2)})" |
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apply (simp add: real_add_def UN_UN_split_split_eq) |
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apply (subst equiv_realrel [THEN UN_equiv_class2]) |
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apply (auto simp add: congruent2_def) |
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apply (blast intro: real_add_congruent2_lemma) |
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done |
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lemma real_add_commute: "(z::real) + w = w + z" |
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apply (rule eq_Abs_REAL [of z]) |
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apply (rule eq_Abs_REAL [of w]) |
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apply (simp add: preal_add_ac real_add) |
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done |
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lemma real_add_assoc: "((z1::real) + z2) + z3 = z1 + (z2 + z3)" |
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apply (rule eq_Abs_REAL [of z1]) |
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apply (rule eq_Abs_REAL [of z2]) |
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apply (rule eq_Abs_REAL [of z3]) |
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apply (simp add: real_add preal_add_assoc) |
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done |
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lemma real_add_zero_left: "(0::real) + z = z" |
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apply (unfold real_of_preal_def real_zero_def) |
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apply (rule eq_Abs_REAL [of z]) |
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apply (simp add: real_add preal_add_ac) |
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done |
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lemma real_add_zero_right: "z + (0::real) = z" |
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by (simp add: real_add_zero_left real_add_commute) |
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instance real :: plus_ac0 |
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by (intro_classes, |
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(assumption | |
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rule real_add_commute real_add_assoc real_add_zero_left)+) |
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subsection{*Additive Inverse on real*} |
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lemma real_minus_congruent: |
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"congruent realrel (%p. (%(x,y). realrel``{(y,x)}) p)" |
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apply (unfold congruent_def, clarify) |
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apply (simp add: preal_add_commute) |
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done |
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lemma real_minus: |
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"- (Abs_REAL(realrel``{(x,y)})) = Abs_REAL(realrel `` {(y,x)})" |
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apply (unfold real_minus_def) |
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apply (rule_tac f = Abs_REAL in arg_cong) |
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apply (simp add: realrel_in_real [THEN Abs_REAL_inverse] |
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UN_equiv_class [OF equiv_realrel real_minus_congruent]) |
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done |
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lemma real_add_minus_left: "(-z) + z = (0::real)" |
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apply (unfold real_zero_def) |
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apply (rule eq_Abs_REAL [of z]) |
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apply (simp add: real_minus real_add preal_add_commute) |
233 |
done |
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subsection{*Congruence property for multiplication*} |
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lemma real_mult_congruent2_lemma: |
239 |
"!!(x1::preal). [| x1 + y2 = x2 + y1 |] ==> |
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x * x1 + y * y1 + (x * y2 + x2 * y) = |
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x * x2 + y * y2 + (x * y1 + x1 * y)" |
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apply (simp add: preal_add_left_commute preal_add_assoc [symmetric] preal_add_mult_distrib2 [symmetric]) |
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apply (rule preal_mult_commute [THEN subst]) |
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apply (rule_tac y1 = x2 in preal_mult_commute [THEN subst]) |
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apply (simp add: preal_add_assoc preal_add_mult_distrib2 [symmetric]) |
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apply (simp add: preal_add_commute) |
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done |
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lemma real_mult_congruent2: |
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"congruent2 realrel (%p1 p2. |
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(%(x1,y1). (%(x2,y2). realrel``{(x1*x2 + y1*y2, x1*y2+x2*y1)}) p2) p1)" |
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apply (rule equiv_realrel [THEN congruent2_commuteI], clarify) |
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apply (unfold split_def) |
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apply (simp add: preal_mult_commute preal_add_commute) |
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apply (auto simp add: real_mult_congruent2_lemma) |
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done |
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lemma real_mult: |
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"Abs_REAL((realrel``{(x1,y1)})) * Abs_REAL((realrel``{(x2,y2)})) = |
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Abs_REAL(realrel `` {(x1*x2+y1*y2,x1*y2+x2*y1)})" |
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apply (unfold real_mult_def) |
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apply (simp add: equiv_realrel [THEN UN_equiv_class2] real_mult_congruent2) |
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done |
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lemma real_mult_commute: "(z::real) * w = w * z" |
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apply (rule eq_Abs_REAL [of z]) |
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apply (rule eq_Abs_REAL [of w]) |
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apply (simp add: real_mult preal_add_ac preal_mult_ac) |
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done |
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lemma real_mult_assoc: "((z1::real) * z2) * z3 = z1 * (z2 * z3)" |
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apply (rule eq_Abs_REAL [of z1]) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
273 |
apply (rule eq_Abs_REAL [of z2]) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
274 |
apply (rule eq_Abs_REAL [of z3]) |
14269 | 275 |
apply (simp add: preal_add_mult_distrib2 real_mult preal_add_ac preal_mult_ac) |
276 |
done |
|
277 |
||
278 |
lemma real_mult_1: "(1::real) * z = z" |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
279 |
apply (unfold real_one_def) |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
280 |
apply (rule eq_Abs_REAL [of z]) |
14334 | 281 |
apply (simp add: real_mult preal_add_mult_distrib2 preal_mult_1_right |
282 |
preal_mult_ac preal_add_ac) |
|
14269 | 283 |
done |
284 |
||
285 |
lemma real_add_mult_distrib: "((z1::real) + z2) * w = (z1 * w) + (z2 * w)" |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
286 |
apply (rule eq_Abs_REAL [of z1]) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
287 |
apply (rule eq_Abs_REAL [of z2]) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
288 |
apply (rule eq_Abs_REAL [of w]) |
14269 | 289 |
apply (simp add: preal_add_mult_distrib2 real_add real_mult preal_add_ac preal_mult_ac) |
290 |
done |
|
291 |
||
14329 | 292 |
text{*one and zero are distinct*} |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
293 |
lemma real_zero_not_eq_one: "0 \<noteq> (1::real)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
294 |
apply (subgoal_tac "preal_of_rat 1 < preal_of_rat 1 + preal_of_rat 1") |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
295 |
prefer 2 apply (simp add: preal_self_less_add_left) |
14269 | 296 |
apply (unfold real_zero_def real_one_def) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
297 |
apply (auto simp add: preal_add_right_cancel_iff) |
14269 | 298 |
done |
299 |
||
14329 | 300 |
subsection{*existence of inverse*} |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
301 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
302 |
lemma real_zero_iff: "Abs_REAL (realrel `` {(x, x)}) = 0" |
14269 | 303 |
apply (unfold real_zero_def) |
304 |
apply (auto simp add: preal_add_commute) |
|
305 |
done |
|
306 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
307 |
text{*Instead of using an existential quantifier and constructing the inverse |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
308 |
within the proof, we could define the inverse explicitly.*} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
309 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
310 |
lemma real_mult_inverse_left_ex: "x \<noteq> 0 ==> \<exists>y. y*x = (1::real)" |
14269 | 311 |
apply (unfold real_zero_def real_one_def) |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
312 |
apply (rule eq_Abs_REAL [of x]) |
14269 | 313 |
apply (cut_tac x = xa and y = y in linorder_less_linear) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
314 |
apply (auto dest!: less_add_left_Ex simp add: real_zero_iff) |
14334 | 315 |
apply (rule_tac |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
316 |
x = "Abs_REAL (realrel `` { (preal_of_rat 1, |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
317 |
inverse (D) + preal_of_rat 1)}) " |
14334 | 318 |
in exI) |
319 |
apply (rule_tac [2] |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
320 |
x = "Abs_REAL (realrel `` { (inverse (D) + preal_of_rat 1, |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
321 |
preal_of_rat 1)})" |
14334 | 322 |
in exI) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
323 |
apply (auto simp add: real_mult preal_mult_1_right |
14329 | 324 |
preal_add_mult_distrib2 preal_add_mult_distrib preal_mult_1 |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
325 |
preal_mult_inverse_right preal_add_ac preal_mult_ac) |
14269 | 326 |
done |
327 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
328 |
lemma real_mult_inverse_left: "x \<noteq> 0 ==> inverse(x)*x = (1::real)" |
14269 | 329 |
apply (unfold real_inverse_def) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
330 |
apply (frule real_mult_inverse_left_ex, safe) |
14269 | 331 |
apply (rule someI2, auto) |
332 |
done |
|
14334 | 333 |
|
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
334 |
|
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
335 |
subsection{*The Real Numbers form a Field*} |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
336 |
|
14334 | 337 |
instance real :: field |
338 |
proof |
|
339 |
fix x y z :: real |
|
340 |
show "(x + y) + z = x + (y + z)" by (rule real_add_assoc) |
|
341 |
show "x + y = y + x" by (rule real_add_commute) |
|
342 |
show "0 + x = x" by simp |
|
343 |
show "- x + x = 0" by (rule real_add_minus_left) |
|
344 |
show "x - y = x + (-y)" by (simp add: real_diff_def) |
|
345 |
show "(x * y) * z = x * (y * z)" by (rule real_mult_assoc) |
|
346 |
show "x * y = y * x" by (rule real_mult_commute) |
|
347 |
show "1 * x = x" by (rule real_mult_1) |
|
348 |
show "(x + y) * z = x * z + y * z" by (simp add: real_add_mult_distrib) |
|
349 |
show "0 \<noteq> (1::real)" by (rule real_zero_not_eq_one) |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
350 |
show "x \<noteq> 0 ==> inverse x * x = 1" by (rule real_mult_inverse_left) |
14334 | 351 |
show "y \<noteq> 0 ==> x / y = x * inverse y" by (simp add: real_divide_def) |
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
352 |
assume eq: "z+x = z+y" |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
353 |
hence "(-z + z) + x = (-z + z) + y" by (simp only: eq real_add_assoc) |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
354 |
thus "x = y" by (simp add: real_add_minus_left) |
14334 | 355 |
qed |
356 |
||
357 |
||
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
358 |
text{*Inverse of zero! Useful to simplify certain equations*} |
14269 | 359 |
|
14334 | 360 |
lemma INVERSE_ZERO: "inverse 0 = (0::real)" |
361 |
apply (unfold real_inverse_def) |
|
362 |
apply (rule someI2) |
|
363 |
apply (auto simp add: zero_neq_one) |
|
14269 | 364 |
done |
14334 | 365 |
|
366 |
lemma DIVISION_BY_ZERO: "a / (0::real) = 0" |
|
367 |
by (simp add: real_divide_def INVERSE_ZERO) |
|
368 |
||
369 |
instance real :: division_by_zero |
|
370 |
proof |
|
371 |
fix x :: real |
|
372 |
show "inverse 0 = (0::real)" by (rule INVERSE_ZERO) |
|
373 |
show "x/0 = 0" by (rule DIVISION_BY_ZERO) |
|
374 |
qed |
|
375 |
||
376 |
||
377 |
(*Pull negations out*) |
|
378 |
declare minus_mult_right [symmetric, simp] |
|
379 |
minus_mult_left [symmetric, simp] |
|
380 |
||
381 |
lemma real_mult_1_right: "z * (1::real) = z" |
|
382 |
by (rule Ring_and_Field.mult_1_right) |
|
14269 | 383 |
|
384 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
385 |
subsection{*The @{text "\<le>"} Ordering*} |
14269 | 386 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
387 |
lemma real_le_refl: "w \<le> (w::real)" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
388 |
apply (rule eq_Abs_REAL [of w]) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
389 |
apply (force simp add: real_le_def) |
14269 | 390 |
done |
391 |
||
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
392 |
text{*The arithmetic decision procedure is not set up for type preal. |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
393 |
This lemma is currently unused, but it could simplify the proofs of the |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
394 |
following two lemmas.*} |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
395 |
lemma preal_eq_le_imp_le: |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
396 |
assumes eq: "a+b = c+d" and le: "c \<le> a" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
397 |
shows "b \<le> (d::preal)" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
398 |
proof - |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
399 |
have "c+d \<le> a+d" by (simp add: prems preal_cancels) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
400 |
hence "a+b \<le> a+d" by (simp add: prems) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
401 |
thus "b \<le> d" by (simp add: preal_cancels) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
402 |
qed |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
403 |
|
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
404 |
lemma real_le_lemma: |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
405 |
assumes l: "u1 + v2 \<le> u2 + v1" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
406 |
and "x1 + v1 = u1 + y1" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
407 |
and "x2 + v2 = u2 + y2" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
408 |
shows "x1 + y2 \<le> x2 + (y1::preal)" |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
409 |
proof - |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
410 |
have "(x1+v1) + (u2+y2) = (u1+y1) + (x2+v2)" by (simp add: prems) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
411 |
hence "(x1+y2) + (u2+v1) = (x2+y1) + (u1+v2)" by (simp add: preal_add_ac) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
412 |
also have "... \<le> (x2+y1) + (u2+v1)" |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
413 |
by (simp add: prems preal_add_le_cancel_left) |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
414 |
finally show ?thesis by (simp add: preal_add_le_cancel_right) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
415 |
qed |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
416 |
|
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
417 |
lemma real_le: |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
418 |
"(Abs_REAL(realrel``{(x1,y1)}) \<le> Abs_REAL(realrel``{(x2,y2)})) = |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
419 |
(x1 + y2 \<le> x2 + y1)" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
420 |
apply (simp add: real_le_def) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
421 |
apply (auto intro: real_le_lemma); |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
422 |
done |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
423 |
|
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
424 |
lemma real_le_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::real)" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
425 |
apply (rule eq_Abs_REAL [of z]) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
426 |
apply (rule eq_Abs_REAL [of w]) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
427 |
apply (simp add: real_le order_antisym) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
428 |
done |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
429 |
|
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
430 |
lemma real_trans_lemma: |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
431 |
assumes "x + v \<le> u + y" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
432 |
and "u + v' \<le> u' + v" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
433 |
and "x2 + v2 = u2 + y2" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
434 |
shows "x + v' \<le> u' + (y::preal)" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
435 |
proof - |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
436 |
have "(x+v') + (u+v) = (x+v) + (u+v')" by (simp add: preal_add_ac) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
437 |
also have "... \<le> (u+y) + (u+v')" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
438 |
by (simp add: preal_add_le_cancel_right prems) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
439 |
also have "... \<le> (u+y) + (u'+v)" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
440 |
by (simp add: preal_add_le_cancel_left prems) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
441 |
also have "... = (u'+y) + (u+v)" by (simp add: preal_add_ac) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
442 |
finally show ?thesis by (simp add: preal_add_le_cancel_right) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
443 |
qed |
14269 | 444 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
445 |
lemma real_le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::real)" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
446 |
apply (rule eq_Abs_REAL [of i]) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
447 |
apply (rule eq_Abs_REAL [of j]) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
448 |
apply (rule eq_Abs_REAL [of k]) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
449 |
apply (simp add: real_le) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
450 |
apply (blast intro: real_trans_lemma) |
14334 | 451 |
done |
452 |
||
453 |
(* Axiom 'order_less_le' of class 'order': *) |
|
454 |
lemma real_less_le: "((w::real) < z) = (w \<le> z & w \<noteq> z)" |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
455 |
by (simp add: real_less_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
456 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
457 |
instance real :: order |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
458 |
proof qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
459 |
(assumption | |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
460 |
rule real_le_refl real_le_trans real_le_anti_sym real_less_le)+ |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
461 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
462 |
(* Axiom 'linorder_linear' of class 'linorder': *) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
463 |
lemma real_le_linear: "(z::real) \<le> w | w \<le> z" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
464 |
apply (rule eq_Abs_REAL [of z]) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
465 |
apply (rule eq_Abs_REAL [of w]) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
466 |
apply (auto simp add: real_le real_zero_def preal_add_ac preal_cancels) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
467 |
apply (cut_tac x="x+ya" and y="xa+y" in linorder_linear) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
468 |
apply (auto ); |
14334 | 469 |
done |
470 |
||
471 |
||
472 |
instance real :: linorder |
|
473 |
by (intro_classes, rule real_le_linear) |
|
474 |
||
475 |
||
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
476 |
lemma real_le_eq_diff: "(x \<le> y) = (x-y \<le> (0::real))" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
477 |
apply (rule eq_Abs_REAL [of x]) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
478 |
apply (rule eq_Abs_REAL [of y]) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
479 |
apply (auto simp add: real_le real_zero_def real_diff_def real_add real_minus |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
480 |
preal_add_ac) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
481 |
apply (simp_all add: preal_add_assoc [symmetric] preal_cancels) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
482 |
done |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
483 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
484 |
lemma real_add_left_mono: "x \<le> y ==> z + x \<le> z + (y::real)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
485 |
apply (auto simp add: real_le_eq_diff [of x] real_le_eq_diff [of "z+x"]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
486 |
apply (subgoal_tac "z + x - (z + y) = (z + -z) + (x - y)") |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
487 |
prefer 2 apply (simp add: diff_minus add_ac, simp) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
488 |
done |
14334 | 489 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
490 |
lemma real_sum_gt_zero_less: "(0 < S + (-W::real)) ==> (W < S)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
491 |
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
492 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
493 |
lemma real_less_sum_gt_zero: "(W < S) ==> (0 < S + (-W::real))" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
494 |
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus) |
14334 | 495 |
|
496 |
lemma real_mult_order: "[| 0 < x; 0 < y |] ==> (0::real) < x * y" |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
497 |
apply (rule eq_Abs_REAL [of x]) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
498 |
apply (rule eq_Abs_REAL [of y]) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
499 |
apply (simp add: linorder_not_le [where 'a = real, symmetric] |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
500 |
linorder_not_le [where 'a = preal] |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
501 |
real_zero_def real_le real_mult) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
502 |
--{*Reduce to the (simpler) @{text "\<le>"} relation *} |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
503 |
apply (auto dest!: less_add_left_Ex |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
504 |
simp add: preal_add_ac preal_mult_ac |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
505 |
preal_add_mult_distrib2 preal_cancels preal_self_less_add_right) |
14334 | 506 |
done |
507 |
||
508 |
lemma real_mult_less_mono2: "[| (0::real) < z; x < y |] ==> z * x < z * y" |
|
509 |
apply (rule real_sum_gt_zero_less) |
|
510 |
apply (drule real_less_sum_gt_zero [of x y]) |
|
511 |
apply (drule real_mult_order, assumption) |
|
512 |
apply (simp add: right_distrib) |
|
513 |
done |
|
514 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
515 |
text{*lemma for proving @{term "0<(1::real)"}*} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
516 |
lemma real_zero_le_one: "0 \<le> (1::real)" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
517 |
apply (simp add: real_zero_def real_one_def real_le |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
518 |
preal_self_less_add_left order_less_imp_le) |
14334 | 519 |
done |
520 |
||
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
521 |
|
14334 | 522 |
subsection{*The Reals Form an Ordered Field*} |
523 |
||
524 |
instance real :: ordered_field |
|
525 |
proof |
|
526 |
fix x y z :: real |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
527 |
show "0 < (1::real)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
528 |
by (simp add: real_less_def real_zero_le_one real_zero_not_eq_one) |
14334 | 529 |
show "x \<le> y ==> z + x \<le> z + y" by (rule real_add_left_mono) |
530 |
show "x < y ==> 0 < z ==> z * x < z * y" by (simp add: real_mult_less_mono2) |
|
531 |
show "\<bar>x\<bar> = (if x < 0 then -x else x)" |
|
532 |
by (auto dest: order_le_less_trans simp add: real_abs_def linorder_not_le) |
|
533 |
qed |
|
534 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
535 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
536 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
537 |
text{*The function @{term real_of_preal} requires many proofs, but it seems |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
538 |
to be essential for proving completeness of the reals from that of the |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
539 |
positive reals.*} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
540 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
541 |
lemma real_of_preal_add: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
542 |
"real_of_preal ((x::preal) + y) = real_of_preal x + real_of_preal y" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
543 |
by (simp add: real_of_preal_def real_add preal_add_mult_distrib preal_mult_1 |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
544 |
preal_add_ac) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
545 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
546 |
lemma real_of_preal_mult: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
547 |
"real_of_preal ((x::preal) * y) = real_of_preal x* real_of_preal y" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
548 |
by (simp add: real_of_preal_def real_mult preal_add_mult_distrib2 |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
549 |
preal_mult_1 preal_mult_1_right preal_add_ac preal_mult_ac) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
550 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
551 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
552 |
text{*Gleason prop 9-4.4 p 127*} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
553 |
lemma real_of_preal_trichotomy: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
554 |
"\<exists>m. (x::real) = real_of_preal m | x = 0 | x = -(real_of_preal m)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
555 |
apply (unfold real_of_preal_def real_zero_def) |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
556 |
apply (rule eq_Abs_REAL [of x]) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
557 |
apply (auto simp add: real_minus preal_add_ac) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
558 |
apply (cut_tac x = x and y = y in linorder_less_linear) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
559 |
apply (auto dest!: less_add_left_Ex simp add: preal_add_assoc [symmetric]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
560 |
apply (auto simp add: preal_add_commute) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
561 |
done |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
562 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
563 |
lemma real_of_preal_leD: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
564 |
"real_of_preal m1 \<le> real_of_preal m2 ==> m1 \<le> m2" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
565 |
apply (unfold real_of_preal_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
566 |
apply (auto simp add: real_le_def preal_add_ac) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
567 |
apply (auto simp add: preal_add_assoc [symmetric] preal_add_right_cancel_iff) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
568 |
apply (auto simp add: preal_add_ac preal_add_le_cancel_left) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
569 |
done |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
570 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
571 |
lemma real_of_preal_lessI: "m1 < m2 ==> real_of_preal m1 < real_of_preal m2" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
572 |
by (auto simp add: real_of_preal_leD linorder_not_le [symmetric]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
573 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
574 |
lemma real_of_preal_lessD: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
575 |
"real_of_preal m1 < real_of_preal m2 ==> m1 < m2" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
576 |
apply (auto simp add: real_less_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
577 |
apply (drule real_of_preal_leD) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
578 |
apply (auto simp add: order_le_less) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
579 |
done |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
580 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
581 |
lemma real_of_preal_less_iff [simp]: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
582 |
"(real_of_preal m1 < real_of_preal m2) = (m1 < m2)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
583 |
by (blast intro: real_of_preal_lessI real_of_preal_lessD) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
584 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
585 |
lemma real_of_preal_le_iff: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
586 |
"(real_of_preal m1 \<le> real_of_preal m2) = (m1 \<le> m2)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
587 |
by (simp add: linorder_not_less [symmetric]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
588 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
589 |
lemma real_of_preal_zero_less: "0 < real_of_preal m" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
590 |
apply (auto simp add: real_zero_def real_of_preal_def real_less_def real_le_def |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
591 |
preal_add_ac preal_cancels) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
592 |
apply (simp_all add: preal_add_assoc [symmetric] preal_cancels) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
593 |
apply (blast intro: preal_self_less_add_left order_less_imp_le) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
594 |
apply (insert preal_not_eq_self [of "preal_of_rat 1" m]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
595 |
apply (simp add: preal_add_ac) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
596 |
done |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
597 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
598 |
lemma real_of_preal_minus_less_zero: "- real_of_preal m < 0" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
599 |
by (simp add: real_of_preal_zero_less) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
600 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
601 |
lemma real_of_preal_not_minus_gt_zero: "~ 0 < - real_of_preal m" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
602 |
apply (cut_tac real_of_preal_minus_less_zero) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
603 |
apply (fast dest: order_less_trans) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
604 |
done |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
605 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
606 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
607 |
subsection{*Theorems About the Ordering*} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
608 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
609 |
text{*obsolete but used a lot*} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
610 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
611 |
lemma real_not_refl2: "x < y ==> x \<noteq> (y::real)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
612 |
by blast |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
613 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
614 |
lemma real_le_imp_less_or_eq: "!!(x::real). x \<le> y ==> x < y | x = y" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
615 |
by (simp add: order_le_less) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
616 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
617 |
lemma real_gt_zero_preal_Ex: "(0 < x) = (\<exists>y. x = real_of_preal y)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
618 |
apply (auto simp add: real_of_preal_zero_less) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
619 |
apply (cut_tac x = x in real_of_preal_trichotomy) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
620 |
apply (blast elim!: real_of_preal_not_minus_gt_zero [THEN notE]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
621 |
done |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
622 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
623 |
lemma real_gt_preal_preal_Ex: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
624 |
"real_of_preal z < x ==> \<exists>y. x = real_of_preal y" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
625 |
by (blast dest!: real_of_preal_zero_less [THEN order_less_trans] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
626 |
intro: real_gt_zero_preal_Ex [THEN iffD1]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
627 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
628 |
lemma real_ge_preal_preal_Ex: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
629 |
"real_of_preal z \<le> x ==> \<exists>y. x = real_of_preal y" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
630 |
by (blast dest: order_le_imp_less_or_eq real_gt_preal_preal_Ex) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
631 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
632 |
lemma real_less_all_preal: "y \<le> 0 ==> \<forall>x. y < real_of_preal x" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
633 |
by (auto elim: order_le_imp_less_or_eq [THEN disjE] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
634 |
intro: real_of_preal_zero_less [THEN [2] order_less_trans] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
635 |
simp add: real_of_preal_zero_less) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
636 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
637 |
lemma real_less_all_real2: "~ 0 < y ==> \<forall>x. y < real_of_preal x" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
638 |
by (blast intro!: real_less_all_preal linorder_not_less [THEN iffD1]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
639 |
|
14334 | 640 |
lemma real_add_less_le_mono: "[| w'<w; z'\<le>z |] ==> w' + z' < w + (z::real)" |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
641 |
by (rule Ring_and_Field.add_less_le_mono) |
14334 | 642 |
|
643 |
lemma real_add_le_less_mono: |
|
644 |
"!!z z'::real. [| w'\<le>w; z'<z |] ==> w' + z' < w + z" |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
645 |
by (rule Ring_and_Field.add_le_less_mono) |
14334 | 646 |
|
647 |
lemma real_zero_less_one: "0 < (1::real)" |
|
648 |
by (rule Ring_and_Field.zero_less_one) |
|
649 |
||
650 |
lemma real_le_square [simp]: "(0::real) \<le> x*x" |
|
651 |
by (rule Ring_and_Field.zero_le_square) |
|
652 |
||
653 |
||
654 |
subsection{*More Lemmas*} |
|
655 |
||
656 |
lemma real_mult_left_cancel: "(c::real) \<noteq> 0 ==> (c*a=c*b) = (a=b)" |
|
657 |
by auto |
|
658 |
||
659 |
lemma real_mult_right_cancel: "(c::real) \<noteq> 0 ==> (a*c=b*c) = (a=b)" |
|
660 |
by auto |
|
661 |
||
662 |
text{*The precondition could be weakened to @{term "0\<le>x"}*} |
|
663 |
lemma real_mult_less_mono: |
|
664 |
"[| u<v; x<y; (0::real) < v; 0 < x |] ==> u*x < v* y" |
|
665 |
by (simp add: Ring_and_Field.mult_strict_mono order_less_imp_le) |
|
666 |
||
667 |
lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)" |
|
668 |
by (force elim: order_less_asym |
|
669 |
simp add: Ring_and_Field.mult_less_cancel_right) |
|
670 |
||
671 |
lemma real_mult_le_cancel_iff1 [simp]: "(0::real) < z ==> (x*z \<le> y*z) = (x\<le>y)" |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
672 |
apply (simp add: mult_le_cancel_right) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
673 |
apply (blast intro: elim: order_less_asym) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
674 |
done |
14334 | 675 |
|
676 |
lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x \<le> z*y) = (x\<le>y)" |
|
677 |
by (force elim: order_less_asym |
|
678 |
simp add: Ring_and_Field.mult_le_cancel_left) |
|
679 |
||
680 |
text{*Only two uses?*} |
|
681 |
lemma real_mult_less_mono': |
|
682 |
"[| x < y; r1 < r2; (0::real) \<le> r1; 0 \<le> x|] ==> r1 * x < r2 * y" |
|
683 |
by (rule Ring_and_Field.mult_strict_mono') |
|
684 |
||
685 |
text{*FIXME: delete or at least combine the next two lemmas*} |
|
686 |
lemma real_sum_squares_cancel: "x * x + y * y = 0 ==> x = (0::real)" |
|
687 |
apply (drule Ring_and_Field.equals_zero_I [THEN sym]) |
|
688 |
apply (cut_tac x = y in real_le_square) |
|
689 |
apply (auto, drule real_le_anti_sym, auto) |
|
690 |
done |
|
691 |
||
692 |
lemma real_sum_squares_cancel2: "x * x + y * y = 0 ==> y = (0::real)" |
|
693 |
apply (rule_tac y = x in real_sum_squares_cancel) |
|
694 |
apply (simp add: real_add_commute) |
|
695 |
done |
|
696 |
||
697 |
lemma real_add_order: "[| 0 < x; 0 < y |] ==> (0::real) < x + y" |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
698 |
by (drule add_strict_mono [of concl: 0 0], assumption, simp) |
14334 | 699 |
|
700 |
lemma real_le_add_order: "[| 0 \<le> x; 0 \<le> y |] ==> (0::real) \<le> x + y" |
|
701 |
apply (drule order_le_imp_less_or_eq)+ |
|
702 |
apply (auto intro: real_add_order order_less_imp_le) |
|
703 |
done |
|
704 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
705 |
lemma real_inverse_unique: "x*y = (1::real) ==> y = inverse x" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
706 |
apply (case_tac "x \<noteq> 0") |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
707 |
apply (rule_tac c1 = x in real_mult_left_cancel [THEN iffD1], auto) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
708 |
done |
14334 | 709 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
710 |
lemma real_inverse_gt_one: "[| (0::real) < x; x < 1 |] ==> 1 < inverse x" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
711 |
by (auto dest: less_imp_inverse_less) |
14334 | 712 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
713 |
lemma real_mult_self_sum_ge_zero: "(0::real) \<le> x*x + y*y" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
714 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
715 |
have "0 + 0 \<le> x*x + y*y" by (blast intro: add_mono zero_le_square) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
716 |
thus ?thesis by simp |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
717 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
718 |
|
14334 | 719 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
720 |
subsection{*Embedding the Integers into the Reals*} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
721 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
722 |
defs (overloaded) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
723 |
real_of_nat_def: "real z == of_nat z" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
724 |
real_of_int_def: "real z == of_int z" |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
725 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
726 |
lemma real_of_int_zero [simp]: "real (0::int) = 0" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
727 |
by (simp add: real_of_int_def) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
728 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
729 |
lemma real_of_one [simp]: "real (1::int) = (1::real)" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
730 |
by (simp add: real_of_int_def) |
14334 | 731 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
732 |
lemma real_of_int_add: "real (x::int) + real y = real (x + y)" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
733 |
by (simp add: real_of_int_def) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
734 |
declare real_of_int_add [symmetric, simp] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
735 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
736 |
lemma real_of_int_minus: "-real (x::int) = real (-x)" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
737 |
by (simp add: real_of_int_def) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
738 |
declare real_of_int_minus [symmetric, simp] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
739 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
740 |
lemma real_of_int_diff: "real (x::int) - real y = real (x - y)" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
741 |
by (simp add: real_of_int_def) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
742 |
declare real_of_int_diff [symmetric, simp] |
14334 | 743 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
744 |
lemma real_of_int_mult: "real (x::int) * real y = real (x * y)" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
745 |
by (simp add: real_of_int_def) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
746 |
declare real_of_int_mult [symmetric, simp] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
747 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
748 |
lemma real_of_int_zero_cancel [simp]: "(real x = 0) = (x = (0::int))" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
749 |
by (simp add: real_of_int_def) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
750 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
751 |
lemma real_of_int_inject [iff]: "(real (x::int) = real y) = (x = y)" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
752 |
by (simp add: real_of_int_def) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
753 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
754 |
lemma real_of_int_less_iff [iff]: "(real (x::int) < real y) = (x < y)" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
755 |
by (simp add: real_of_int_def) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
756 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
757 |
lemma real_of_int_le_iff [simp]: "(real (x::int) \<le> real y) = (x \<le> y)" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
758 |
by (simp add: real_of_int_def) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
759 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
760 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
761 |
subsection{*Embedding the Naturals into the Reals*} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
762 |
|
14334 | 763 |
lemma real_of_nat_zero [simp]: "real (0::nat) = 0" |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
764 |
by (simp add: real_of_nat_def) |
14334 | 765 |
|
766 |
lemma real_of_nat_one [simp]: "real (Suc 0) = (1::real)" |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
767 |
by (simp add: real_of_nat_def) |
14334 | 768 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
769 |
lemma real_of_nat_add [simp]: "real (m + n) = real (m::nat) + real n" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
770 |
by (simp add: real_of_nat_def) |
14334 | 771 |
|
772 |
(*Not for addsimps: often the LHS is used to represent a positive natural*) |
|
773 |
lemma real_of_nat_Suc: "real (Suc n) = real n + (1::real)" |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
774 |
by (simp add: real_of_nat_def) |
14334 | 775 |
|
776 |
lemma real_of_nat_less_iff [iff]: |
|
777 |
"(real (n::nat) < real m) = (n < m)" |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
778 |
by (simp add: real_of_nat_def) |
14334 | 779 |
|
780 |
lemma real_of_nat_le_iff [iff]: "(real (n::nat) \<le> real m) = (n \<le> m)" |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
781 |
by (simp add: real_of_nat_def) |
14334 | 782 |
|
783 |
lemma real_of_nat_ge_zero [iff]: "0 \<le> real (n::nat)" |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
784 |
by (simp add: real_of_nat_def zero_le_imp_of_nat) |
14334 | 785 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
786 |
lemma real_of_nat_Suc_gt_zero: "0 < real (Suc n)" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
787 |
by (simp add: real_of_nat_def del: of_nat_Suc) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
788 |
|
14334 | 789 |
lemma real_of_nat_mult [simp]: "real (m * n) = real (m::nat) * real n" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
790 |
by (simp add: real_of_nat_def) |
14334 | 791 |
|
792 |
lemma real_of_nat_inject [iff]: "(real (n::nat) = real m) = (n = m)" |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
793 |
by (simp add: real_of_nat_def) |
14334 | 794 |
|
795 |
lemma real_of_nat_zero_iff: "(real (n::nat) = 0) = (n = 0)" |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
796 |
by (simp add: real_of_nat_def) |
14334 | 797 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
798 |
lemma real_of_nat_diff: "n \<le> m ==> real (m - n) = real (m::nat) - real n" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
799 |
by (simp add: add: real_of_nat_def) |
14334 | 800 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
801 |
lemma real_of_nat_gt_zero_cancel_iff [simp]: "(0 < real (n::nat)) = (0 < n)" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
802 |
by (simp add: add: real_of_nat_def) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
803 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
804 |
lemma real_of_nat_le_zero_cancel_iff [simp]: "(real (n::nat) \<le> 0) = (n = 0)" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
805 |
by (simp add: add: real_of_nat_def) |
14334 | 806 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
807 |
lemma not_real_of_nat_less_zero [simp]: "~ real (n::nat) < 0" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
808 |
by (simp add: add: real_of_nat_def) |
14334 | 809 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
810 |
lemma real_of_nat_ge_zero_cancel_iff [simp]: "(0 \<le> real (n::nat)) = (0 \<le> n)" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
811 |
by (simp add: add: real_of_nat_def) |
14334 | 812 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
813 |
lemma real_of_int_real_of_nat: "real (int n) = real n" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
814 |
by (simp add: real_of_nat_def real_of_int_def int_eq_of_nat) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
815 |
|
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
816 |
|
14334 | 817 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
818 |
text{*Still needed for binary arith*} |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
819 |
lemma real_of_nat_real_of_int: "~neg z ==> real (nat z) = real z" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
820 |
proof (simp add: not_neg_eq_ge_0 real_of_nat_def real_of_int_def) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
821 |
assume "0 \<le> z" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
822 |
hence eq: "of_nat (nat z) = z" |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
823 |
by (simp add: nat_0_le int_eq_of_nat[symmetric]) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
824 |
have "of_nat (nat z) = of_int (of_nat (nat z))" by simp |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
825 |
also have "... = of_int z" by (simp add: eq) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
826 |
finally show "of_nat (nat z) = of_int z" . |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset
|
827 |
qed |
14334 | 828 |
|
829 |
ML |
|
830 |
{* |
|
831 |
val real_abs_def = thm "real_abs_def"; |
|
832 |
||
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
833 |
val real_le_def = thm "real_le_def"; |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
834 |
val real_diff_def = thm "real_diff_def"; |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
835 |
val real_divide_def = thm "real_divide_def"; |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
836 |
|
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
837 |
val realrel_iff = thm"realrel_iff"; |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
838 |
val realrel_refl = thm"realrel_refl"; |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
839 |
val equiv_realrel = thm"equiv_realrel"; |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
840 |
val equiv_realrel_iff = thm"equiv_realrel_iff"; |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
841 |
val realrel_in_real = thm"realrel_in_real"; |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
842 |
val inj_on_Abs_REAL = thm"inj_on_Abs_REAL"; |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
843 |
val eq_realrelD = thm"eq_realrelD"; |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
844 |
val inj_Rep_REAL = thm"inj_Rep_REAL"; |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
845 |
val inj_real_of_preal = thm"inj_real_of_preal"; |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
846 |
val eq_Abs_REAL = thm"eq_Abs_REAL"; |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
847 |
val real_minus_congruent = thm"real_minus_congruent"; |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
848 |
val real_minus = thm"real_minus"; |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
849 |
val real_add = thm"real_add"; |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
850 |
val real_add_commute = thm"real_add_commute"; |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
851 |
val real_add_assoc = thm"real_add_assoc"; |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
852 |
val real_add_zero_left = thm"real_add_zero_left"; |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
853 |
val real_add_zero_right = thm"real_add_zero_right"; |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
854 |
|
14334 | 855 |
val real_mult = thm"real_mult"; |
856 |
val real_mult_commute = thm"real_mult_commute"; |
|
857 |
val real_mult_assoc = thm"real_mult_assoc"; |
|
858 |
val real_mult_1 = thm"real_mult_1"; |
|
859 |
val real_mult_1_right = thm"real_mult_1_right"; |
|
860 |
val preal_le_linear = thm"preal_le_linear"; |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
861 |
val real_mult_inverse_left = thm"real_mult_inverse_left"; |
14334 | 862 |
val real_not_refl2 = thm"real_not_refl2"; |
863 |
val real_of_preal_add = thm"real_of_preal_add"; |
|
864 |
val real_of_preal_mult = thm"real_of_preal_mult"; |
|
865 |
val real_of_preal_trichotomy = thm"real_of_preal_trichotomy"; |
|
866 |
val real_of_preal_minus_less_zero = thm"real_of_preal_minus_less_zero"; |
|
867 |
val real_of_preal_not_minus_gt_zero = thm"real_of_preal_not_minus_gt_zero"; |
|
868 |
val real_of_preal_zero_less = thm"real_of_preal_zero_less"; |
|
869 |
val real_le_imp_less_or_eq = thm"real_le_imp_less_or_eq"; |
|
870 |
val real_le_refl = thm"real_le_refl"; |
|
871 |
val real_le_linear = thm"real_le_linear"; |
|
872 |
val real_le_trans = thm"real_le_trans"; |
|
873 |
val real_le_anti_sym = thm"real_le_anti_sym"; |
|
874 |
val real_less_le = thm"real_less_le"; |
|
875 |
val real_less_sum_gt_zero = thm"real_less_sum_gt_zero"; |
|
876 |
val real_gt_zero_preal_Ex = thm "real_gt_zero_preal_Ex"; |
|
877 |
val real_gt_preal_preal_Ex = thm "real_gt_preal_preal_Ex"; |
|
878 |
val real_ge_preal_preal_Ex = thm "real_ge_preal_preal_Ex"; |
|
879 |
val real_less_all_preal = thm "real_less_all_preal"; |
|
880 |
val real_less_all_real2 = thm "real_less_all_real2"; |
|
881 |
val real_of_preal_le_iff = thm "real_of_preal_le_iff"; |
|
882 |
val real_mult_order = thm "real_mult_order"; |
|
883 |
val real_zero_less_one = thm "real_zero_less_one"; |
|
884 |
val real_add_less_le_mono = thm "real_add_less_le_mono"; |
|
885 |
val real_add_le_less_mono = thm "real_add_le_less_mono"; |
|
886 |
val real_add_order = thm "real_add_order"; |
|
887 |
val real_le_add_order = thm "real_le_add_order"; |
|
888 |
val real_le_square = thm "real_le_square"; |
|
889 |
val real_mult_less_mono2 = thm "real_mult_less_mono2"; |
|
890 |
||
891 |
val real_mult_less_iff1 = thm "real_mult_less_iff1"; |
|
892 |
val real_mult_le_cancel_iff1 = thm "real_mult_le_cancel_iff1"; |
|
893 |
val real_mult_le_cancel_iff2 = thm "real_mult_le_cancel_iff2"; |
|
894 |
val real_mult_less_mono = thm "real_mult_less_mono"; |
|
895 |
val real_mult_less_mono' = thm "real_mult_less_mono'"; |
|
896 |
val real_sum_squares_cancel = thm "real_sum_squares_cancel"; |
|
897 |
val real_sum_squares_cancel2 = thm "real_sum_squares_cancel2"; |
|
898 |
||
899 |
val real_mult_left_cancel = thm"real_mult_left_cancel"; |
|
900 |
val real_mult_right_cancel = thm"real_mult_right_cancel"; |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
901 |
val real_inverse_unique = thm "real_inverse_unique"; |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
902 |
val real_inverse_gt_one = thm "real_inverse_gt_one"; |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
903 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
904 |
val real_of_int_zero = thm"real_of_int_zero"; |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
905 |
val real_of_one = thm"real_of_one"; |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
906 |
val real_of_int_add = thm"real_of_int_add"; |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
907 |
val real_of_int_minus = thm"real_of_int_minus"; |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
908 |
val real_of_int_diff = thm"real_of_int_diff"; |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
909 |
val real_of_int_mult = thm"real_of_int_mult"; |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
910 |
val real_of_int_real_of_nat = thm"real_of_int_real_of_nat"; |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
911 |
val real_of_int_inject = thm"real_of_int_inject"; |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
912 |
val real_of_int_less_iff = thm"real_of_int_less_iff"; |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
913 |
val real_of_int_le_iff = thm"real_of_int_le_iff"; |
14334 | 914 |
val real_of_nat_zero = thm "real_of_nat_zero"; |
915 |
val real_of_nat_one = thm "real_of_nat_one"; |
|
916 |
val real_of_nat_add = thm "real_of_nat_add"; |
|
917 |
val real_of_nat_Suc = thm "real_of_nat_Suc"; |
|
918 |
val real_of_nat_less_iff = thm "real_of_nat_less_iff"; |
|
919 |
val real_of_nat_le_iff = thm "real_of_nat_le_iff"; |
|
920 |
val real_of_nat_ge_zero = thm "real_of_nat_ge_zero"; |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset
|
921 |
val real_of_nat_Suc_gt_zero = thm "real_of_nat_Suc_gt_zero"; |
14334 | 922 |
val real_of_nat_mult = thm "real_of_nat_mult"; |
923 |
val real_of_nat_inject = thm "real_of_nat_inject"; |
|
924 |
val real_of_nat_diff = thm "real_of_nat_diff"; |
|
925 |
val real_of_nat_zero_iff = thm "real_of_nat_zero_iff"; |
|
926 |
val real_of_nat_gt_zero_cancel_iff = thm "real_of_nat_gt_zero_cancel_iff"; |
|
927 |
val real_of_nat_le_zero_cancel_iff = thm "real_of_nat_le_zero_cancel_iff"; |
|
928 |
val not_real_of_nat_less_zero = thm "not_real_of_nat_less_zero"; |
|
929 |
val real_of_nat_ge_zero_cancel_iff = thm "real_of_nat_ge_zero_cancel_iff"; |
|
930 |
*} |
|
10752
c4f1bf2acf4c
tidying, and separation of HOL-Hyperreal from HOL-Real
paulson
parents:
10648
diff
changeset
|
931 |
|
5588 | 932 |
end |