author | paulson |
Fri, 09 Jan 2004 10:46:18 +0100 | |
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parent 14341 | a09441bd4f1e |
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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 constants denoting the Nat and Real subsets of enclosing |
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types such as hypreal and complex*) |
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Nats :: "'a set" |
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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_prat(prat_of_pnat 1), |
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preal_of_prat(prat_of_pnat 1))})" |
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real_one_def: |
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"1 == Abs_REAL(realrel`` |
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{(preal_of_prat(prat_of_pnat 1) + preal_of_prat(prat_of_pnat 1), |
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preal_of_prat(prat_of_pnat 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_prat(prat_of_pnat 1), |
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preal_of_prat(prat_of_pnat 1))})" |
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real_of_posnat :: "nat => real" |
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"real_of_posnat n == real_of_preal(preal_of_prat(prat_of_pnat(pnat_of_nat n)))" |
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defs (overloaded) |
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real_of_nat_def: "real n == real_of_posnat n + (- 1)" |
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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: |
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"P<Q == \<exists>x1 y1 x2 y2. x1 + y2 < x2 + y1 & |
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(x1,y1)\<in>Rep_REAL(P) & (x2,y2)\<in>Rep_REAL(Q)" |
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real_le_def: |
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"P \<le> (Q::real) == ~(Q < P)" |
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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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Nats :: "'a set" ("\<nat>") |
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subsection{*Proving that realrel is an equivalence relation*} |
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lemma preal_trans_lemma: |
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"[| (x1::preal) + y2 = x2 + y1; x2 + y3 = x3 + y2 |] |
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==> x1 + y3 = x3 + y1" |
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apply (rule_tac C = y2 in preal_add_right_cancel) |
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apply (rotate_tac 1, drule sym) |
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apply (simp add: preal_add_ac) |
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apply (rule preal_add_left_commute [THEN subst]) |
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apply (rule_tac x1 = x1 in preal_add_assoc [THEN subst]) |
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apply (simp add: preal_add_ac) |
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done |
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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 (unfold equiv_def refl_def sym_def trans_def realrel_def) |
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apply (fast elim!: sym 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) |
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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_tac z = z in eq_Abs_REAL) |
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apply (rule_tac z = w in eq_Abs_REAL) |
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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_tac z = z1 in eq_Abs_REAL) |
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apply (rule_tac z = z2 in eq_Abs_REAL) |
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apply (rule_tac z = z3 in eq_Abs_REAL) |
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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_tac z = z in eq_Abs_REAL) |
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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_tac z = z in eq_Abs_REAL) |
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apply (simp add: real_minus real_add preal_add_commute) |
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done |
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subsection{*Congruence property for multiplication*} |
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lemma real_mult_congruent2_lemma: |
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"!!(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_tac z = z in eq_Abs_REAL) |
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apply (rule_tac z = w in eq_Abs_REAL) |
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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_tac z = z1 in eq_Abs_REAL) |
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apply (rule_tac z = z2 in eq_Abs_REAL) |
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apply (rule_tac z = z3 in eq_Abs_REAL) |
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apply (simp add: preal_add_mult_distrib2 real_mult preal_add_ac preal_mult_ac) |
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done |
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lemma real_mult_1: "(1::real) * z = z" |
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apply (unfold real_one_def pnat_one_def) |
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apply (rule_tac z = z in eq_Abs_REAL) |
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apply (simp add: real_mult preal_add_mult_distrib2 preal_mult_1_right |
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preal_mult_ac preal_add_ac) |
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done |
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lemma real_add_mult_distrib: "((z1::real) + z2) * w = (z1 * w) + (z2 * w)" |
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apply (rule_tac z = z1 in eq_Abs_REAL) |
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apply (rule_tac z = z2 in eq_Abs_REAL) |
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apply (rule_tac z = w in eq_Abs_REAL) |
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apply (simp add: preal_add_mult_distrib2 real_add real_mult preal_add_ac preal_mult_ac) |
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done |
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text{*one and zero are distinct*} |
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lemma real_zero_not_eq_one: "0 ~= (1::real)" |
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apply (unfold real_zero_def real_one_def) |
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apply (auto simp add: preal_self_less_add_left [THEN preal_not_refl2]) |
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done |
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subsection{*existence of inverse*} |
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(** lemma -- alternative definition of 0 **) |
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lemma real_zero_iff: "0 = Abs_REAL (realrel `` {(x, x)})" |
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apply (unfold real_zero_def) |
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apply (auto simp add: preal_add_commute) |
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done |
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lemma real_mult_inv_left_ex: "x ~= 0 ==> \<exists>y. y*x = (1::real)" |
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apply (unfold real_zero_def real_one_def) |
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apply (rule_tac z = x in eq_Abs_REAL) |
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apply (cut_tac x = xa and y = y in linorder_less_linear) |
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apply (auto dest!: preal_less_add_left_Ex simp add: real_zero_iff [symmetric]) |
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apply (rule_tac |
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x = "Abs_REAL (realrel `` { (preal_of_prat (prat_of_pnat 1), |
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pinv (D) + preal_of_prat (prat_of_pnat 1))}) " |
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in exI) |
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apply (rule_tac [2] |
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x = "Abs_REAL (realrel `` { (pinv (D) + preal_of_prat (prat_of_pnat 1), |
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preal_of_prat (prat_of_pnat 1))})" |
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in exI) |
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apply (auto simp add: real_mult pnat_one_def preal_mult_1_right |
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preal_add_mult_distrib2 preal_add_mult_distrib preal_mult_1 |
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preal_mult_inv_right preal_add_ac preal_mult_ac) |
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done |
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lemma real_mult_inv_left: "x ~= 0 ==> inverse(x)*x = (1::real)" |
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apply (unfold real_inverse_def) |
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apply (frule real_mult_inv_left_ex, safe) |
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apply (rule someI2, auto) |
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done |
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|
334 |
subsection{*The Real Numbers form a Field*} |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
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changeset
|
335 |
|
14334 | 336 |
instance real :: field |
337 |
proof |
|
338 |
fix x y z :: real |
|
339 |
show "(x + y) + z = x + (y + z)" by (rule real_add_assoc) |
|
340 |
show "x + y = y + x" by (rule real_add_commute) |
|
341 |
show "0 + x = x" by simp |
|
342 |
show "- x + x = 0" by (rule real_add_minus_left) |
|
343 |
show "x - y = x + (-y)" by (simp add: real_diff_def) |
|
344 |
show "(x * y) * z = x * (y * z)" by (rule real_mult_assoc) |
|
345 |
show "x * y = y * x" by (rule real_mult_commute) |
|
346 |
show "1 * x = x" by (rule real_mult_1) |
|
347 |
show "(x + y) * z = x * z + y * z" by (simp add: real_add_mult_distrib) |
|
348 |
show "0 \<noteq> (1::real)" by (rule real_zero_not_eq_one) |
|
349 |
show "x \<noteq> 0 ==> inverse x * x = 1" by (rule real_mult_inv_left) |
|
350 |
show "y \<noteq> 0 ==> x / y = x * inverse y" by (simp add: real_divide_def) |
|
14341
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|
351 |
assume eq: "z+x = z+y" |
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Ring_and_Field now requires axiom add_left_imp_eq for semirings.
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|
352 |
hence "(-z + z) + x = (-z + z) + y" by (simp only: eq real_add_assoc) |
a09441bd4f1e
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|
353 |
thus "x = y" by (simp add: real_add_minus_left) |
14334 | 354 |
qed |
355 |
||
356 |
||
14341
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|
357 |
text{*Inverse of zero! Useful to simplify certain equations*} |
14269 | 358 |
|
14334 | 359 |
lemma INVERSE_ZERO: "inverse 0 = (0::real)" |
360 |
apply (unfold real_inverse_def) |
|
361 |
apply (rule someI2) |
|
362 |
apply (auto simp add: zero_neq_one) |
|
14269 | 363 |
done |
14334 | 364 |
|
365 |
lemma DIVISION_BY_ZERO: "a / (0::real) = 0" |
|
366 |
by (simp add: real_divide_def INVERSE_ZERO) |
|
367 |
||
368 |
instance real :: division_by_zero |
|
369 |
proof |
|
370 |
fix x :: real |
|
371 |
show "inverse 0 = (0::real)" by (rule INVERSE_ZERO) |
|
372 |
show "x/0 = 0" by (rule DIVISION_BY_ZERO) |
|
373 |
qed |
|
374 |
||
375 |
||
376 |
(*Pull negations out*) |
|
377 |
declare minus_mult_right [symmetric, simp] |
|
378 |
minus_mult_left [symmetric, simp] |
|
379 |
||
380 |
text{*Used in RealBin*} |
|
381 |
lemma real_minus_mult_commute: "(-x) * y = x * (- y :: real)" |
|
382 |
by simp |
|
383 |
||
384 |
lemma real_mult_1_right: "z * (1::real) = z" |
|
385 |
by (rule Ring_and_Field.mult_1_right) |
|
14269 | 386 |
|
387 |
||
14329 | 388 |
subsection{*Theorems for Ordering*} |
389 |
||
390 |
(* real_less is a strict order: irreflexive *) |
|
14269 | 391 |
|
14329 | 392 |
text{*lemmas*} |
393 |
lemma preal_lemma_eq_rev_sum: |
|
394 |
"!!(x::preal). [| x = y; x1 = y1 |] ==> x + y1 = x1 + y" |
|
14269 | 395 |
by (simp add: preal_add_commute) |
396 |
||
14329 | 397 |
lemma preal_add_left_commute_cancel: |
398 |
"!!(b::preal). x + (b + y) = x1 + (b + y1) ==> x + y = x1 + y1" |
|
14269 | 399 |
by (simp add: preal_add_ac) |
400 |
||
14329 | 401 |
lemma preal_lemma_for_not_refl: |
402 |
"!!(x::preal). [| x + y2a = x2a + y; |
|
14269 | 403 |
x + y2b = x2b + y |] |
404 |
==> x2a + y2b = x2b + y2a" |
|
405 |
apply (drule preal_lemma_eq_rev_sum, assumption) |
|
406 |
apply (erule_tac V = "x + y2b = x2b + y" in thin_rl) |
|
407 |
apply (simp add: preal_add_ac) |
|
408 |
apply (drule preal_add_left_commute_cancel) |
|
409 |
apply (simp add: preal_add_ac) |
|
410 |
done |
|
411 |
||
412 |
lemma real_less_not_refl: "~ (R::real) < R" |
|
413 |
apply (rule_tac z = R in eq_Abs_REAL) |
|
414 |
apply (auto simp add: real_less_def) |
|
415 |
apply (drule preal_lemma_for_not_refl, assumption, auto) |
|
416 |
done |
|
417 |
||
418 |
(*** y < y ==> P ***) |
|
419 |
lemmas real_less_irrefl = real_less_not_refl [THEN notE, standard] |
|
420 |
declare real_less_irrefl [elim!] |
|
421 |
||
422 |
lemma real_not_refl2: "!!(x::real). x < y ==> x ~= y" |
|
423 |
by (auto simp add: real_less_not_refl) |
|
424 |
||
425 |
(* lemma re-arranging and eliminating terms *) |
|
426 |
lemma preal_lemma_trans: "!! (a::preal). [| a + b = c + d; |
|
427 |
x2b + d + (c + y2e) < a + y2b + (x2e + b) |] |
|
428 |
==> x2b + y2e < x2e + y2b" |
|
429 |
apply (simp add: preal_add_ac) |
|
430 |
apply (rule_tac C = "c+d" in preal_add_left_less_cancel) |
|
431 |
apply (simp add: preal_add_assoc [symmetric]) |
|
432 |
done |
|
433 |
||
434 |
(** A MESS! heavy re-writing involved*) |
|
435 |
lemma real_less_trans: "!!(R1::real). [| R1 < R2; R2 < R3 |] ==> R1 < R3" |
|
436 |
apply (rule_tac z = R1 in eq_Abs_REAL) |
|
437 |
apply (rule_tac z = R2 in eq_Abs_REAL) |
|
438 |
apply (rule_tac z = R3 in eq_Abs_REAL) |
|
439 |
apply (auto simp add: real_less_def) |
|
440 |
apply (rule exI)+ |
|
441 |
apply (rule conjI, rule_tac [2] conjI) |
|
442 |
prefer 2 apply blast |
|
443 |
prefer 2 apply blast |
|
444 |
apply (drule preal_lemma_for_not_refl, assumption) |
|
445 |
apply (blast dest: preal_add_less_mono intro: preal_lemma_trans) |
|
446 |
done |
|
447 |
||
448 |
lemma real_less_not_sym: "!! (R1::real). R1 < R2 ==> ~ (R2 < R1)" |
|
449 |
apply (rule notI) |
|
450 |
apply (drule real_less_trans, assumption) |
|
451 |
apply (simp add: real_less_not_refl) |
|
452 |
done |
|
453 |
||
454 |
(* [| x < y; ~P ==> y < x |] ==> P *) |
|
455 |
lemmas real_less_asym = real_less_not_sym [THEN contrapos_np, standard] |
|
456 |
||
457 |
lemma real_of_preal_add: |
|
458 |
"real_of_preal ((z1::preal) + z2) = |
|
459 |
real_of_preal z1 + real_of_preal z2" |
|
460 |
apply (unfold real_of_preal_def) |
|
461 |
apply (simp add: real_add preal_add_mult_distrib preal_mult_1 add: preal_add_ac) |
|
462 |
done |
|
463 |
||
464 |
lemma real_of_preal_mult: |
|
465 |
"real_of_preal ((z1::preal) * z2) = |
|
466 |
real_of_preal z1* real_of_preal z2" |
|
467 |
apply (unfold real_of_preal_def) |
|
468 |
apply (simp (no_asm_use) add: real_mult preal_add_mult_distrib2 preal_mult_1 preal_mult_1_right pnat_one_def preal_add_ac preal_mult_ac) |
|
469 |
done |
|
470 |
||
471 |
lemma real_of_preal_ExI: |
|
472 |
"!!(x::preal). y < x ==> |
|
473 |
\<exists>m. Abs_REAL (realrel `` {(x,y)}) = real_of_preal m" |
|
474 |
apply (unfold real_of_preal_def) |
|
475 |
apply (auto dest!: preal_less_add_left_Ex simp add: preal_add_ac) |
|
476 |
done |
|
477 |
||
478 |
lemma real_of_preal_ExD: |
|
479 |
"!!(x::preal). \<exists>m. Abs_REAL (realrel `` {(x,y)}) = |
|
480 |
real_of_preal m ==> y < x" |
|
481 |
apply (unfold real_of_preal_def) |
|
482 |
apply (auto simp add: preal_add_commute preal_add_assoc) |
|
483 |
apply (simp add: preal_add_assoc [symmetric] preal_self_less_add_left) |
|
484 |
done |
|
485 |
||
14329 | 486 |
lemma real_of_preal_iff: |
487 |
"(\<exists>m. Abs_REAL (realrel `` {(x,y)}) = real_of_preal m) = (y < x)" |
|
14269 | 488 |
by (blast intro!: real_of_preal_ExI real_of_preal_ExD) |
489 |
||
14329 | 490 |
text{*Gleason prop 9-4.4 p 127*} |
14269 | 491 |
lemma real_of_preal_trichotomy: |
492 |
"\<exists>m. (x::real) = real_of_preal m | x = 0 | x = -(real_of_preal m)" |
|
493 |
apply (unfold real_of_preal_def real_zero_def) |
|
494 |
apply (rule_tac z = x in eq_Abs_REAL) |
|
495 |
apply (auto simp add: real_minus preal_add_ac) |
|
496 |
apply (cut_tac x = x and y = y in linorder_less_linear) |
|
497 |
apply (auto dest!: preal_less_add_left_Ex simp add: preal_add_assoc [symmetric]) |
|
498 |
apply (auto simp add: preal_add_commute) |
|
499 |
done |
|
500 |
||
14329 | 501 |
lemma real_of_preal_trichotomyE: |
502 |
"!!P. [| !!m. x = real_of_preal m ==> P; |
|
14269 | 503 |
x = 0 ==> P; |
504 |
!!m. x = -(real_of_preal m) ==> P |] ==> P" |
|
505 |
apply (cut_tac x = x in real_of_preal_trichotomy, auto) |
|
506 |
done |
|
507 |
||
508 |
lemma real_of_preal_lessD: |
|
509 |
"real_of_preal m1 < real_of_preal m2 ==> m1 < m2" |
|
510 |
apply (unfold real_of_preal_def) |
|
511 |
apply (auto simp add: real_less_def preal_add_ac) |
|
512 |
apply (auto simp add: preal_add_assoc [symmetric]) |
|
513 |
apply (auto simp add: preal_add_ac) |
|
514 |
done |
|
515 |
||
516 |
lemma real_of_preal_lessI: "m1 < m2 ==> real_of_preal m1 < real_of_preal m2" |
|
517 |
apply (drule preal_less_add_left_Ex) |
|
518 |
apply (auto simp add: real_of_preal_add real_of_preal_def real_less_def) |
|
519 |
apply (rule exI)+ |
|
520 |
apply (rule conjI, rule_tac [2] conjI) |
|
521 |
apply (rule_tac [2] refl)+ |
|
522 |
apply (simp add: preal_self_less_add_left del: preal_add_less_iff2) |
|
523 |
done |
|
524 |
||
14329 | 525 |
lemma real_of_preal_less_iff1: |
526 |
"(real_of_preal m1 < real_of_preal m2) = (m1 < m2)" |
|
14269 | 527 |
by (blast intro: real_of_preal_lessI real_of_preal_lessD) |
528 |
||
529 |
declare real_of_preal_less_iff1 [simp] |
|
530 |
||
531 |
lemma real_of_preal_minus_less_self: "- real_of_preal m < real_of_preal m" |
|
532 |
apply (auto simp add: real_of_preal_def real_less_def real_minus) |
|
533 |
apply (rule exI)+ |
|
534 |
apply (rule conjI, rule_tac [2] conjI) |
|
535 |
apply (rule_tac [2] refl)+ |
|
536 |
apply (simp (no_asm_use) add: preal_add_ac) |
|
537 |
apply (simp (no_asm_use) add: preal_self_less_add_right preal_add_assoc [symmetric]) |
|
538 |
done |
|
539 |
||
540 |
lemma real_of_preal_minus_less_zero: "- real_of_preal m < 0" |
|
541 |
apply (unfold real_zero_def) |
|
542 |
apply (auto simp add: real_of_preal_def real_less_def real_minus) |
|
543 |
apply (rule exI)+ |
|
544 |
apply (rule conjI, rule_tac [2] conjI) |
|
545 |
apply (rule_tac [2] refl)+ |
|
546 |
apply (simp (no_asm_use) add: preal_self_less_add_right preal_add_ac) |
|
547 |
done |
|
548 |
||
549 |
lemma real_of_preal_not_minus_gt_zero: "~ 0 < - real_of_preal m" |
|
550 |
apply (cut_tac real_of_preal_minus_less_zero) |
|
551 |
apply (fast dest: real_less_trans elim: real_less_irrefl) |
|
552 |
done |
|
553 |
||
554 |
lemma real_of_preal_zero_less: "0 < real_of_preal m" |
|
555 |
apply (unfold real_zero_def) |
|
556 |
apply (auto simp add: real_of_preal_def real_less_def real_minus) |
|
557 |
apply (rule exI)+ |
|
558 |
apply (rule conjI, rule_tac [2] conjI) |
|
559 |
apply (rule_tac [2] refl)+ |
|
560 |
apply (simp (no_asm_use) add: preal_self_less_add_right preal_add_ac) |
|
561 |
done |
|
562 |
||
563 |
lemma real_of_preal_not_less_zero: "~ real_of_preal m < 0" |
|
564 |
apply (cut_tac real_of_preal_zero_less) |
|
565 |
apply (blast dest: real_less_trans elim: real_less_irrefl) |
|
566 |
done |
|
567 |
||
568 |
lemma real_minus_minus_zero_less: "0 < - (- real_of_preal m)" |
|
569 |
by (simp add: real_of_preal_zero_less) |
|
570 |
||
571 |
(* another lemma *) |
|
572 |
lemma real_of_preal_sum_zero_less: |
|
573 |
"0 < real_of_preal m + real_of_preal m1" |
|
574 |
apply (unfold real_zero_def) |
|
575 |
apply (auto simp add: real_of_preal_def real_less_def real_add) |
|
576 |
apply (rule exI)+ |
|
577 |
apply (rule conjI, rule_tac [2] conjI) |
|
578 |
apply (rule_tac [2] refl)+ |
|
579 |
apply (simp (no_asm_use) add: preal_add_ac) |
|
580 |
apply (simp (no_asm_use) add: preal_self_less_add_right preal_add_assoc [symmetric]) |
|
581 |
done |
|
582 |
||
583 |
lemma real_of_preal_minus_less_all: "- real_of_preal m < real_of_preal m1" |
|
584 |
apply (auto simp add: real_of_preal_def real_less_def real_minus) |
|
585 |
apply (rule exI)+ |
|
586 |
apply (rule conjI, rule_tac [2] conjI) |
|
587 |
apply (rule_tac [2] refl)+ |
|
588 |
apply (simp (no_asm_use) add: preal_add_ac) |
|
589 |
apply (simp (no_asm_use) add: preal_self_less_add_right preal_add_assoc [symmetric]) |
|
590 |
done |
|
591 |
||
592 |
lemma real_of_preal_not_minus_gt_all: "~ real_of_preal m < - real_of_preal m1" |
|
593 |
apply (cut_tac real_of_preal_minus_less_all) |
|
594 |
apply (blast dest: real_less_trans elim: real_less_irrefl) |
|
595 |
done |
|
596 |
||
14329 | 597 |
lemma real_of_preal_minus_less_rev1: |
598 |
"- real_of_preal m1 < - real_of_preal m2 |
|
14269 | 599 |
==> real_of_preal m2 < real_of_preal m1" |
600 |
apply (auto simp add: real_of_preal_def real_less_def real_minus) |
|
601 |
apply (rule exI)+ |
|
602 |
apply (rule conjI, rule_tac [2] conjI) |
|
603 |
apply (rule_tac [2] refl)+ |
|
604 |
apply (auto simp add: preal_add_ac) |
|
605 |
apply (simp add: preal_add_assoc [symmetric]) |
|
606 |
apply (auto simp add: preal_add_ac) |
|
607 |
done |
|
608 |
||
14329 | 609 |
lemma real_of_preal_minus_less_rev2: |
610 |
"real_of_preal m1 < real_of_preal m2 |
|
14269 | 611 |
==> - real_of_preal m2 < - real_of_preal m1" |
612 |
apply (auto simp add: real_of_preal_def real_less_def real_minus) |
|
613 |
apply (rule exI)+ |
|
614 |
apply (rule conjI, rule_tac [2] conjI) |
|
615 |
apply (rule_tac [2] refl)+ |
|
616 |
apply (auto simp add: preal_add_ac) |
|
617 |
apply (simp add: preal_add_assoc [symmetric]) |
|
618 |
apply (auto simp add: preal_add_ac) |
|
619 |
done |
|
620 |
||
14329 | 621 |
lemma real_of_preal_minus_less_rev_iff: |
622 |
"(- real_of_preal m1 < - real_of_preal m2) = |
|
14269 | 623 |
(real_of_preal m2 < real_of_preal m1)" |
624 |
apply (blast intro!: real_of_preal_minus_less_rev1 real_of_preal_minus_less_rev2) |
|
625 |
done |
|
626 |
||
14270 | 627 |
|
628 |
subsection{*Linearity of the Ordering*} |
|
629 |
||
14269 | 630 |
lemma real_linear: "(x::real) < y | x = y | y < x" |
631 |
apply (rule_tac x = x in real_of_preal_trichotomyE) |
|
632 |
apply (rule_tac [!] x = y in real_of_preal_trichotomyE) |
|
14270 | 633 |
apply (auto dest!: preal_le_anti_sym |
634 |
simp add: preal_less_le_iff real_of_preal_minus_less_zero |
|
14334 | 635 |
real_of_preal_zero_less real_of_preal_minus_less_all |
636 |
real_of_preal_minus_less_rev_iff) |
|
14269 | 637 |
done |
638 |
||
639 |
lemma real_neq_iff: "!!w::real. (w ~= z) = (w<z | z<w)" |
|
640 |
by (cut_tac real_linear, blast) |
|
641 |
||
642 |
||
14329 | 643 |
lemma real_linear_less2: |
644 |
"!!(R1::real). [| R1 < R2 ==> P; R1 = R2 ==> P; |
|
14269 | 645 |
R2 < R1 ==> P |] ==> P" |
646 |
apply (cut_tac x = R1 and y = R2 in real_linear, auto) |
|
647 |
done |
|
648 |
||
649 |
lemma real_minus_zero_less_iff: "(0 < -R) = (R < (0::real))" |
|
650 |
apply (rule_tac x = R in real_of_preal_trichotomyE) |
|
651 |
apply (auto simp add: real_of_preal_not_minus_gt_zero real_of_preal_not_less_zero real_of_preal_zero_less real_of_preal_minus_less_zero) |
|
652 |
done |
|
653 |
declare real_minus_zero_less_iff [simp] |
|
654 |
||
655 |
lemma real_minus_zero_less_iff2: "(-R < 0) = ((0::real) < R)" |
|
656 |
apply (rule_tac x = R in real_of_preal_trichotomyE) |
|
657 |
apply (auto simp add: real_of_preal_not_minus_gt_zero real_of_preal_not_less_zero real_of_preal_zero_less real_of_preal_minus_less_zero) |
|
658 |
done |
|
659 |
declare real_minus_zero_less_iff2 [simp] |
|
660 |
||
661 |
||
14334 | 662 |
subsection{*Properties of Less-Than Or Equals*} |
663 |
||
664 |
lemma real_le_imp_less_or_eq: "!!(x::real). x \<le> y ==> x < y | x = y" |
|
665 |
apply (unfold real_le_def) |
|
666 |
apply (cut_tac real_linear) |
|
667 |
apply (blast elim: real_less_irrefl real_less_asym) |
|
668 |
done |
|
669 |
||
670 |
lemma real_less_or_eq_imp_le: "z<w | z=w ==> z \<le>(w::real)" |
|
671 |
apply (unfold real_le_def) |
|
672 |
apply (cut_tac real_linear) |
|
673 |
apply (fast elim: real_less_irrefl real_less_asym) |
|
674 |
done |
|
675 |
||
676 |
lemma real_le_less: "(x \<le> (y::real)) = (x < y | x=y)" |
|
677 |
by (blast intro!: real_less_or_eq_imp_le dest!: real_le_imp_less_or_eq) |
|
678 |
||
679 |
lemma real_le_refl: "w \<le> (w::real)" |
|
680 |
by (simp add: real_le_less) |
|
681 |
||
682 |
lemma real_le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::real)" |
|
683 |
apply (drule real_le_imp_less_or_eq) |
|
684 |
apply (drule real_le_imp_less_or_eq) |
|
685 |
apply (rule real_less_or_eq_imp_le) |
|
686 |
apply (blast intro: real_less_trans) |
|
687 |
done |
|
688 |
||
689 |
lemma real_le_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::real)" |
|
690 |
apply (drule real_le_imp_less_or_eq) |
|
691 |
apply (drule real_le_imp_less_or_eq) |
|
692 |
apply (fast elim: real_less_irrefl real_less_asym) |
|
693 |
done |
|
694 |
||
695 |
(* Axiom 'order_less_le' of class 'order': *) |
|
696 |
lemma real_less_le: "((w::real) < z) = (w \<le> z & w \<noteq> z)" |
|
697 |
apply (simp add: real_le_def real_neq_iff) |
|
698 |
apply (blast elim!: real_less_asym) |
|
699 |
done |
|
700 |
||
701 |
instance real :: order |
|
702 |
by (intro_classes, |
|
703 |
(assumption | |
|
704 |
rule real_le_refl real_le_trans real_le_anti_sym real_less_le)+) |
|
705 |
||
706 |
(* Axiom 'linorder_linear' of class 'linorder': *) |
|
707 |
lemma real_le_linear: "(z::real) \<le> w | w \<le> z" |
|
708 |
apply (simp add: real_le_less) |
|
709 |
apply (cut_tac real_linear, blast) |
|
710 |
done |
|
711 |
||
712 |
instance real :: linorder |
|
713 |
by (intro_classes, rule real_le_linear) |
|
714 |
||
715 |
||
716 |
subsection{*Theorems About the Ordering*} |
|
717 |
||
718 |
lemma real_gt_zero_preal_Ex: "(0 < x) = (\<exists>y. x = real_of_preal y)" |
|
719 |
apply (auto simp add: real_of_preal_zero_less) |
|
720 |
apply (cut_tac x = x in real_of_preal_trichotomy) |
|
721 |
apply (blast elim!: real_less_irrefl real_of_preal_not_minus_gt_zero [THEN notE]) |
|
722 |
done |
|
723 |
||
724 |
lemma real_gt_preal_preal_Ex: |
|
725 |
"real_of_preal z < x ==> \<exists>y. x = real_of_preal y" |
|
726 |
by (blast dest!: real_of_preal_zero_less [THEN real_less_trans] |
|
727 |
intro: real_gt_zero_preal_Ex [THEN iffD1]) |
|
728 |
||
729 |
lemma real_ge_preal_preal_Ex: |
|
730 |
"real_of_preal z \<le> x ==> \<exists>y. x = real_of_preal y" |
|
731 |
by (blast dest: order_le_imp_less_or_eq real_gt_preal_preal_Ex) |
|
732 |
||
733 |
lemma real_less_all_preal: "y \<le> 0 ==> \<forall>x. y < real_of_preal x" |
|
734 |
by (auto elim: order_le_imp_less_or_eq [THEN disjE] |
|
735 |
intro: real_of_preal_zero_less [THEN [2] real_less_trans] |
|
736 |
simp add: real_of_preal_zero_less) |
|
737 |
||
738 |
lemma real_less_all_real2: "~ 0 < y ==> \<forall>x. y < real_of_preal x" |
|
739 |
by (blast intro!: real_less_all_preal linorder_not_less [THEN iffD1]) |
|
740 |
||
741 |
lemma real_of_preal_le_iff: |
|
742 |
"(real_of_preal m1 \<le> real_of_preal m2) = (m1 \<le> m2)" |
|
14335 | 743 |
by (auto intro!: preal_le_iff_less_or_eq [THEN iffD1] |
744 |
simp add: linorder_not_less [symmetric]) |
|
14334 | 745 |
|
746 |
||
747 |
subsection{*Monotonicity of Addition*} |
|
748 |
||
749 |
lemma real_mult_order: "[| 0 < x; 0 < y |] ==> (0::real) < x * y" |
|
750 |
apply (auto simp add: real_gt_zero_preal_Ex) |
|
751 |
apply (rule_tac x = "y*ya" in exI) |
|
752 |
apply (simp (no_asm_use) add: real_of_preal_mult) |
|
753 |
done |
|
754 |
||
755 |
(*Alternative definition for real_less*) |
|
756 |
lemma real_less_add_positive_left_Ex: "R < S ==> \<exists>T::real. 0 < T & R + T = S" |
|
757 |
apply (rule_tac x = R in real_of_preal_trichotomyE) |
|
758 |
apply (rule_tac [!] x = S in real_of_preal_trichotomyE) |
|
14335 | 759 |
apply (auto dest!: preal_less_add_left_Ex |
760 |
simp add: real_of_preal_not_minus_gt_all real_of_preal_add |
|
761 |
real_of_preal_not_less_zero real_less_not_refl |
|
762 |
real_of_preal_not_minus_gt_zero real_of_preal_minus_less_rev_iff) |
|
14334 | 763 |
apply (rule real_of_preal_zero_less) |
764 |
apply (rule_tac [1] x = "real_of_preal m+real_of_preal ma" in exI) |
|
765 |
apply (rule_tac [2] x = "real_of_preal D" in exI) |
|
14335 | 766 |
apply (auto simp add: real_of_preal_minus_less_rev_iff real_of_preal_zero_less |
767 |
real_of_preal_sum_zero_less real_add_assoc) |
|
14334 | 768 |
apply (simp add: real_add_assoc [symmetric]) |
769 |
done |
|
770 |
||
771 |
lemma real_less_sum_gt_zero: "(W < S) ==> (0 < S + (-W::real))" |
|
772 |
apply (drule real_less_add_positive_left_Ex) |
|
773 |
apply (auto simp add: add_ac) |
|
774 |
done |
|
775 |
||
776 |
lemma real_lemma_change_eq_subj: "!!S::real. T = S + W ==> S = T + (-W)" |
|
777 |
by (simp add: add_ac) |
|
778 |
||
779 |
(* FIXME: long! *) |
|
780 |
lemma real_sum_gt_zero_less: "(0 < S + (-W::real)) ==> (W < S)" |
|
781 |
apply (rule ccontr) |
|
782 |
apply (drule linorder_not_less [THEN iffD1, THEN real_le_imp_less_or_eq]) |
|
783 |
apply (auto simp add: real_less_not_refl) |
|
784 |
apply (drule real_less_add_positive_left_Ex, clarify, simp) |
|
785 |
apply (drule real_lemma_change_eq_subj, auto) |
|
786 |
apply (drule real_less_sum_gt_zero) |
|
787 |
apply (auto elim: real_less_asym simp add: add_left_commute [of W] add_ac) |
|
788 |
done |
|
789 |
||
790 |
lemma real_mult_less_mono2: "[| (0::real) < z; x < y |] ==> z * x < z * y" |
|
791 |
apply (rule real_sum_gt_zero_less) |
|
792 |
apply (drule real_less_sum_gt_zero [of x y]) |
|
793 |
apply (drule real_mult_order, assumption) |
|
794 |
apply (simp add: right_distrib) |
|
795 |
done |
|
796 |
||
797 |
lemma real_less_sum_gt_0_iff: "(0 < S + (-W::real)) = (W < S)" |
|
798 |
by (blast intro: real_less_sum_gt_zero real_sum_gt_zero_less) |
|
799 |
||
800 |
lemma real_less_eq_diff: "(x<y) = (x-y < (0::real))" |
|
801 |
apply (unfold real_diff_def) |
|
802 |
apply (subst real_minus_zero_less_iff [symmetric]) |
|
803 |
apply (simp add: real_add_commute real_less_sum_gt_0_iff) |
|
804 |
done |
|
805 |
||
806 |
lemma real_less_eqI: "(x::real) - y = x' - y' ==> (x<y) = (x'<y')" |
|
807 |
apply (subst real_less_eq_diff) |
|
808 |
apply (rule_tac y1 = y in real_less_eq_diff [THEN ssubst], simp) |
|
809 |
done |
|
810 |
||
811 |
lemma real_le_eqI: "(x::real) - y = x' - y' ==> (y\<le>x) = (y'\<le>x')" |
|
812 |
apply (drule real_less_eqI) |
|
813 |
apply (simp add: real_le_def) |
|
814 |
done |
|
815 |
||
816 |
lemma real_add_left_mono: "x \<le> y ==> z + x \<le> z + (y::real)" |
|
817 |
apply (rule real_le_eqI [THEN iffD1]) |
|
818 |
prefer 2 apply assumption |
|
819 |
apply (simp add: real_diff_def add_ac) |
|
820 |
done |
|
821 |
||
822 |
||
823 |
subsection{*The Reals Form an Ordered Field*} |
|
824 |
||
825 |
instance real :: ordered_field |
|
826 |
proof |
|
827 |
fix x y z :: real |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
828 |
show "0 < (1::real)" |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
829 |
by (force intro: real_gt_zero_preal_Ex [THEN iffD2] |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
830 |
simp add: real_one_def real_of_preal_def) |
14334 | 831 |
show "x \<le> y ==> z + x \<le> z + y" by (rule real_add_left_mono) |
832 |
show "x < y ==> 0 < z ==> z * x < z * y" by (simp add: real_mult_less_mono2) |
|
833 |
show "\<bar>x\<bar> = (if x < 0 then -x else x)" |
|
834 |
by (auto dest: order_le_less_trans simp add: real_abs_def linorder_not_le) |
|
835 |
qed |
|
836 |
||
837 |
text{*These two need to be proved in @{text Ring_and_Field}*} |
|
838 |
lemma real_add_less_le_mono: "[| w'<w; z'\<le>z |] ==> w' + z' < w + (z::real)" |
|
839 |
apply (erule add_strict_right_mono [THEN order_less_le_trans]) |
|
840 |
apply (erule add_left_mono) |
|
841 |
done |
|
842 |
||
843 |
lemma real_add_le_less_mono: |
|
844 |
"!!z z'::real. [| w'\<le>w; z'<z |] ==> w' + z' < w + z" |
|
845 |
apply (erule add_right_mono [THEN order_le_less_trans]) |
|
846 |
apply (erule add_strict_left_mono) |
|
847 |
done |
|
848 |
||
849 |
lemma real_zero_less_one: "0 < (1::real)" |
|
850 |
by (rule Ring_and_Field.zero_less_one) |
|
851 |
||
852 |
lemma real_le_square [simp]: "(0::real) \<le> x*x" |
|
853 |
by (rule Ring_and_Field.zero_le_square) |
|
854 |
||
855 |
||
856 |
subsection{*More Lemmas*} |
|
857 |
||
858 |
lemma real_mult_left_cancel: "(c::real) \<noteq> 0 ==> (c*a=c*b) = (a=b)" |
|
859 |
by auto |
|
860 |
||
861 |
lemma real_mult_right_cancel: "(c::real) \<noteq> 0 ==> (a*c=b*c) = (a=b)" |
|
862 |
by auto |
|
863 |
||
864 |
text{*The precondition could be weakened to @{term "0\<le>x"}*} |
|
865 |
lemma real_mult_less_mono: |
|
866 |
"[| u<v; x<y; (0::real) < v; 0 < x |] ==> u*x < v* y" |
|
867 |
by (simp add: Ring_and_Field.mult_strict_mono order_less_imp_le) |
|
868 |
||
869 |
lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)" |
|
870 |
by (force elim: order_less_asym |
|
871 |
simp add: Ring_and_Field.mult_less_cancel_right) |
|
872 |
||
873 |
lemma real_mult_le_cancel_iff1 [simp]: "(0::real) < z ==> (x*z \<le> y*z) = (x\<le>y)" |
|
874 |
by (auto simp add: real_le_def) |
|
875 |
||
876 |
lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x \<le> z*y) = (x\<le>y)" |
|
877 |
by (force elim: order_less_asym |
|
878 |
simp add: Ring_and_Field.mult_le_cancel_left) |
|
879 |
||
880 |
text{*Only two uses?*} |
|
881 |
lemma real_mult_less_mono': |
|
882 |
"[| x < y; r1 < r2; (0::real) \<le> r1; 0 \<le> x|] ==> r1 * x < r2 * y" |
|
883 |
by (rule Ring_and_Field.mult_strict_mono') |
|
884 |
||
885 |
text{*FIXME: delete or at least combine the next two lemmas*} |
|
886 |
lemma real_sum_squares_cancel: "x * x + y * y = 0 ==> x = (0::real)" |
|
887 |
apply (drule Ring_and_Field.equals_zero_I [THEN sym]) |
|
888 |
apply (cut_tac x = y in real_le_square) |
|
889 |
apply (auto, drule real_le_anti_sym, auto) |
|
890 |
done |
|
891 |
||
892 |
lemma real_sum_squares_cancel2: "x * x + y * y = 0 ==> y = (0::real)" |
|
893 |
apply (rule_tac y = x in real_sum_squares_cancel) |
|
894 |
apply (simp add: real_add_commute) |
|
895 |
done |
|
896 |
||
897 |
||
898 |
lemma real_add_order: "[| 0 < x; 0 < y |] ==> (0::real) < x + y" |
|
899 |
apply (drule add_strict_mono [of concl: 0 0], assumption) |
|
900 |
apply simp |
|
901 |
done |
|
902 |
||
903 |
lemma real_le_add_order: "[| 0 \<le> x; 0 \<le> y |] ==> (0::real) \<le> x + y" |
|
904 |
apply (drule order_le_imp_less_or_eq)+ |
|
905 |
apply (auto intro: real_add_order order_less_imp_le) |
|
906 |
done |
|
907 |
||
908 |
||
909 |
subsection{*An Embedding of the Naturals in the Reals*} |
|
910 |
||
911 |
lemma real_of_posnat_one: "real_of_posnat 0 = (1::real)" |
|
912 |
by (simp add: real_of_posnat_def pnat_one_iff [symmetric] |
|
913 |
real_of_preal_def symmetric real_one_def) |
|
914 |
||
915 |
lemma real_of_posnat_two: "real_of_posnat (Suc 0) = (1::real) + (1::real)" |
|
916 |
by (simp add: real_of_posnat_def real_of_preal_def real_one_def pnat_two_eq |
|
917 |
real_add |
|
918 |
prat_of_pnat_add [symmetric] preal_of_prat_add [symmetric] |
|
919 |
pnat_add_ac) |
|
920 |
||
921 |
lemma real_of_posnat_add: |
|
922 |
"real_of_posnat n1 + real_of_posnat n2 = real_of_posnat (n1 + n2) + (1::real)" |
|
923 |
apply (unfold real_of_posnat_def) |
|
924 |
apply (simp (no_asm_use) add: real_of_posnat_one [symmetric] real_of_posnat_def real_of_preal_add [symmetric] preal_of_prat_add [symmetric] prat_of_pnat_add [symmetric] pnat_of_nat_add) |
|
925 |
done |
|
926 |
||
927 |
lemma real_of_posnat_add_one: |
|
928 |
"real_of_posnat (n + 1) = real_of_posnat n + (1::real)" |
|
929 |
apply (rule_tac a1 = " (1::real) " in add_right_cancel [THEN iffD1]) |
|
930 |
apply (rule real_of_posnat_add [THEN subst]) |
|
931 |
apply (simp (no_asm_use) add: real_of_posnat_two real_add_assoc) |
|
932 |
done |
|
933 |
||
934 |
lemma real_of_posnat_Suc: |
|
935 |
"real_of_posnat (Suc n) = real_of_posnat n + (1::real)" |
|
936 |
by (subst real_of_posnat_add_one [symmetric], simp) |
|
937 |
||
938 |
lemma inj_real_of_posnat: "inj(real_of_posnat)" |
|
939 |
apply (rule inj_onI) |
|
940 |
apply (unfold real_of_posnat_def) |
|
941 |
apply (drule inj_real_of_preal [THEN injD]) |
|
942 |
apply (drule inj_preal_of_prat [THEN injD]) |
|
943 |
apply (drule inj_prat_of_pnat [THEN injD]) |
|
944 |
apply (erule inj_pnat_of_nat [THEN injD]) |
|
945 |
done |
|
946 |
||
947 |
lemma real_of_nat_zero [simp]: "real (0::nat) = 0" |
|
948 |
by (simp add: real_of_nat_def real_of_posnat_one) |
|
949 |
||
950 |
lemma real_of_nat_one [simp]: "real (Suc 0) = (1::real)" |
|
951 |
by (simp add: real_of_nat_def real_of_posnat_two real_add_assoc) |
|
952 |
||
953 |
lemma real_of_nat_add [simp]: |
|
954 |
"real (m + n) = real (m::nat) + real n" |
|
955 |
apply (simp add: real_of_nat_def add_ac) |
|
956 |
apply (simp add: real_of_posnat_add add_assoc [symmetric]) |
|
957 |
apply (simp add: add_commute) |
|
958 |
apply (simp add: add_assoc [symmetric]) |
|
959 |
done |
|
960 |
||
961 |
(*Not for addsimps: often the LHS is used to represent a positive natural*) |
|
962 |
lemma real_of_nat_Suc: "real (Suc n) = real n + (1::real)" |
|
963 |
by (simp add: real_of_nat_def real_of_posnat_Suc add_ac) |
|
964 |
||
965 |
lemma real_of_nat_less_iff [iff]: |
|
966 |
"(real (n::nat) < real m) = (n < m)" |
|
967 |
by (auto simp add: real_of_nat_def real_of_posnat_def) |
|
968 |
||
969 |
lemma real_of_nat_le_iff [iff]: "(real (n::nat) \<le> real m) = (n \<le> m)" |
|
970 |
by (simp add: linorder_not_less [symmetric]) |
|
971 |
||
972 |
lemma inj_real_of_nat: "inj (real :: nat => real)" |
|
973 |
apply (rule inj_onI) |
|
974 |
apply (auto intro!: inj_real_of_posnat [THEN injD] |
|
975 |
simp add: real_of_nat_def add_right_cancel) |
|
976 |
done |
|
977 |
||
978 |
lemma real_of_nat_ge_zero [iff]: "0 \<le> real (n::nat)" |
|
979 |
apply (induct_tac "n") |
|
980 |
apply (auto simp add: real_of_nat_Suc) |
|
981 |
apply (drule real_add_le_less_mono) |
|
982 |
apply (rule zero_less_one) |
|
983 |
apply (simp add: order_less_imp_le) |
|
984 |
done |
|
985 |
||
986 |
lemma real_of_nat_mult [simp]: "real (m * n) = real (m::nat) * real n" |
|
987 |
apply (induct_tac "m") |
|
988 |
apply (auto simp add: real_of_nat_Suc left_distrib add_commute) |
|
989 |
done |
|
990 |
||
991 |
lemma real_of_nat_inject [iff]: "(real (n::nat) = real m) = (n = m)" |
|
992 |
by (auto dest: inj_real_of_nat [THEN injD]) |
|
993 |
||
994 |
lemma real_of_nat_diff [rule_format]: |
|
995 |
"n \<le> m --> real (m - n) = real (m::nat) - real n" |
|
996 |
apply (induct_tac "m", simp) |
|
997 |
apply (simp add: real_diff_def Suc_diff_le le_Suc_eq real_of_nat_Suc add_ac) |
|
998 |
apply (simp add: add_left_commute [of _ "- 1"]) |
|
999 |
done |
|
1000 |
||
1001 |
lemma real_of_nat_zero_iff: "(real (n::nat) = 0) = (n = 0)" |
|
1002 |
proof |
|
1003 |
assume "real n = 0" |
|
1004 |
have "real n = real (0::nat)" by simp |
|
1005 |
then show "n = 0" by (simp only: real_of_nat_inject) |
|
1006 |
next |
|
1007 |
show "n = 0 \<Longrightarrow> real n = 0" by simp |
|
1008 |
qed |
|
1009 |
||
1010 |
lemma real_of_nat_neg_int [simp]: "neg z ==> real (nat z) = 0" |
|
1011 |
by (simp add: neg_nat real_of_nat_zero) |
|
1012 |
||
1013 |
||
1014 |
lemma real_inverse_unique: "x*y = (1::real) ==> y = inverse x" |
|
1015 |
apply (case_tac "x \<noteq> 0") |
|
1016 |
apply (rule_tac c1 = x in real_mult_left_cancel [THEN iffD1], auto) |
|
1017 |
done |
|
1018 |
||
1019 |
lemma real_inverse_gt_one: "[| (0::real) < x; x < 1 |] ==> 1 < inverse x" |
|
1020 |
by (auto dest: less_imp_inverse_less) |
|
1021 |
||
1022 |
lemma real_of_nat_gt_zero_cancel_iff: "(0 < real (n::nat)) = (0 < n)" |
|
1023 |
by (rule real_of_nat_less_iff [THEN subst], auto) |
|
1024 |
declare real_of_nat_gt_zero_cancel_iff [simp] |
|
1025 |
||
1026 |
lemma real_of_nat_le_zero_cancel_iff: "(real (n::nat) <= 0) = (n = 0)" |
|
1027 |
apply (rule real_of_nat_zero [THEN subst]) |
|
1028 |
apply (subst real_of_nat_le_iff, auto) |
|
1029 |
done |
|
1030 |
declare real_of_nat_le_zero_cancel_iff [simp] |
|
1031 |
||
1032 |
lemma not_real_of_nat_less_zero: "~ real (n::nat) < 0" |
|
1033 |
apply (simp (no_asm) add: real_le_def [symmetric] real_of_nat_ge_zero) |
|
1034 |
done |
|
1035 |
declare not_real_of_nat_less_zero [simp] |
|
1036 |
||
1037 |
lemma real_of_nat_ge_zero_cancel_iff [simp]: |
|
1038 |
"(0 <= real (n::nat)) = (0 <= n)" |
|
1039 |
apply (simp add: real_le_def le_def) |
|
1040 |
done |
|
1041 |
||
1042 |
lemma real_mult_self_sum_ge_zero: "(0::real) \<le> x*x + y*y" |
|
1043 |
proof - |
|
1044 |
have "0 + 0 \<le> x*x + y*y" by (blast intro: add_mono zero_le_square) |
|
1045 |
thus ?thesis by simp |
|
1046 |
qed |
|
1047 |
||
1048 |
||
1049 |
ML |
|
1050 |
{* |
|
1051 |
val real_abs_def = thm "real_abs_def"; |
|
1052 |
||
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
1053 |
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
|
1054 |
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
|
1055 |
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
|
1056 |
val real_of_nat_def = thm "real_of_nat_def"; |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
1057 |
|
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
1058 |
val preal_trans_lemma = thm"preal_trans_lemma"; |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
1059 |
val realrel_iff = thm"realrel_iff"; |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
1060 |
val realrel_refl = thm"realrel_refl"; |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
1061 |
val equiv_realrel = thm"equiv_realrel"; |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
1062 |
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
|
1063 |
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
|
1064 |
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
|
1065 |
val eq_realrelD = thm"eq_realrelD"; |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
1066 |
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
|
1067 |
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
|
1068 |
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
|
1069 |
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
|
1070 |
val real_minus = thm"real_minus"; |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
1071 |
val real_add = thm"real_add"; |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset
|
1072 |
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
|
1073 |
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
|
1074 |
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
|
1075 |
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
|
1076 |
|
14334 | 1077 |
val real_less_eq_diff = thm "real_less_eq_diff"; |
1078 |
||
1079 |
val real_mult = thm"real_mult"; |
|
1080 |
val real_mult_commute = thm"real_mult_commute"; |
|
1081 |
val real_mult_assoc = thm"real_mult_assoc"; |
|
1082 |
val real_mult_1 = thm"real_mult_1"; |
|
1083 |
val real_mult_1_right = thm"real_mult_1_right"; |
|
1084 |
val real_minus_mult_commute = thm"real_minus_mult_commute"; |
|
1085 |
val preal_le_linear = thm"preal_le_linear"; |
|
1086 |
val real_mult_inv_left = thm"real_mult_inv_left"; |
|
1087 |
val real_less_not_refl = thm"real_less_not_refl"; |
|
1088 |
val real_less_irrefl = thm"real_less_irrefl"; |
|
1089 |
val real_not_refl2 = thm"real_not_refl2"; |
|
1090 |
val preal_lemma_trans = thm"preal_lemma_trans"; |
|
1091 |
val real_less_trans = thm"real_less_trans"; |
|
1092 |
val real_less_not_sym = thm"real_less_not_sym"; |
|
1093 |
val real_less_asym = thm"real_less_asym"; |
|
1094 |
val real_of_preal_add = thm"real_of_preal_add"; |
|
1095 |
val real_of_preal_mult = thm"real_of_preal_mult"; |
|
1096 |
val real_of_preal_ExI = thm"real_of_preal_ExI"; |
|
1097 |
val real_of_preal_ExD = thm"real_of_preal_ExD"; |
|
1098 |
val real_of_preal_iff = thm"real_of_preal_iff"; |
|
1099 |
val real_of_preal_trichotomy = thm"real_of_preal_trichotomy"; |
|
1100 |
val real_of_preal_trichotomyE = thm"real_of_preal_trichotomyE"; |
|
1101 |
val real_of_preal_lessD = thm"real_of_preal_lessD"; |
|
1102 |
val real_of_preal_lessI = thm"real_of_preal_lessI"; |
|
1103 |
val real_of_preal_less_iff1 = thm"real_of_preal_less_iff1"; |
|
1104 |
val real_of_preal_minus_less_self = thm"real_of_preal_minus_less_self"; |
|
1105 |
val real_of_preal_minus_less_zero = thm"real_of_preal_minus_less_zero"; |
|
1106 |
val real_of_preal_not_minus_gt_zero = thm"real_of_preal_not_minus_gt_zero"; |
|
1107 |
val real_of_preal_zero_less = thm"real_of_preal_zero_less"; |
|
1108 |
val real_of_preal_not_less_zero = thm"real_of_preal_not_less_zero"; |
|
1109 |
val real_minus_minus_zero_less = thm"real_minus_minus_zero_less"; |
|
1110 |
val real_of_preal_sum_zero_less = thm"real_of_preal_sum_zero_less"; |
|
1111 |
val real_of_preal_minus_less_all = thm"real_of_preal_minus_less_all"; |
|
1112 |
val real_of_preal_not_minus_gt_all = thm"real_of_preal_not_minus_gt_all"; |
|
1113 |
val real_of_preal_minus_less_rev1 = thm"real_of_preal_minus_less_rev1"; |
|
1114 |
val real_of_preal_minus_less_rev2 = thm"real_of_preal_minus_less_rev2"; |
|
1115 |
val real_linear = thm"real_linear"; |
|
1116 |
val real_neq_iff = thm"real_neq_iff"; |
|
1117 |
val real_linear_less2 = thm"real_linear_less2"; |
|
1118 |
val real_le_imp_less_or_eq = thm"real_le_imp_less_or_eq"; |
|
1119 |
val real_less_or_eq_imp_le = thm"real_less_or_eq_imp_le"; |
|
1120 |
val real_le_less = thm"real_le_less"; |
|
1121 |
val real_le_refl = thm"real_le_refl"; |
|
1122 |
val real_le_linear = thm"real_le_linear"; |
|
1123 |
val real_le_trans = thm"real_le_trans"; |
|
1124 |
val real_le_anti_sym = thm"real_le_anti_sym"; |
|
1125 |
val real_less_le = thm"real_less_le"; |
|
1126 |
val real_less_sum_gt_zero = thm"real_less_sum_gt_zero"; |
|
1127 |
val real_sum_gt_zero_less = thm"real_sum_gt_zero_less"; |
|
1128 |
||
1129 |
val real_gt_zero_preal_Ex = thm "real_gt_zero_preal_Ex"; |
|
1130 |
val real_gt_preal_preal_Ex = thm "real_gt_preal_preal_Ex"; |
|
1131 |
val real_ge_preal_preal_Ex = thm "real_ge_preal_preal_Ex"; |
|
1132 |
val real_less_all_preal = thm "real_less_all_preal"; |
|
1133 |
val real_less_all_real2 = thm "real_less_all_real2"; |
|
1134 |
val real_of_preal_le_iff = thm "real_of_preal_le_iff"; |
|
1135 |
val real_mult_order = thm "real_mult_order"; |
|
1136 |
val real_zero_less_one = thm "real_zero_less_one"; |
|
1137 |
val real_add_less_le_mono = thm "real_add_less_le_mono"; |
|
1138 |
val real_add_le_less_mono = thm "real_add_le_less_mono"; |
|
1139 |
val real_add_order = thm "real_add_order"; |
|
1140 |
val real_le_add_order = thm "real_le_add_order"; |
|
1141 |
val real_le_square = thm "real_le_square"; |
|
1142 |
val real_mult_less_mono2 = thm "real_mult_less_mono2"; |
|
1143 |
||
1144 |
val real_mult_less_iff1 = thm "real_mult_less_iff1"; |
|
1145 |
val real_mult_le_cancel_iff1 = thm "real_mult_le_cancel_iff1"; |
|
1146 |
val real_mult_le_cancel_iff2 = thm "real_mult_le_cancel_iff2"; |
|
1147 |
val real_mult_less_mono = thm "real_mult_less_mono"; |
|
1148 |
val real_mult_less_mono' = thm "real_mult_less_mono'"; |
|
1149 |
val real_sum_squares_cancel = thm "real_sum_squares_cancel"; |
|
1150 |
val real_sum_squares_cancel2 = thm "real_sum_squares_cancel2"; |
|
1151 |
||
1152 |
val real_mult_left_cancel = thm"real_mult_left_cancel"; |
|
1153 |
val real_mult_right_cancel = thm"real_mult_right_cancel"; |
|
1154 |
val real_of_posnat_one = thm "real_of_posnat_one"; |
|
1155 |
val real_of_posnat_two = thm "real_of_posnat_two"; |
|
1156 |
val real_of_posnat_add = thm "real_of_posnat_add"; |
|
1157 |
val real_of_posnat_add_one = thm "real_of_posnat_add_one"; |
|
1158 |
val real_of_nat_zero = thm "real_of_nat_zero"; |
|
1159 |
val real_of_nat_one = thm "real_of_nat_one"; |
|
1160 |
val real_of_nat_add = thm "real_of_nat_add"; |
|
1161 |
val real_of_nat_Suc = thm "real_of_nat_Suc"; |
|
1162 |
val real_of_nat_less_iff = thm "real_of_nat_less_iff"; |
|
1163 |
val real_of_nat_le_iff = thm "real_of_nat_le_iff"; |
|
1164 |
val inj_real_of_nat = thm "inj_real_of_nat"; |
|
1165 |
val real_of_nat_ge_zero = thm "real_of_nat_ge_zero"; |
|
1166 |
val real_of_nat_mult = thm "real_of_nat_mult"; |
|
1167 |
val real_of_nat_inject = thm "real_of_nat_inject"; |
|
1168 |
val real_of_nat_diff = thm "real_of_nat_diff"; |
|
1169 |
val real_of_nat_zero_iff = thm "real_of_nat_zero_iff"; |
|
1170 |
val real_of_nat_neg_int = thm "real_of_nat_neg_int"; |
|
1171 |
val real_inverse_unique = thm "real_inverse_unique"; |
|
1172 |
val real_inverse_gt_one = thm "real_inverse_gt_one"; |
|
1173 |
val real_of_nat_gt_zero_cancel_iff = thm "real_of_nat_gt_zero_cancel_iff"; |
|
1174 |
val real_of_nat_le_zero_cancel_iff = thm "real_of_nat_le_zero_cancel_iff"; |
|
1175 |
val not_real_of_nat_less_zero = thm "not_real_of_nat_less_zero"; |
|
1176 |
val real_of_nat_ge_zero_cancel_iff = thm "real_of_nat_ge_zero_cancel_iff"; |
|
1177 |
*} |
|
10752
c4f1bf2acf4c
tidying, and separation of HOL-Hyperreal from HOL-Real
paulson
parents:
10648
diff
changeset
|
1178 |
|
5588 | 1179 |
end |