# HG changeset patch # User paulson # Date 1075802796 -3600 # Node ID 67a628beb9812f52e5d9b8c5eb97caf2183d07cf # Parent 51ddf8963c95a8638d5209ba71146a1859b08f0e tidying of the complex numbers diff -r 51ddf8963c95 -r 67a628beb981 src/HOL/Complex/CLim.ML --- a/src/HOL/Complex/CLim.ML Tue Feb 03 10:19:21 2004 +0100 +++ b/src/HOL/Complex/CLim.ML Tue Feb 03 11:06:36 2004 +0100 @@ -284,7 +284,7 @@ (*** NSCLIM_zero, CLIM_zero, etc. ***) Goal "f -- a --NSC> l ==> (%x. f(x) - l) -- a --NSC> 0"; -by (res_inst_tac [("z1","l")] (complex_add_minus_right_zero RS subst) 1); +by (res_inst_tac [("a1","l")] (right_minus RS subst) 1); by (rewtac complex_diff_def); by (rtac NSCLIM_add 1 THEN Auto_tac); qed "NSCLIM_zero"; @@ -701,7 +701,7 @@ by (Step_tac 1); by (dres_inst_tac [("x","x - a")] spec 1); by (dres_inst_tac [("x","x + a")] spec 2); -by (auto_tac (claset(), simpset() addsimps complex_add_ac)); +by (auto_tac (claset(), simpset() addsimps add_ac)); qed "CDERIV_CLIM_iff"; Goalw [cderiv_def] "(CDERIV f x :> D) = \ @@ -755,10 +755,10 @@ [mem_cinfmal_iff RS sym,hcomplex_add_commute])); by (dres_inst_tac [("c","xa + - hcomplex_of_complex x")] capprox_mult1 1); by (auto_tac (claset() addIs [CInfinitesimal_subset_CFinite - RS subsetD],simpset() addsimps [hcomplex_mult_assoc])); + RS subsetD],simpset() addsimps [mult_assoc])); by (dres_inst_tac [("x3","D")] (CFinite_hcomplex_of_complex RSN (2,CInfinitesimal_CFinite_mult) RS (mem_cinfmal_iff RS iffD1)) 1); -by (blast_tac (claset() addIs [capprox_trans,hcomplex_mult_commute RS subst, +by (blast_tac (claset() addIs [capprox_trans,mult_commute RS subst, (capprox_minus_iff RS iffD2)]) 1); qed "NSCDERIV_isNSContc"; @@ -833,10 +833,9 @@ by (dres_inst_tac [("D","Db")] lemma_nscderiv2 1); by (dtac (capprox_minus_iff RS iffD2 RS (bex_CInfinitesimal_iff2 RS iffD2)) 4); by (auto_tac (claset() addSIs [capprox_add_mono1], - simpset() addsimps [hcomplex_add_mult_distrib, right_distrib, - hcomplex_mult_commute, hcomplex_add_assoc])); -by (res_inst_tac [("w1","hcomplex_of_complex Db * hcomplex_of_complex (f x)")] - (hcomplex_add_commute RS subst) 1); + simpset() addsimps [left_distrib, right_distrib, mult_commute, add_assoc])); +by (res_inst_tac [("b1","hcomplex_of_complex Db * hcomplex_of_complex (f x)")] + (add_commute RS subst) 1); by (auto_tac (claset() addSIs [CInfinitesimal_add_capprox_self2 RS capprox_sym, CInfinitesimal_add, CInfinitesimal_mult, CInfinitesimal_hcomplex_of_complex_mult, @@ -853,7 +852,7 @@ Goal "NSCDERIV f x :> D ==> NSCDERIV (%x. c * f x) x :> c*D"; by (asm_full_simp_tac (simpset() addsimps [times_divide_eq_right RS sym, NSCDERIV_NSCLIM_iff, - minus_mult_right, complex_add_mult_distrib2 RS sym, + minus_mult_right, right_distrib RS sym, complex_diff_def] delsimps [times_divide_eq_right, minus_mult_right RS sym]) 1); by (etac (NSCLIM_const RS NSCLIM_mult) 1); @@ -866,9 +865,8 @@ Goal "NSCDERIV f x :> D ==> NSCDERIV (%x. -(f x)) x :> -D"; by (asm_full_simp_tac (simpset() addsimps [NSCDERIV_NSCLIM_iff,complex_diff_def]) 1); -by (res_inst_tac [("t","f x")] (complex_minus_minus RS subst) 1); -by (asm_simp_tac (simpset() addsimps [complex_minus_add_distrib RS sym] - delsimps [complex_minus_add_distrib, complex_minus_minus] +by (res_inst_tac [("t","f x")] (minus_minus RS subst) 1); +by (asm_simp_tac (simpset() addsimps [minus_add_distrib RS sym] delsimps [minus_add_distrib, minus_minus] ) 1); @@ -1032,12 +1030,12 @@ by (dtac (CDERIV_Id RS CDERIV_mult) 2); by (auto_tac (claset(), simpset() addsimps [complex_of_real_add RS sym, - complex_add_mult_distrib,real_of_nat_Suc] + left_distrib,real_of_nat_Suc] delsimps [complex_of_real_add])); by (case_tac "n" 1); by (auto_tac (claset(), - simpset() addsimps [complex_mult_assoc, complex_add_commute])); -by (auto_tac (claset(),simpset() addsimps [complex_mult_commute])); + simpset() addsimps [mult_assoc, add_commute])); +by (auto_tac (claset(),simpset() addsimps [mult_commute])); qed "CDERIV_pow"; Addsimps [CDERIV_pow,simplify (simpset()) CDERIV_pow]; @@ -1095,10 +1093,9 @@ Goal "[| CDERIV f x :> d; f(x) ~= 0 |] \ \ ==> CDERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ 2)))"; -by (rtac (complex_mult_commute RS subst) 1); -by (asm_simp_tac (simpset() addsimps [complex_minus_mult_eq1, - power_inverse] delsimps [complexpow_Suc, minus_mult_left RS sym, -complex_minus_mult_eq1 RS sym]) 1); +by (rtac (mult_commute RS subst) 1); +by (asm_simp_tac (simpset() addsimps [minus_mult_left, + power_inverse] delsimps [complexpow_Suc, minus_mult_left RS sym]) 1); by (fold_goals_tac [o_def]); by (blast_tac (claset() addSIs [CDERIV_chain,CDERIV_inverse]) 1); qed "CDERIV_inverse_fun"; @@ -1126,8 +1123,8 @@ by (dtac CDERIV_mult 2); by (REPEAT(assume_tac 1)); by (asm_full_simp_tac - (simpset() addsimps [complex_divide_def, complex_add_mult_distrib2, - power_inverse,complex_minus_mult_eq1] @ complex_mult_ac + (simpset() addsimps [complex_divide_def, right_distrib, + power_inverse,minus_mult_left] @ mult_ac delsimps [complexpow_Suc, minus_mult_right RS sym, minus_mult_left RS sym]) 1); qed "CDERIV_quotient"; @@ -1161,7 +1158,7 @@ by (Step_tac 1); by (res_inst_tac [("x","%z. if z = x then l else (f(z) - f(x)) / (z - x)")] exI 1); -by (auto_tac (claset(),simpset() addsimps [complex_mult_assoc, +by (auto_tac (claset(),simpset() addsimps [mult_assoc, CLAIM "z ~= x ==> z - x ~= (0::complex)"])); by (auto_tac (claset(),simpset() addsimps [isContc_iff,CDERIV_iff])); by (ALLGOALS(rtac (CLIM_equal RS iffD1))); diff -r 51ddf8963c95 -r 67a628beb981 src/HOL/Complex/CSeries.ML --- a/src/HOL/Complex/CSeries.ML Tue Feb 03 10:19:21 2004 +0100 +++ b/src/HOL/Complex/CSeries.ML Tue Feb 03 11:06:36 2004 +0100 @@ -37,7 +37,7 @@ by (induct_tac "n" 1); by (Auto_tac); by (auto_tac (claset(),simpset() addsimps - [complex_add_mult_distrib2])); + [right_distrib])); qed "sumc_mult"; Goal "n < p --> sumc 0 n f + sumc n p f = sumc 0 p f"; @@ -49,7 +49,7 @@ Goal "n < p ==> sumc 0 p f + \ \ - sumc 0 n f = sumc n p f"; by (dres_inst_tac [("f1","f")] (sumc_split_add RS sym) 1); -by (asm_simp_tac (simpset() addsimps complex_add_ac) 1); +by (asm_simp_tac (simpset() addsimps add_ac) 1); qed "sumc_split_add_minus"; Goal "cmod(sumc m n f) <= sumr m n (%i. cmod(f i))"; @@ -67,7 +67,7 @@ Goal "sumc 0 n (%i. r) = complex_of_real (real n) * r"; by (induct_tac "n" 1); by (auto_tac (claset(), - simpset() addsimps [complex_add_mult_distrib, + simpset() addsimps [left_distrib, complex_of_real_add RS sym, real_of_nat_Suc])); qed "sumc_const"; @@ -109,7 +109,7 @@ by (induct_tac "na" 1); by (auto_tac (claset(),simpset() addsimps [left_distrib, Suc_diff_n, real_of_nat_Suc,complex_of_real_add RS sym, - complex_add_mult_distrib])); + left_distrib])); qed_spec_mp "sumc_interval_const"; Goal "(ALL n. m <= n --> f n = r) & m <= na \ @@ -117,7 +117,7 @@ by (induct_tac "na" 1); by (auto_tac (claset(),simpset() addsimps [left_distrib, Suc_diff_n, real_of_nat_Suc,complex_of_real_add RS sym, - complex_add_mult_distrib])); + left_distrib])); qed_spec_mp "sumc_interval_const2"; (*** diff -r 51ddf8963c95 -r 67a628beb981 src/HOL/Complex/Complex.thy --- a/src/HOL/Complex/Complex.thy Tue Feb 03 10:19:21 2004 +0100 +++ b/src/HOL/Complex/Complex.thy Tue Feb 03 11:06:36 2004 +0100 @@ -1,13 +1,15 @@ (* Title: Complex.thy Author: Jacques D. Fleuriot Copyright: 2001 University of Edinburgh - Description: Complex numbers *) +header {* Complex numbers *} + theory Complex = HLog: -typedef complex = "{p::(real*real). True}" - by blast +subsection {* Representation of complex numbers *} + +datatype complex = Complex real real instance complex :: zero .. instance complex :: one .. @@ -18,18 +20,19 @@ instance complex :: power .. consts - "ii" :: complex ("ii") + "ii" :: complex ("\") + +consts Re :: "complex => real" +primrec "Re (Complex x y) = x" + +consts Im :: "complex => real" +primrec "Im (Complex x y) = y" + +lemma complex_surj [simp]: "Complex (Re z) (Im z) = z" + by (induct z) simp constdefs - (*--- real and Imaginary parts ---*) - - Re :: "complex => real" - "Re(z) == fst(Rep_complex z)" - - Im :: "complex => real" - "Im(z) == snd(Rep_complex z)" - (*----------- modulus ------------*) cmod :: "complex => real" @@ -38,12 +41,12 @@ (*----- injection from reals -----*) complex_of_real :: "real => complex" - "complex_of_real r == Abs_complex(r,0::real)" + "complex_of_real r == Complex r 0" (*------- complex conjugate ------*) cnj :: "complex => complex" - "cnj z == Abs_complex(Re z, -Im z)" + "cnj z == Complex (Re z) (-Im z)" (*------------ Argand -------------*) @@ -57,41 +60,29 @@ defs (overloaded) complex_zero_def: - "0 == Abs_complex(0::real,0)" + "0 == Complex 0 0" complex_one_def: - "1 == Abs_complex(1,0::real)" - - (*------ imaginary unit ----------*) + "1 == Complex 1 0" - i_def: - "ii == Abs_complex(0::real,1)" - - (*----------- negation -----------*) + i_def: "ii == Complex 0 1" - complex_minus_def: - "- (z::complex) == Abs_complex(-Re z, -Im z)" - + complex_minus_def: "- z == Complex (- Re z) (- Im z)" - (*----------- inverse -----------*) complex_inverse_def: - "inverse (z::complex) == Abs_complex(Re(z)/(Re(z) ^ 2 + Im(z) ^ 2), - -Im(z)/(Re(z) ^ 2 + Im(z) ^ 2))" + "inverse z == + Complex (Re z / ((Re z)\ + (Im z)\)) (- Im z / ((Re z)\ + (Im z)\))" complex_add_def: - "w + (z::complex) == Abs_complex(Re(w) + Re(z),Im(w) + Im(z))" + "z + w == Complex (Re z + Re w) (Im z + Im w)" complex_diff_def: - "w - (z::complex) == w + -(z::complex)" + "z - w == z + - (w::complex)" - complex_mult_def: - "w * (z::complex) == Abs_complex(Re(w) * Re(z) - Im(w) * Im(z), - Re(w) * Im(z) + Im(w) * Re(z))" + complex_mult_def: + "z * w == Complex (Re z * Re w - Im z * Im w) (Re z * Im w + Im z * Re w)" - - (*----------- division ----------*) - complex_divide_def: - "w / (z::complex) == w * inverse z" + complex_divide_def: "w / (z::complex) == w * inverse z" constdefs @@ -109,429 +100,152 @@ "expi z == complex_of_real(exp (Re z)) * cis (Im z)" -lemma inj_Rep_complex: "inj Rep_complex" -apply (rule inj_on_inverseI) -apply (rule Rep_complex_inverse) -done - -lemma inj_Abs_complex: "inj Abs_complex" -apply (rule inj_on_inverseI) -apply (rule Abs_complex_inverse) -apply (simp (no_asm) add: complex_def) -done -declare inj_Abs_complex [THEN injD, simp] - -lemma Abs_complex_cancel_iff: "(Abs_complex x = Abs_complex y) = (x = y)" -by (auto dest: inj_Abs_complex [THEN injD]) -declare Abs_complex_cancel_iff [simp] +lemma complex_equality [intro?]: "Re z = Re w ==> Im z = Im w ==> z = w" + by (induct z, induct w) simp -lemma pair_mem_complex: "(x,y) : complex" -by (unfold complex_def, auto) -declare pair_mem_complex [simp] - -lemma Abs_complex_inverse2: "Rep_complex (Abs_complex (x,y)) = (x,y)" -apply (simp (no_asm) add: Abs_complex_inverse) -done -declare Abs_complex_inverse2 [simp] - -lemma eq_Abs_complex: "(!!x y. z = Abs_complex(x,y) ==> P) ==> P" -apply (rule_tac p = "Rep_complex z" in PairE) -apply (drule_tac f = Abs_complex in arg_cong) -apply (force simp add: Rep_complex_inverse) -done - -lemma Re: "Re(Abs_complex(x,y)) = x" -apply (unfold Re_def) -apply (simp (no_asm)) -done +lemma Re: "Re(Complex x y) = x" +by simp declare Re [simp] -lemma Im: "Im(Abs_complex(x,y)) = y" -apply (unfold Im_def) -apply (simp (no_asm)) -done +lemma Im: "Im(Complex x y) = y" +by simp declare Im [simp] -lemma Abs_complex_cancel: "Abs_complex(Re(z),Im(z)) = z" -apply (rule_tac z = z in eq_Abs_complex) -apply (simp (no_asm_simp)) -done -declare Abs_complex_cancel [simp] - lemma complex_Re_Im_cancel_iff: "(w=z) = (Re(w) = Re(z) & Im(w) = Im(z))" -apply (rule_tac z = w in eq_Abs_complex) -apply (rule_tac z = z in eq_Abs_complex) -apply (auto dest: inj_Abs_complex [THEN injD]) -done +by (induct w, induct z, simp) lemma complex_Re_zero: "Re 0 = 0" -apply (unfold complex_zero_def) -apply (simp (no_asm)) -done +by (simp add: complex_zero_def) lemma complex_Im_zero: "Im 0 = 0" -apply (unfold complex_zero_def) -apply (simp (no_asm)) -done +by (simp add: complex_zero_def) declare complex_Re_zero [simp] complex_Im_zero [simp] lemma complex_Re_one: "Re 1 = 1" -apply (unfold complex_one_def) -apply (simp (no_asm)) -done +by (simp add: complex_one_def) declare complex_Re_one [simp] lemma complex_Im_one: "Im 1 = 0" -apply (unfold complex_one_def) -apply (simp (no_asm)) -done +by (simp add: complex_one_def) declare complex_Im_one [simp] lemma complex_Re_i: "Re(ii) = 0" -by (unfold i_def, auto) +by (simp add: i_def) declare complex_Re_i [simp] lemma complex_Im_i: "Im(ii) = 1" -by (unfold i_def, auto) +by (simp add: i_def) declare complex_Im_i [simp] lemma Re_complex_of_real_zero: "Re(complex_of_real 0) = 0" -apply (unfold complex_of_real_def) -apply (simp (no_asm)) -done +by (simp add: complex_of_real_def) declare Re_complex_of_real_zero [simp] lemma Im_complex_of_real_zero: "Im(complex_of_real 0) = 0" -apply (unfold complex_of_real_def) -apply (simp (no_asm)) -done +by (simp add: complex_of_real_def) declare Im_complex_of_real_zero [simp] lemma Re_complex_of_real_one: "Re(complex_of_real 1) = 1" -apply (unfold complex_of_real_def) -apply (simp (no_asm)) -done +by (simp add: complex_of_real_def) declare Re_complex_of_real_one [simp] lemma Im_complex_of_real_one: "Im(complex_of_real 1) = 0" -apply (unfold complex_of_real_def) -apply (simp (no_asm)) -done +by (simp add: complex_of_real_def) declare Im_complex_of_real_one [simp] lemma Re_complex_of_real: "Re(complex_of_real z) = z" -by (unfold complex_of_real_def, auto) +by (simp add: complex_of_real_def) declare Re_complex_of_real [simp] lemma Im_complex_of_real: "Im(complex_of_real z) = 0" -by (unfold complex_of_real_def, auto) +by (simp add: complex_of_real_def) declare Im_complex_of_real [simp] subsection{*Negation*} -lemma complex_minus: "- Abs_complex(x,y) = Abs_complex(-x,-y)" -apply (unfold complex_minus_def) -apply (simp (no_asm)) -done +lemma complex_minus: "- (Complex x y) = Complex (-x) (-y)" +by (simp add: complex_minus_def) lemma complex_Re_minus: "Re (-z) = - Re z" -apply (unfold Re_def) -apply (rule_tac z = z in eq_Abs_complex) -apply (auto simp add: complex_minus) -done +by (simp add: complex_minus_def) lemma complex_Im_minus: "Im (-z) = - Im z" -apply (unfold Im_def) -apply (rule_tac z = z in eq_Abs_complex) -apply (auto simp add: complex_minus) -done - -lemma complex_minus_minus: "- (- z) = (z::complex)" -apply (unfold complex_minus_def) -apply (simp (no_asm)) -done -declare complex_minus_minus [simp] - -lemma inj_complex_minus: "inj(%r::complex. -r)" -apply (rule inj_onI) -apply (drule_tac f = uminus in arg_cong, simp) -done +by (simp add: complex_minus_def) lemma complex_minus_zero: "-(0::complex) = 0" -apply (unfold complex_zero_def) -apply (simp (no_asm) add: complex_minus) -done +by (simp add: complex_minus_def complex_zero_def) declare complex_minus_zero [simp] lemma complex_minus_zero_iff: "(-x = 0) = (x = (0::complex))" -apply (rule_tac z = x in eq_Abs_complex) -apply (auto dest: inj_Abs_complex [THEN injD] - simp add: complex_zero_def complex_minus) -done +by (induct x, simp add: complex_minus_def complex_zero_def) declare complex_minus_zero_iff [simp] -lemma complex_minus_zero_iff2: "(0 = -x) = (x = (0::real))" -by (auto dest: sym) -declare complex_minus_zero_iff2 [simp] - -lemma complex_minus_not_zero_iff: "(-x \ 0) = (x \ (0::complex))" -by auto - subsection{*Addition*} -lemma complex_add: - "Abs_complex(x1,y1) + Abs_complex(x2,y2) = Abs_complex(x1+x2,y1+y2)" -apply (unfold complex_add_def) -apply (simp (no_asm)) -done +lemma complex_add: "Complex x1 y1 + Complex x2 y2 = Complex (x1+x2) (y1+y2)" +by (simp add: complex_add_def) lemma complex_Re_add: "Re(x + y) = Re(x) + Re(y)" -apply (unfold Re_def) -apply (rule_tac z = x in eq_Abs_complex) -apply (rule_tac z = y in eq_Abs_complex) -apply (auto simp add: complex_add) -done +by (simp add: complex_add_def) lemma complex_Im_add: "Im(x + y) = Im(x) + Im(y)" -apply (unfold Im_def) -apply (rule_tac z = x in eq_Abs_complex) -apply (rule_tac z = y in eq_Abs_complex) -apply (auto simp add: complex_add) -done +by (simp add: complex_add_def) lemma complex_add_commute: "(u::complex) + v = v + u" -apply (unfold complex_add_def) -apply (simp (no_asm) add: real_add_commute) -done +by (simp add: complex_add_def add_commute) lemma complex_add_assoc: "((u::complex) + v) + w = u + (v + w)" -apply (unfold complex_add_def) -apply (simp (no_asm) add: real_add_assoc) -done - -lemma complex_add_left_commute: "(x::complex) + (y + z) = y + (x + z)" -apply (unfold complex_add_def) -apply (simp (no_asm) add: add_left_commute) -done - -lemmas complex_add_ac = complex_add_assoc complex_add_commute - complex_add_left_commute +by (simp add: complex_add_def add_assoc) lemma complex_add_zero_left: "(0::complex) + z = z" -apply (unfold complex_add_def complex_zero_def) -apply (simp (no_asm)) -done -declare complex_add_zero_left [simp] +by (simp add: complex_add_def complex_zero_def) lemma complex_add_zero_right: "z + (0::complex) = z" -apply (unfold complex_add_def complex_zero_def) -apply (simp (no_asm)) -done -declare complex_add_zero_right [simp] - -lemma complex_add_minus_right_zero: - "z + -z = (0::complex)" -apply (unfold complex_add_def complex_minus_def complex_zero_def) -apply (simp (no_asm)) -done -declare complex_add_minus_right_zero [simp] - -lemma complex_add_minus_left: - "-z + z = (0::complex)" -apply (unfold complex_add_def complex_minus_def complex_zero_def) -apply (simp (no_asm)) -done - -lemma complex_add_minus_cancel: "z + (- z + w) = (w::complex)" -apply (simp (no_asm) add: complex_add_assoc [symmetric]) -done - -lemma complex_minus_add_cancel: "(-z) + (z + w) = (w::complex)" -by (simp add: complex_add_minus_left complex_add_assoc [symmetric]) - -declare complex_add_minus_cancel [simp] complex_minus_add_cancel [simp] - -lemma complex_add_minus_eq_minus: "x + y = (0::complex) ==> x = -y" -by (auto simp add: complex_Re_Im_cancel_iff complex_Re_add complex_Im_add complex_Re_minus complex_Im_minus) +by (simp add: complex_add_def complex_zero_def) -lemma complex_minus_add_distrib: "-(x + y) = -x + -(y::complex)" -apply (rule_tac z = x in eq_Abs_complex) -apply (rule_tac z = y in eq_Abs_complex) -apply (auto simp add: complex_minus complex_add) -done -declare complex_minus_add_distrib [simp] - -lemma complex_add_left_cancel: "((x::complex) + y = x + z) = (y = z)" -apply safe -apply (drule_tac f = "%t.-x + t" in arg_cong) -apply (simp add: complex_add_minus_left complex_add_assoc [symmetric]) -done -declare complex_add_left_cancel [iff] - -lemma complex_add_right_cancel: "(y + (x::complex)= z + x) = (y = z)" -apply (simp (no_asm) add: complex_add_commute) -done -declare complex_add_right_cancel [simp] - -lemma complex_eq_minus_iff: "((x::complex) = y) = (0 = x + - y)" -apply safe -apply (rule_tac [2] x1 = "-y" in complex_add_right_cancel [THEN iffD1], auto) -done - -lemma complex_eq_minus_iff2: "((x::complex) = y) = (x + - y = 0)" -apply safe -apply (rule_tac [2] x1 = "-y" in complex_add_right_cancel [THEN iffD1], auto) -done - -lemma complex_diff_0: "(0::complex) - x = -x" -apply (simp (no_asm) add: complex_diff_def) -done - -lemma complex_diff_0_right: "x - (0::complex) = x" -apply (simp (no_asm) add: complex_diff_def) -done - -lemma complex_diff_self: "x - x = (0::complex)" -apply (simp (no_asm) add: complex_diff_def) -done - -declare complex_diff_0 [simp] complex_diff_0_right [simp] complex_diff_self [simp] +lemma complex_add_minus_left: "-z + z = (0::complex)" +by (simp add: complex_add_def complex_minus_def complex_zero_def) lemma complex_diff: - "Abs_complex(x1,y1) - Abs_complex(x2,y2) = Abs_complex(x1-x2,y1-y2)" -apply (unfold complex_diff_def) -apply (simp (no_asm) add: complex_add complex_minus) -done - -lemma complex_diff_eq_eq: "((x::complex) - y = z) = (x = z + y)" -by (auto simp add: complex_add_minus_left complex_diff_def complex_add_assoc) - + "Complex x1 y1 - Complex x2 y2 = Complex (x1-x2) (y1-y2)" +by (simp add: complex_add_def complex_minus_def complex_diff_def) subsection{*Multiplication*} lemma complex_mult: - "Abs_complex(x1,y1) * Abs_complex(x2,y2) = - Abs_complex(x1*x2 - y1*y2,x1*y2 + y1*x2)" -apply (unfold complex_mult_def) -apply (simp (no_asm)) -done + "Complex x1 y1 * Complex x2 y2 = Complex (x1*x2 - y1*y2) (x1*y2 + y1*x2)" +by (simp add: complex_mult_def) lemma complex_mult_commute: "(w::complex) * z = z * w" -apply (unfold complex_mult_def) -apply (simp (no_asm) add: real_mult_commute real_add_commute) -done +by (simp add: complex_mult_def mult_commute add_commute) lemma complex_mult_assoc: "((u::complex) * v) * w = u * (v * w)" -apply (unfold complex_mult_def) -apply (simp (no_asm) add: complex_Re_Im_cancel_iff real_mult_assoc right_diff_distrib right_distrib left_diff_distrib left_distrib mult_left_commute) -done - -lemma complex_mult_left_commute: "(x::complex) * (y * z) = y * (x * z)" -apply (unfold complex_mult_def) -apply (simp (no_asm) add: complex_Re_Im_cancel_iff mult_left_commute right_diff_distrib right_distrib) -done - -lemmas complex_mult_ac = complex_mult_assoc complex_mult_commute - complex_mult_left_commute +by (simp add: complex_mult_def mult_ac add_ac + right_diff_distrib right_distrib left_diff_distrib left_distrib) lemma complex_mult_one_left: "(1::complex) * z = z" -apply (unfold complex_one_def) -apply (rule_tac z = z in eq_Abs_complex) -apply (simp (no_asm_simp) add: complex_mult) -done -declare complex_mult_one_left [simp] +by (simp add: complex_mult_def complex_one_def) lemma complex_mult_one_right: "z * (1::complex) = z" -apply (simp (no_asm) add: complex_mult_commute) -done -declare complex_mult_one_right [simp] - -lemma complex_mult_zero_left: "(0::complex) * z = 0" -apply (unfold complex_zero_def) -apply (rule_tac z = z in eq_Abs_complex) -apply (simp (no_asm_simp) add: complex_mult) -done -declare complex_mult_zero_left [simp] - -lemma complex_mult_zero_right: "z * 0 = (0::complex)" -apply (simp (no_asm) add: complex_mult_commute) -done -declare complex_mult_zero_right [simp] - -lemma complex_divide_zero: "0 / z = (0::complex)" -by (unfold complex_divide_def, auto) -declare complex_divide_zero [simp] - -lemma complex_minus_mult_eq1: "-(x * y) = -x * (y::complex)" -apply (rule_tac z = x in eq_Abs_complex) -apply (rule_tac z = y in eq_Abs_complex) -apply (auto simp add: complex_mult complex_minus real_diff_def) -done - -lemma complex_minus_mult_eq2: "-(x * y) = x * -(y::complex)" -apply (rule_tac z = x in eq_Abs_complex) -apply (rule_tac z = y in eq_Abs_complex) -apply (auto simp add: complex_mult complex_minus real_diff_def) -done - -lemma complex_add_mult_distrib: "((z1::complex) + z2) * w = (z1 * w) + (z2 * w)" -apply (rule_tac z = z1 in eq_Abs_complex) -apply (rule_tac z = z2 in eq_Abs_complex) -apply (rule_tac z = w in eq_Abs_complex) -apply (auto simp add: complex_mult complex_add left_distrib real_diff_def add_ac) -done - -lemma complex_add_mult_distrib2: "(w::complex) * (z1 + z2) = (w * z1) + (w * z2)" -apply (rule_tac z1 = "z1 + z2" in complex_mult_commute [THEN ssubst]) -apply (simp (no_asm) add: complex_add_mult_distrib) -apply (simp (no_asm) add: complex_mult_commute) -done - -lemma complex_zero_not_eq_one: "(0::complex) \ 1" -apply (unfold complex_zero_def complex_one_def) -apply (simp (no_asm) add: complex_Re_Im_cancel_iff) -done -declare complex_zero_not_eq_one [simp] -declare complex_zero_not_eq_one [THEN not_sym, simp] +by (simp add: complex_mult_def complex_one_def) subsection{*Inverse*} lemma complex_inverse: - "inverse (Abs_complex(x,y)) = - Abs_complex(x/(x ^ 2 + y ^ 2),-y/(x ^ 2 + y ^ 2))" -apply (unfold complex_inverse_def) -apply (simp (no_asm)) -done - -lemma COMPLEX_INVERSE_ZERO: "inverse 0 = (0::complex)" -by (unfold complex_inverse_def complex_zero_def, auto) - -lemma COMPLEX_DIVISION_BY_ZERO: "a / (0::complex) = 0" -apply (simp (no_asm) add: complex_divide_def COMPLEX_INVERSE_ZERO) -done - -instance complex :: division_by_zero -proof - fix x :: complex - show "inverse 0 = (0::complex)" by (rule COMPLEX_INVERSE_ZERO) - show "x/0 = 0" by (rule COMPLEX_DIVISION_BY_ZERO) -qed + "inverse (Complex x y) = Complex (x/(x ^ 2 + y ^ 2)) (-y/(x ^ 2 + y ^ 2))" +by (simp add: complex_inverse_def) lemma complex_mult_inv_left: "z \ (0::complex) ==> inverse(z) * z = 1" -apply (rule_tac z = z in eq_Abs_complex) +apply (induct z) +apply (rename_tac x y) apply (auto simp add: complex_mult complex_inverse complex_one_def complex_zero_def add_divide_distrib [symmetric] power2_eq_square mult_ac) apply (drule_tac y = y in real_sum_squares_not_zero) apply (drule_tac [2] x = x in real_sum_squares_not_zero2, auto) done -declare complex_mult_inv_left [simp] - -lemma complex_mult_inv_right: "z \ (0::complex) ==> z * inverse(z) = 1" -by (auto intro: complex_mult_commute [THEN subst]) -declare complex_mult_inv_right [simp] subsection {* The field of complex numbers *} @@ -556,14 +270,14 @@ show "1 * z = z" by (rule complex_mult_one_left) show "0 \ (1::complex)" - by (rule complex_zero_not_eq_one) + by (simp add: complex_zero_def complex_one_def) show "(u + v) * w = u * w + v * w" - by (rule complex_add_mult_distrib) + by (simp add: complex_mult_def complex_add_def left_distrib real_diff_def add_ac) show "z+u = z+v ==> u=v" proof - assume eq: "z+u = z+v" hence "(-z + z) + u = (-z + z) + v" by (simp only: eq complex_add_assoc) - thus "u = v" by (simp add: complex_add_minus_left) + thus "u = v" by (simp add: complex_add_minus_left complex_add_zero_left) qed assume neq: "w \ 0" thus "z / w = z * inverse w" @@ -572,40 +286,33 @@ by (simp add: neq complex_mult_inv_left) qed +instance complex :: division_by_zero +proof + show inv: "inverse 0 = (0::complex)" + by (simp add: complex_inverse_def complex_zero_def) + fix x :: complex + show "x/0 = 0" + by (simp add: complex_divide_def inv) +qed -lemma complex_minus_mult_commute: "-x * y = x * -(y::complex)" -apply (simp) -done subsection{*Embedding Properties for @{term complex_of_real} Map*} -lemma inj_complex_of_real: "inj complex_of_real" -apply (rule inj_onI) -apply (auto dest: inj_Abs_complex [THEN injD] simp add: complex_of_real_def) -done - -lemma complex_of_real_one: - "complex_of_real 1 = 1" -apply (unfold complex_one_def complex_of_real_def) -apply (rule refl) -done +lemma complex_of_real_one: "complex_of_real 1 = 1" +by (simp add: complex_one_def complex_of_real_def) declare complex_of_real_one [simp] -lemma complex_of_real_zero: - "complex_of_real 0 = 0" -apply (unfold complex_zero_def complex_of_real_def) -apply (rule refl) -done +lemma complex_of_real_zero: "complex_of_real 0 = 0" +by (simp add: complex_zero_def complex_of_real_def) declare complex_of_real_zero [simp] lemma complex_of_real_eq_iff: "(complex_of_real x = complex_of_real y) = (x = y)" -by (auto dest: inj_complex_of_real [THEN injD]) +by (simp add: complex_of_real_def) declare complex_of_real_eq_iff [iff] lemma complex_of_real_minus: "complex_of_real(-x) = - complex_of_real x" -apply (simp (no_asm) add: complex_of_real_def complex_minus) -done +by (simp add: complex_of_real_def complex_minus) lemma complex_of_real_inverse: "complex_of_real(inverse x) = inverse(complex_of_real x)" @@ -615,133 +322,93 @@ done lemma complex_of_real_add: - "complex_of_real x + complex_of_real y = complex_of_real (x + y)" -apply (simp (no_asm) add: complex_add complex_of_real_def) -done + "complex_of_real x + complex_of_real y = complex_of_real (x + y)" +by (simp add: complex_add complex_of_real_def) lemma complex_of_real_diff: - "complex_of_real x - complex_of_real y = complex_of_real (x - y)" -apply (simp (no_asm) add: complex_of_real_minus [symmetric] complex_diff_def complex_of_real_add) -done + "complex_of_real x - complex_of_real y = complex_of_real (x - y)" +by (simp add: complex_of_real_minus [symmetric] complex_diff_def complex_of_real_add) lemma complex_of_real_mult: - "complex_of_real x * complex_of_real y = complex_of_real (x * y)" -apply (simp (no_asm) add: complex_mult complex_of_real_def) -done + "complex_of_real x * complex_of_real y = complex_of_real (x * y)" +by (simp add: complex_mult complex_of_real_def) lemma complex_of_real_divide: "complex_of_real x / complex_of_real y = complex_of_real(x/y)" -apply (unfold complex_divide_def) -apply (case_tac "y=0") -apply (simp (no_asm_simp) add: DIVISION_BY_ZERO COMPLEX_INVERSE_ZERO) -apply (simp (no_asm_simp) add: complex_of_real_mult [symmetric] complex_of_real_inverse real_divide_def) +apply (simp add: complex_divide_def) +apply (case_tac "y=0", simp) +apply (simp add: complex_of_real_mult [symmetric] complex_of_real_inverse real_divide_def) done -lemma complex_mod: "cmod (Abs_complex(x,y)) = sqrt(x ^ 2 + y ^ 2)" -apply (unfold cmod_def) -apply (simp (no_asm)) -done +lemma complex_mod: "cmod (Complex x y) = sqrt(x ^ 2 + y ^ 2)" +by (simp add: cmod_def) lemma complex_mod_zero: "cmod(0) = 0" -apply (unfold cmod_def) -apply (simp (no_asm)) -done +by (simp add: cmod_def) declare complex_mod_zero [simp] lemma complex_mod_one [simp]: "cmod(1) = 1" by (simp add: cmod_def power2_eq_square) lemma complex_mod_complex_of_real: "cmod(complex_of_real x) = abs x" -apply (simp add: complex_of_real_def power2_eq_square complex_mod) -done +by (simp add: complex_of_real_def power2_eq_square complex_mod) declare complex_mod_complex_of_real [simp] lemma complex_of_real_abs: "complex_of_real (abs x) = complex_of_real(cmod(complex_of_real x))" -by (simp) +by simp subsection{*Conjugation is an Automorphism*} -lemma complex_cnj: "cnj (Abs_complex(x,y)) = Abs_complex(x,-y)" -apply (unfold cnj_def) -apply (simp (no_asm)) -done - -lemma inj_cnj: "inj cnj" -apply (rule inj_onI) -apply (auto simp add: cnj_def Abs_complex_cancel_iff complex_Re_Im_cancel_iff) -done +lemma complex_cnj: "cnj (Complex x y) = Complex x (-y)" +by (simp add: cnj_def) lemma complex_cnj_cancel_iff: "(cnj x = cnj y) = (x = y)" -by (auto dest: inj_cnj [THEN injD]) +by (simp add: cnj_def complex_Re_Im_cancel_iff) declare complex_cnj_cancel_iff [simp] lemma complex_cnj_cnj: "cnj (cnj z) = z" -apply (unfold cnj_def) -apply (simp (no_asm)) -done +by (simp add: cnj_def) declare complex_cnj_cnj [simp] lemma complex_cnj_complex_of_real: - "cnj (complex_of_real x) = complex_of_real x" -apply (unfold complex_of_real_def) -apply (simp (no_asm) add: complex_cnj) -done + "cnj (complex_of_real x) = complex_of_real x" +by (simp add: complex_of_real_def complex_cnj) declare complex_cnj_complex_of_real [simp] lemma complex_mod_cnj: "cmod (cnj z) = cmod z" -apply (rule_tac z = z in eq_Abs_complex) -apply (simp (no_asm_simp) add: complex_cnj complex_mod power2_eq_square) -done +by (induct z, simp add: complex_cnj complex_mod power2_eq_square) declare complex_mod_cnj [simp] lemma complex_cnj_minus: "cnj (-z) = - cnj z" -apply (unfold cnj_def) -apply (simp (no_asm) add: complex_minus complex_Re_minus complex_Im_minus) -done +by (simp add: cnj_def complex_minus complex_Re_minus complex_Im_minus) lemma complex_cnj_inverse: "cnj(inverse z) = inverse(cnj z)" -apply (rule_tac z = z in eq_Abs_complex) -apply (simp (no_asm_simp) add: complex_cnj complex_inverse power2_eq_square) -done +by (induct z, simp add: complex_cnj complex_inverse power2_eq_square) lemma complex_cnj_add: "cnj(w + z) = cnj(w) + cnj(z)" -apply (rule_tac z = w in eq_Abs_complex) -apply (rule_tac z = z in eq_Abs_complex) -apply (simp (no_asm_simp) add: complex_cnj complex_add) -done +by (induct w, induct z, simp add: complex_cnj complex_add) lemma complex_cnj_diff: "cnj(w - z) = cnj(w) - cnj(z)" -apply (unfold complex_diff_def) -apply (simp (no_asm) add: complex_cnj_add complex_cnj_minus) -done +by (simp add: complex_diff_def complex_cnj_add complex_cnj_minus) lemma complex_cnj_mult: "cnj(w * z) = cnj(w) * cnj(z)" -apply (rule_tac z = w in eq_Abs_complex) -apply (rule_tac z = z in eq_Abs_complex) -apply (simp (no_asm_simp) add: complex_cnj complex_mult) -done +by (induct w, induct z, simp add: complex_cnj complex_mult) lemma complex_cnj_divide: "cnj(w / z) = (cnj w)/(cnj z)" -apply (unfold complex_divide_def) -apply (simp (no_asm) add: complex_cnj_mult complex_cnj_inverse) -done +by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse) lemma complex_cnj_one: "cnj 1 = 1" -apply (unfold cnj_def complex_one_def) -apply (simp (no_asm)) -done +by (simp add: cnj_def complex_one_def) declare complex_cnj_one [simp] lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re(z))" -apply (rule_tac z = z in eq_Abs_complex) -apply (simp (no_asm_simp) add: complex_add complex_cnj complex_of_real_def) -done +by (induct z, simp add: complex_add complex_cnj complex_of_real_def) lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im(z)) * ii" -apply (rule_tac z = z in eq_Abs_complex) +apply (induct z) apply (simp add: complex_add complex_cnj complex_of_real_def complex_diff_def complex_minus i_def complex_mult) done @@ -750,81 +417,62 @@ by (simp add: cnj_def complex_zero_def) lemma complex_cnj_zero_iff: "(cnj z = 0) = (z = 0)" -apply (rule_tac z = z in eq_Abs_complex) -apply (auto simp add: complex_zero_def complex_cnj) -done +by (induct z, simp add: complex_zero_def complex_cnj) declare complex_cnj_zero_iff [iff] lemma complex_mult_cnj: "z * cnj z = complex_of_real (Re(z) ^ 2 + Im(z) ^ 2)" -apply (rule_tac z = z in eq_Abs_complex) -apply (auto simp add: complex_cnj complex_mult complex_of_real_def power2_eq_square) -done +by (induct z, simp add: complex_cnj complex_mult complex_of_real_def power2_eq_square) subsection{*Algebra*} lemma complex_add_left_cancel_zero: "(x + y = x) = (y = (0::complex))" -apply (unfold complex_zero_def) -apply (rule_tac z = x in eq_Abs_complex) -apply (rule_tac z = y in eq_Abs_complex) -apply (auto simp add: complex_add) -done +by (induct x, induct y, simp add: complex_zero_def complex_add) declare complex_add_left_cancel_zero [simp] -lemma complex_diff_mult_distrib: - "((z1::complex) - z2) * w = (z1 * w) - (z2 * w)" -apply (unfold complex_diff_def) -apply (simp (no_asm) add: complex_add_mult_distrib) -done +lemma complex_diff_mult_distrib: "((z1::complex) - z2) * w = (z1 * w) - (z2 * w)" +by (simp add: complex_diff_def left_distrib) -lemma complex_diff_mult_distrib2: - "(w::complex) * (z1 - z2) = (w * z1) - (w * z2)" -apply (unfold complex_diff_def) -apply (simp (no_asm) add: complex_add_mult_distrib2) -done +lemma complex_diff_mult_distrib2: "(w::complex) * (z1 - z2) = (w * z1) - (w * z2)" +by (simp add: complex_diff_def right_distrib) subsection{*Modulus*} lemma complex_mod_eq_zero_cancel: "(cmod x = 0) = (x = 0)" -apply (rule_tac z = x in eq_Abs_complex) -apply (auto intro: real_sum_squares_cancel real_sum_squares_cancel2 simp add: complex_mod complex_zero_def power2_eq_square) +apply (induct x) +apply (auto intro: real_sum_squares_cancel real_sum_squares_cancel2 + simp add: complex_mod complex_zero_def power2_eq_square) done declare complex_mod_eq_zero_cancel [simp] -lemma complex_mod_complex_of_real_of_nat: "cmod (complex_of_real(real (n::nat))) = real n" -apply (simp (no_asm)) -done +lemma complex_mod_complex_of_real_of_nat: + "cmod (complex_of_real(real (n::nat))) = real n" +by simp declare complex_mod_complex_of_real_of_nat [simp] lemma complex_mod_minus: "cmod (-x) = cmod(x)" -apply (rule_tac z = x in eq_Abs_complex) -apply (simp (no_asm_simp) add: complex_mod complex_minus power2_eq_square) -done +by (induct x, simp add: complex_mod complex_minus power2_eq_square) declare complex_mod_minus [simp] lemma complex_mod_mult_cnj: "cmod(z * cnj(z)) = cmod(z) ^ 2" -apply (rule_tac z = z in eq_Abs_complex) -apply (simp (no_asm_simp) add: complex_mod complex_cnj complex_mult); -apply (simp (no_asm) add: power2_eq_square real_abs_def) +apply (induct z, simp add: complex_mod complex_cnj complex_mult) +apply (simp add: power2_eq_square real_abs_def) done -lemma complex_mod_squared: "cmod(Abs_complex(x,y)) ^ 2 = x ^ 2 + y ^ 2" -by (unfold cmod_def, auto) +lemma complex_mod_squared: "cmod(Complex x y) ^ 2 = x ^ 2 + y ^ 2" +by (simp add: cmod_def) lemma complex_mod_ge_zero: "0 \ cmod x" -apply (unfold cmod_def) -apply (auto intro: real_sqrt_ge_zero) -done +by (simp add: cmod_def) declare complex_mod_ge_zero [simp] lemma abs_cmod_cancel: "abs(cmod x) = cmod x" -by (auto intro: abs_eqI1) +by (simp add: abs_if linorder_not_less) declare abs_cmod_cancel [simp] lemma complex_mod_mult: "cmod(x*y) = cmod(x) * cmod(y)" -apply (rule_tac z = x in eq_Abs_complex) -apply (rule_tac z = y in eq_Abs_complex) +apply (induct x, induct y) apply (auto simp add: complex_mult complex_mod real_sqrt_mult_distrib2 [symmetric] simp del: realpow_Suc) apply (rule_tac n = 1 in power_inject_base) apply (auto simp add: power2_eq_square [symmetric] simp del: realpow_Suc) @@ -832,38 +480,30 @@ done lemma complex_mod_add_squared_eq: "cmod(x + y) ^ 2 = cmod(x) ^ 2 + cmod(y) ^ 2 + 2 * Re(x * cnj y)" -apply (rule_tac z = x in eq_Abs_complex) -apply (rule_tac z = y in eq_Abs_complex) +apply (induct x, induct y) apply (auto simp add: complex_add complex_mod_squared complex_mult complex_cnj real_diff_def simp del: realpow_Suc) apply (auto simp add: right_distrib left_distrib power2_eq_square mult_ac add_ac) done lemma complex_Re_mult_cnj_le_cmod: "Re(x * cnj y) \ cmod(x * cnj y)" -apply (rule_tac z = x in eq_Abs_complex) -apply (rule_tac z = y in eq_Abs_complex) +apply (induct x, induct y) apply (auto simp add: complex_mod complex_mult complex_cnj real_diff_def simp del: realpow_Suc) done declare complex_Re_mult_cnj_le_cmod [simp] lemma complex_Re_mult_cnj_le_cmod2: "Re(x * cnj y) \ cmod(x * y)" -apply (cut_tac x = x and y = y in complex_Re_mult_cnj_le_cmod) -apply (simp add: complex_mod_mult) -done +by (insert complex_Re_mult_cnj_le_cmod [of x y], simp add: complex_mod_mult) declare complex_Re_mult_cnj_le_cmod2 [simp] lemma real_sum_squared_expand: "((x::real) + y) ^ 2 = x ^ 2 + y ^ 2 + 2 * x * y" -apply (simp (no_asm) add: left_distrib right_distrib power2_eq_square) -done +by (simp add: left_distrib right_distrib power2_eq_square) lemma complex_mod_triangle_squared: "cmod (x + y) ^ 2 \ (cmod(x) + cmod(y)) ^ 2" -apply (simp (no_asm) add: real_sum_squared_expand complex_mod_add_squared_eq real_mult_assoc complex_mod_mult [symmetric]) -done +by (simp add: real_sum_squared_expand complex_mod_add_squared_eq real_mult_assoc complex_mod_mult [symmetric]) declare complex_mod_triangle_squared [simp] lemma complex_mod_minus_le_complex_mod: "- cmod x \ cmod x" -apply (rule order_trans [OF _ complex_mod_ge_zero]) -apply (simp (no_asm)) -done +by (rule order_trans [OF _ complex_mod_ge_zero], simp) declare complex_mod_minus_le_complex_mod [simp] lemma complex_mod_triangle_ineq: "cmod (x + y) \ cmod(x) + cmod(y)" @@ -874,15 +514,11 @@ declare complex_mod_triangle_ineq [simp] lemma complex_mod_triangle_ineq2: "cmod(b + a) - cmod b \ cmod a" -apply (cut_tac x1 = b and y1 = a and c = "-cmod b" - in complex_mod_triangle_ineq [THEN add_right_mono]) -apply (simp (no_asm)) -done +by (insert complex_mod_triangle_ineq [THEN add_right_mono, of b a"-cmod b"], simp) declare complex_mod_triangle_ineq2 [simp] lemma complex_mod_diff_commute: "cmod (x - y) = cmod (y - x)" -apply (rule_tac z = x in eq_Abs_complex) -apply (rule_tac z = y in eq_Abs_complex) +apply (induct x, induct y) apply (auto simp add: complex_diff complex_mod right_diff_distrib power2_eq_square left_diff_distrib add_ac mult_ac) done @@ -901,18 +537,16 @@ apply (rule complex_mod_minus [THEN subst]) apply (rule order_trans) apply (rule_tac [2] complex_mod_triangle_ineq) -apply (auto simp add: complex_add_ac) +apply (auto simp add: add_ac) done declare complex_mod_diff_ineq [simp] lemma complex_Re_le_cmod: "Re z \ cmod z" -apply (rule_tac z = z in eq_Abs_complex) -apply (auto simp add: complex_mod simp del: realpow_Suc) -done +by (induct z, simp add: complex_mod del: realpow_Suc) declare complex_Re_le_cmod [simp] lemma complex_mod_gt_zero: "z \ 0 ==> 0 < cmod z" -apply (cut_tac x = z in complex_mod_ge_zero) +apply (insert complex_mod_ge_zero [of z]) apply (drule order_le_imp_less_or_eq, auto) done @@ -920,22 +554,16 @@ subsection{*A Few More Theorems*} lemma complex_mod_inverse: "cmod(inverse x) = inverse(cmod x)" -apply (case_tac "x=0", simp add: COMPLEX_INVERSE_ZERO) +apply (case_tac "x=0", simp) apply (rule_tac c1 = "cmod x" in real_mult_left_cancel [THEN iffD1]) apply (auto simp add: complex_mod_mult [symmetric]) done -lemma complex_mod_divide: - "cmod(x/y) = cmod(x)/(cmod y)" -apply (unfold complex_divide_def real_divide_def) -apply (auto simp add: complex_mod_mult complex_mod_inverse) -done +lemma complex_mod_divide: "cmod(x/y) = cmod(x)/(cmod y)" +by (simp add: complex_divide_def real_divide_def, simp add: complex_mod_mult complex_mod_inverse) -lemma complex_inverse_divide: - "inverse(x/y) = y/(x::complex)" -apply (unfold complex_divide_def) -apply (auto simp add: inverse_mult_distrib complex_mult_commute) -done +lemma complex_inverse_divide: "inverse(x/y) = y/(x::complex)" +by (simp add: complex_divide_def inverse_mult_distrib mult_commute) declare complex_inverse_divide [simp] @@ -977,34 +605,29 @@ by (simp add: i_def complex_mult complex_one_def complex_minus numeral_2_eq_2) lemma complex_i_not_zero [simp]: "ii \ 0" -by (unfold i_def complex_zero_def, auto) +by (simp add: i_def complex_zero_def) subsection{*The Function @{term sgn}*} lemma sgn_zero: "sgn 0 = 0" -apply (unfold sgn_def) -apply (simp (no_asm)) -done +by (simp add: sgn_def) declare sgn_zero [simp] lemma sgn_one: "sgn 1 = 1" -apply (unfold sgn_def) -apply (simp (no_asm)) -done +by (simp add: sgn_def) declare sgn_one [simp] lemma sgn_minus: "sgn (-z) = - sgn(z)" -by (unfold sgn_def, auto) +by (simp add: sgn_def) lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)" -apply (unfold sgn_def) -apply (simp (no_asm)) +apply (simp add: sgn_def) done lemma complex_split: "\x y. z = complex_of_real(x) + ii * complex_of_real(y)" -apply (rule_tac z = z in eq_Abs_complex) +apply (induct z) apply (auto simp add: complex_of_real_def i_def complex_mult complex_add) done @@ -1017,59 +640,46 @@ declare Im_complex_i [simp] lemma i_mult_eq: "ii * ii = complex_of_real (-1)" -apply (unfold i_def complex_of_real_def) -apply (auto simp add: complex_mult complex_add) -done +by (simp add: i_def complex_of_real_def complex_mult complex_add) lemma i_mult_eq2: "ii * ii = -(1::complex)" -apply (unfold i_def complex_one_def) -apply (simp (no_asm) add: complex_mult complex_minus) -done +by (simp add: i_def complex_one_def complex_mult complex_minus) declare i_mult_eq2 [simp] lemma cmod_i: "cmod (complex_of_real(x) + ii * complex_of_real(y)) = sqrt (x ^ 2 + y ^ 2)" -apply (auto simp add: complex_mult complex_add i_def complex_of_real_def cmod_def) -done +by (simp add: complex_mult complex_add i_def complex_of_real_def cmod_def) lemma complex_eq_Re_eq: "complex_of_real xa + ii * complex_of_real ya = complex_of_real xb + ii * complex_of_real yb ==> xa = xb" -apply (unfold complex_of_real_def i_def) -apply (auto simp add: complex_mult complex_add) -done +by (simp add: complex_of_real_def i_def complex_mult complex_add) lemma complex_eq_Im_eq: "complex_of_real xa + ii * complex_of_real ya = complex_of_real xb + ii * complex_of_real yb ==> ya = yb" -apply (unfold complex_of_real_def i_def) -apply (auto simp add: complex_mult complex_add) -done +by (simp add: complex_of_real_def i_def complex_mult complex_add) lemma complex_eq_cancel_iff: "(complex_of_real xa + ii * complex_of_real ya = complex_of_real xb + ii * complex_of_real yb) = ((xa = xb) & (ya = yb))" -apply (auto intro: complex_eq_Im_eq complex_eq_Re_eq) -done +by (auto intro: complex_eq_Im_eq complex_eq_Re_eq) declare complex_eq_cancel_iff [iff] lemma complex_eq_cancel_iffA: "(complex_of_real xa + complex_of_real ya * ii = - complex_of_real xb + complex_of_real yb * ii ) = ((xa = xb) & (ya = yb))" -apply (auto simp add: complex_mult_commute) -done + complex_of_real xb + complex_of_real yb * ii) = ((xa = xb) & (ya = yb))" +by (simp add: mult_commute) declare complex_eq_cancel_iffA [iff] lemma complex_eq_cancel_iffB: "(complex_of_real xa + complex_of_real ya * ii = complex_of_real xb + ii * complex_of_real yb) = ((xa = xb) & (ya = yb))" -apply (auto simp add: complex_mult_commute) -done +by (auto simp add: mult_commute) declare complex_eq_cancel_iffB [iff] lemma complex_eq_cancel_iffC: "(complex_of_real xa + ii * complex_of_real ya = complex_of_real xb + complex_of_real yb * ii) = ((xa = xb) & (ya = yb))" -apply (auto simp add: complex_mult_commute) -done +by (auto simp add: mult_commute) declare complex_eq_cancel_iffC [iff] lemma complex_eq_cancel_iff2: "(complex_of_real x + ii * complex_of_real y = @@ -1081,8 +691,7 @@ lemma complex_eq_cancel_iff2a: "(complex_of_real x + complex_of_real y * ii = complex_of_real xa) = (x = xa & y = 0)" -apply (auto simp add: complex_mult_commute) -done +by (auto simp add: mult_commute) declare complex_eq_cancel_iff2a [simp] lemma complex_eq_cancel_iff3: "(complex_of_real x + ii * complex_of_real y = @@ -1094,39 +703,31 @@ lemma complex_eq_cancel_iff3a: "(complex_of_real x + complex_of_real y * ii = ii * complex_of_real ya) = (x = 0 & y = ya)" -apply (auto simp add: complex_mult_commute) -done +by (auto simp add: mult_commute) declare complex_eq_cancel_iff3a [simp] lemma complex_split_Re_zero: "complex_of_real x + ii * complex_of_real y = 0 ==> x = 0" -apply (unfold complex_of_real_def i_def complex_zero_def) -apply (auto simp add: complex_mult complex_add) -done +by (simp add: complex_of_real_def i_def complex_zero_def complex_mult complex_add) lemma complex_split_Im_zero: "complex_of_real x + ii * complex_of_real y = 0 ==> y = 0" -apply (unfold complex_of_real_def i_def complex_zero_def) -apply (auto simp add: complex_mult complex_add) -done +by (simp add: complex_of_real_def i_def complex_zero_def complex_mult complex_add) -lemma Re_sgn: - "Re(sgn z) = Re(z)/cmod z" -apply (unfold sgn_def complex_divide_def) -apply (rule_tac z = z in eq_Abs_complex) -apply (auto simp add: complex_of_real_inverse [symmetric]) -apply (auto simp add: complex_of_real_def complex_mult real_divide_def) +lemma Re_sgn: "Re(sgn z) = Re(z)/cmod z" +apply (induct z) +apply (simp add: sgn_def complex_divide_def complex_of_real_inverse [symmetric]) +apply (simp add: complex_of_real_def complex_mult real_divide_def) done declare Re_sgn [simp] lemma Im_sgn: "Im(sgn z) = Im(z)/cmod z" -apply (unfold sgn_def complex_divide_def) -apply (rule_tac z = z in eq_Abs_complex) -apply (auto simp add: complex_of_real_inverse [symmetric]) -apply (auto simp add: complex_of_real_def complex_mult real_divide_def) +apply (induct z) +apply (simp add: sgn_def complex_divide_def complex_of_real_inverse [symmetric]) +apply (simp add: complex_of_real_def complex_mult real_divide_def) done declare Im_sgn [simp] @@ -1134,9 +735,8 @@ "inverse(complex_of_real x + ii * complex_of_real y) = complex_of_real(x/(x ^ 2 + y ^ 2)) - ii * complex_of_real(y/(x ^ 2 + y ^ 2))" -apply (unfold complex_of_real_def i_def) -apply (auto simp add: complex_mult complex_add complex_diff_def complex_minus complex_inverse real_divide_def) -done +by (simp add: complex_of_real_def i_def complex_mult complex_add + complex_diff_def complex_minus complex_inverse real_divide_def) (*----------------------------------------------------------------------------*) (* Many of the theorems below need to be moved elsewhere e.g. Transc. Also *) @@ -1146,47 +746,37 @@ lemma complex_of_real_zero_iff [simp]: "(complex_of_real y = 0) = (y = 0)" by (auto simp add: complex_zero_def complex_of_real_def) -lemma Re_mult_i_eq: - "Re (ii * complex_of_real y) = 0" -apply (unfold i_def complex_of_real_def) -apply (auto simp add: complex_mult) -done +lemma Re_mult_i_eq: "Re (ii * complex_of_real y) = 0" +by (simp add: i_def complex_of_real_def complex_mult) declare Re_mult_i_eq [simp] -lemma Im_mult_i_eq: - "Im (ii * complex_of_real y) = y" -apply (unfold i_def complex_of_real_def) -apply (auto simp add: complex_mult) -done +lemma Im_mult_i_eq: "Im (ii * complex_of_real y) = y" +by (simp add: i_def complex_of_real_def complex_mult) declare Im_mult_i_eq [simp] -lemma complex_mod_mult_i: - "cmod (ii * complex_of_real y) = abs y" -apply (unfold i_def complex_of_real_def) -apply (auto simp add: complex_mult complex_mod power2_eq_square) -done +lemma complex_mod_mult_i: "cmod (ii * complex_of_real y) = abs y" +by (simp add: i_def complex_of_real_def complex_mult complex_mod power2_eq_square) declare complex_mod_mult_i [simp] lemma cos_arg_i_mult_zero_pos: "0 < y ==> cos (arg(ii * complex_of_real y)) = 0" -apply (unfold arg_def) -apply (auto simp add: abs_eqI2) +apply (simp add: arg_def abs_if) apply (rule_tac a = "pi/2" in someI2, auto) apply (rule order_less_trans [of _ 0], auto) done lemma cos_arg_i_mult_zero_neg: "y < 0 ==> cos (arg(ii * complex_of_real y)) = 0" -apply (unfold arg_def) -apply (auto simp add: abs_minus_eqI2) +apply (simp add: arg_def abs_if) apply (rule_tac a = "- pi/2" in someI2, auto) apply (rule order_trans [of _ 0], auto) done lemma cos_arg_i_mult_zero [simp] : "y \ 0 ==> cos (arg(ii * complex_of_real y)) = 0" -by (cut_tac x = y and y = 0 in linorder_less_linear, - auto simp add: cos_arg_i_mult_zero_pos cos_arg_i_mult_zero_neg) +apply (insert linorder_less_linear [of y 0]) +apply (auto simp add: cos_arg_i_mult_zero_pos cos_arg_i_mult_zero_neg) +done subsection{*Finally! Polar Form for Complex Numbers*} @@ -1198,97 +788,75 @@ done lemma rcis_Ex: "\r a. z = rcis r a" -apply (unfold rcis_def cis_def) +apply (simp add: rcis_def cis_def) apply (rule complex_split_polar) done lemma Re_complex_polar: "Re(complex_of_real r * (complex_of_real(cos a) + ii * complex_of_real(sin a))) = r * cos a" -apply (auto simp add: complex_add_mult_distrib2 complex_of_real_mult complex_mult_ac) -done +by (auto simp add: right_distrib complex_of_real_mult mult_ac) declare Re_complex_polar [simp] lemma Re_rcis: "Re(rcis r a) = r * cos a" -by (unfold rcis_def cis_def, auto) +by (simp add: rcis_def cis_def) declare Re_rcis [simp] lemma Im_complex_polar [simp]: "Im(complex_of_real r * (complex_of_real(cos a) + ii * complex_of_real(sin a))) = r * sin a" -by (auto simp add: complex_add_mult_distrib2 complex_of_real_mult mult_ac) +by (auto simp add: right_distrib complex_of_real_mult mult_ac) lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a" -by (unfold rcis_def cis_def, auto) +by (simp add: rcis_def cis_def) lemma complex_mod_complex_polar [simp]: "cmod (complex_of_real r * (complex_of_real(cos a) + ii * complex_of_real(sin a))) = abs r" -by (auto simp add: complex_add_mult_distrib2 cmod_i complex_of_real_mult +by (auto simp add: right_distrib cmod_i complex_of_real_mult right_distrib [symmetric] power_mult_distrib mult_ac simp del: realpow_Suc) lemma complex_mod_rcis: "cmod(rcis r a) = abs r" -by (unfold rcis_def cis_def, auto) +by (simp add: rcis_def cis_def) declare complex_mod_rcis [simp] lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))" -apply (unfold cmod_def) +apply (simp add: cmod_def) apply (rule real_sqrt_eq_iff [THEN iffD2]) apply (auto simp add: complex_mult_cnj) done lemma complex_Re_cnj: "Re(cnj z) = Re z" -apply (rule_tac z = z in eq_Abs_complex) -apply (auto simp add: complex_cnj) -done +by (induct z, simp add: complex_cnj) declare complex_Re_cnj [simp] lemma complex_Im_cnj: "Im(cnj z) = - Im z" -apply (rule_tac z = z in eq_Abs_complex) -apply (auto simp add: complex_cnj) -done +by (induct z, simp add: complex_cnj) declare complex_Im_cnj [simp] lemma complex_In_mult_cnj_zero: "Im (z * cnj z) = 0" -apply (rule_tac z = z in eq_Abs_complex) -apply (auto simp add: complex_cnj complex_mult) -done +by (induct z, simp add: complex_cnj complex_mult) declare complex_In_mult_cnj_zero [simp] lemma complex_Re_mult: "[| Im w = 0; Im z = 0 |] ==> Re(w * z) = Re(w) * Re(z)" -apply (rule_tac z = z in eq_Abs_complex) -apply (rule_tac z = w in eq_Abs_complex) -apply (auto simp add: complex_mult) -done +by (induct z, induct w, simp add: complex_mult) lemma complex_Re_mult_complex_of_real: "Re (z * complex_of_real c) = Re(z) * c" -apply (unfold complex_of_real_def) -apply (rule_tac z = z in eq_Abs_complex) -apply (auto simp add: complex_mult) -done +by (induct z, simp add: complex_of_real_def complex_mult) declare complex_Re_mult_complex_of_real [simp] lemma complex_Im_mult_complex_of_real: "Im (z * complex_of_real c) = Im(z) * c" -apply (unfold complex_of_real_def) -apply (rule_tac z = z in eq_Abs_complex) -apply (auto simp add: complex_mult) -done +by (induct z, simp add: complex_of_real_def complex_mult) declare complex_Im_mult_complex_of_real [simp] lemma complex_Re_mult_complex_of_real2: "Re (complex_of_real c * z) = c * Re(z)" -apply (unfold complex_of_real_def) -apply (rule_tac z = z in eq_Abs_complex) -apply (auto simp add: complex_mult) -done +by (induct z, simp add: complex_of_real_def complex_mult) declare complex_Re_mult_complex_of_real2 [simp] lemma complex_Im_mult_complex_of_real2: "Im (complex_of_real c * z) = c * Im(z)" -apply (unfold complex_of_real_def) -apply (rule_tac z = z in eq_Abs_complex) -apply (auto simp add: complex_mult) -done +by (induct z, simp add: complex_of_real_def complex_mult) declare complex_Im_mult_complex_of_real2 [simp] (*---------------------------------------------------------------------------*) @@ -1296,64 +864,51 @@ (*---------------------------------------------------------------------------*) lemma cis_rcis_eq: "cis a = rcis 1 a" -apply (unfold rcis_def) -apply (simp (no_asm)) -done +by (simp add: rcis_def) lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)" -apply (unfold rcis_def cis_def) -apply (auto simp add: cos_add sin_add complex_add_mult_distrib2 complex_add_mult_distrib complex_mult_ac complex_add_ac) -apply (auto simp add: complex_add_mult_distrib2 [symmetric] complex_mult_assoc [symmetric] complex_of_real_mult complex_of_real_add complex_add_assoc [symmetric] i_mult_eq simp del: i_mult_eq2) -apply (auto simp add: complex_add_ac) +apply (simp add: rcis_def cis_def cos_add sin_add right_distrib left_distrib + mult_ac add_ac) +apply (auto simp add: right_distrib [symmetric] complex_mult_assoc [symmetric] complex_of_real_mult complex_of_real_add complex_add_assoc [symmetric] i_mult_eq simp del: i_mult_eq2) +apply (auto simp add: add_ac) apply (auto simp add: complex_add_assoc [symmetric] complex_of_real_add right_distrib real_diff_def mult_ac add_ac) done lemma cis_mult: "cis a * cis b = cis (a + b)" -apply (simp (no_asm) add: cis_rcis_eq rcis_mult) -done +by (simp add: cis_rcis_eq rcis_mult) lemma cis_zero: "cis 0 = 1" -by (unfold cis_def, auto) +by (simp add: cis_def) declare cis_zero [simp] lemma cis_zero2: "cis 0 = complex_of_real 1" -by (unfold cis_def, auto) +by (simp add: cis_def) declare cis_zero2 [simp] lemma rcis_zero_mod: "rcis 0 a = 0" -apply (unfold rcis_def) -apply (simp (no_asm)) -done +by (simp add: rcis_def) declare rcis_zero_mod [simp] lemma rcis_zero_arg: "rcis r 0 = complex_of_real r" -apply (unfold rcis_def) -apply (simp (no_asm)) -done +by (simp add: rcis_def) declare rcis_zero_arg [simp] lemma complex_of_real_minus_one: "complex_of_real (-(1::real)) = -(1::complex)" -apply (unfold complex_of_real_def complex_one_def) -apply (simp (no_asm) add: complex_minus) +apply (simp add: complex_of_real_def complex_one_def complex_minus) done lemma complex_i_mult_minus: "ii * (ii * x) = - x" -apply (simp (no_asm) add: complex_mult_assoc [symmetric]) -done +by (simp add: complex_mult_assoc [symmetric]) declare complex_i_mult_minus [simp] -lemma complex_i_mult_minus2: "ii * ii * x = - x" -apply (simp (no_asm)) -done -declare complex_i_mult_minus2 [simp] lemma cis_real_of_nat_Suc_mult: "cis (real (Suc n) * a) = cis a * cis (real n * a)" -apply (unfold cis_def) -apply (auto simp add: real_of_nat_Suc left_distrib cos_add sin_add complex_add_mult_distrib complex_add_mult_distrib2 complex_of_real_add complex_of_real_mult complex_mult_ac complex_add_ac) -apply (auto simp add: complex_add_mult_distrib2 [symmetric] complex_mult_assoc [symmetric] i_mult_eq complex_of_real_mult complex_of_real_add complex_add_assoc [symmetric] complex_of_real_minus [symmetric] real_diff_def mult_ac simp del: i_mult_eq2) +apply (simp add: cis_def) +apply (auto simp add: real_of_nat_Suc left_distrib cos_add sin_add left_distrib right_distrib complex_of_real_add complex_of_real_mult mult_ac add_ac) +apply (auto simp add: right_distrib [symmetric] complex_mult_assoc [symmetric] i_mult_eq complex_of_real_mult complex_of_real_add complex_add_assoc [symmetric] complex_of_real_minus [symmetric] real_diff_def mult_ac simp del: i_mult_eq2) done lemma DeMoivre: "(cis a) ^ n = cis (real n * a)" @@ -1363,14 +918,11 @@ lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)" -apply (unfold rcis_def) -apply (auto simp add: power_mult_distrib DeMoivre complex_of_real_pow) +apply (simp add: rcis_def power_mult_distrib DeMoivre complex_of_real_pow) done lemma cis_inverse: "inverse(cis a) = cis (-a)" -apply (unfold cis_def) -apply (auto simp add: complex_inverse_complex_split complex_of_real_minus complex_diff_def) -done +by (simp add: cis_def complex_inverse_complex_split complex_of_real_minus complex_diff_def) declare cis_inverse [simp] lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)" @@ -1381,23 +933,20 @@ done lemma cis_divide: "cis a / cis b = cis (a - b)" -apply (unfold complex_divide_def) -apply (auto simp add: cis_mult real_diff_def) -done +by (simp add: complex_divide_def cis_mult real_diff_def) lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)" -apply (unfold complex_divide_def) -apply (case_tac "r2=0") -apply (simp (no_asm_simp) add: DIVISION_BY_ZERO COMPLEX_INVERSE_ZERO) -apply (auto simp add: rcis_inverse rcis_mult real_diff_def) +apply (simp add: complex_divide_def) +apply (case_tac "r2=0", simp) +apply (simp add: rcis_inverse rcis_mult real_diff_def) done lemma Re_cis: "Re(cis a) = cos a" -by (unfold cis_def, auto) +by (simp add: cis_def) declare Re_cis [simp] lemma Im_cis: "Im(cis a) = sin a" -by (unfold cis_def, auto) +by (simp add: cis_def) declare Im_cis [simp] lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)" @@ -1409,65 +958,40 @@ lemma expi_Im_split: "expi (ii * complex_of_real y) = complex_of_real (cos y) + ii * complex_of_real (sin y)" -by (unfold expi_def cis_def, auto) +by (simp add: expi_def cis_def) lemma expi_Im_cis: "expi (ii * complex_of_real y) = cis y" -by (unfold expi_def, auto) +by (simp add: expi_def) lemma expi_add: "expi(a + b) = expi(a) * expi(b)" -apply (unfold expi_def) -apply (auto simp add: complex_Re_add exp_add complex_Im_add cis_mult [symmetric] complex_of_real_mult complex_mult_ac) -done +by (simp add: expi_def complex_Re_add exp_add complex_Im_add cis_mult [symmetric] complex_of_real_mult mult_ac) lemma expi_complex_split: "expi(complex_of_real x + ii * complex_of_real y) = complex_of_real (exp(x)) * cis y" -by (unfold expi_def, auto) +by (simp add: expi_def) lemma expi_zero: "expi (0::complex) = 1" -by (unfold expi_def, auto) +by (simp add: expi_def) declare expi_zero [simp] lemma complex_Re_mult_eq: "Re (w * z) = Re w * Re z - Im w * Im z" -apply (rule_tac z = z in eq_Abs_complex) -apply (rule_tac z = w in eq_Abs_complex) -apply (auto simp add: complex_mult) -done +by (induct z, induct w, simp add: complex_mult) lemma complex_Im_mult_eq: "Im (w * z) = Re w * Im z + Im w * Re z" -apply (rule_tac z = z in eq_Abs_complex) -apply (rule_tac z = w in eq_Abs_complex) -apply (auto simp add: complex_mult) +apply (induct z, induct w, simp add: complex_mult) done lemma complex_expi_Ex: "\a r. z = complex_of_real r * expi a" -apply (cut_tac z = z in rcis_Ex) +apply (insert rcis_Ex [of z]) apply (auto simp add: expi_def rcis_def complex_mult_assoc [symmetric] complex_of_real_mult) apply (rule_tac x = "ii * complex_of_real a" in exI, auto) done -(**** -Goal "[| - pi < a; a \ pi |] ==> (-pi < a & a \ 0) | (0 \ a & a \ pi)" -by Auto_tac -qed "lemma_split_interval"; - -Goalw [arg_def] - "[| r \ 0; - pi < a; a \ pi |] \ -\ ==> arg(complex_of_real r * \ -\ (complex_of_real(cos a) + ii * complex_of_real(sin a))) = a"; -by Auto_tac -by (cut_inst_tac [("x","0"),("y","r")] linorder_less_linear 1); -by (auto_tac (claset(),simpset() addsimps (map (full_rename_numerals thy) - [rabs_eqI2,rabs_minus_eqI2,real_minus_rinv]) [real_divide_def, - minus_mult_right RS sym] mult_ac)); -by (auto_tac (claset(),simpset() addsimps [real_mult_assoc RS sym])); -by (dtac lemma_split_interval 1 THEN safe) -****) - ML {* @@ -1490,15 +1014,8 @@ val complexpow_0 = thm"complexpow_0"; val complexpow_Suc = thm"complexpow_Suc"; -val inj_Rep_complex = thm"inj_Rep_complex"; -val inj_Abs_complex = thm"inj_Abs_complex"; -val Abs_complex_cancel_iff = thm"Abs_complex_cancel_iff"; -val pair_mem_complex = thm"pair_mem_complex"; -val Abs_complex_inverse2 = thm"Abs_complex_inverse2"; -val eq_Abs_complex = thm"eq_Abs_complex"; val Re = thm"Re"; val Im = thm"Im"; -val Abs_complex_cancel = thm"Abs_complex_cancel"; val complex_Re_Im_cancel_iff = thm"complex_Re_Im_cancel_iff"; val complex_Re_zero = thm"complex_Re_zero"; val complex_Im_zero = thm"complex_Im_zero"; @@ -1515,55 +1032,20 @@ val complex_minus = thm"complex_minus"; val complex_Re_minus = thm"complex_Re_minus"; val complex_Im_minus = thm"complex_Im_minus"; -val complex_minus_minus = thm"complex_minus_minus"; -val inj_complex_minus = thm"inj_complex_minus"; val complex_minus_zero = thm"complex_minus_zero"; val complex_minus_zero_iff = thm"complex_minus_zero_iff"; -val complex_minus_zero_iff2 = thm"complex_minus_zero_iff2"; -val complex_minus_not_zero_iff = thm"complex_minus_not_zero_iff"; val complex_add = thm"complex_add"; val complex_Re_add = thm"complex_Re_add"; val complex_Im_add = thm"complex_Im_add"; val complex_add_commute = thm"complex_add_commute"; val complex_add_assoc = thm"complex_add_assoc"; -val complex_add_left_commute = thm"complex_add_left_commute"; val complex_add_zero_left = thm"complex_add_zero_left"; val complex_add_zero_right = thm"complex_add_zero_right"; -val complex_add_minus_right_zero = thm"complex_add_minus_right_zero"; -val complex_add_minus_cancel = thm"complex_add_minus_cancel"; -val complex_minus_add_cancel = thm"complex_minus_add_cancel"; -val complex_add_minus_eq_minus = thm"complex_add_minus_eq_minus"; -val complex_minus_add_distrib = thm"complex_minus_add_distrib"; -val complex_add_left_cancel = thm"complex_add_left_cancel"; -val complex_add_right_cancel = thm"complex_add_right_cancel"; -val complex_eq_minus_iff = thm"complex_eq_minus_iff"; -val complex_eq_minus_iff2 = thm"complex_eq_minus_iff2"; -val complex_diff_0 = thm"complex_diff_0"; -val complex_diff_0_right = thm"complex_diff_0_right"; -val complex_diff_self = thm"complex_diff_self"; val complex_diff = thm"complex_diff"; -val complex_diff_eq_eq = thm"complex_diff_eq_eq"; val complex_mult = thm"complex_mult"; -val complex_mult_commute = thm"complex_mult_commute"; -val complex_mult_assoc = thm"complex_mult_assoc"; -val complex_mult_left_commute = thm"complex_mult_left_commute"; val complex_mult_one_left = thm"complex_mult_one_left"; val complex_mult_one_right = thm"complex_mult_one_right"; -val complex_mult_zero_left = thm"complex_mult_zero_left"; -val complex_mult_zero_right = thm"complex_mult_zero_right"; -val complex_divide_zero = thm"complex_divide_zero"; -val complex_minus_mult_eq1 = thm"complex_minus_mult_eq1"; -val complex_minus_mult_eq2 = thm"complex_minus_mult_eq2"; -val complex_minus_mult_commute = thm"complex_minus_mult_commute"; -val complex_add_mult_distrib = thm"complex_add_mult_distrib"; -val complex_add_mult_distrib2 = thm"complex_add_mult_distrib2"; -val complex_zero_not_eq_one = thm"complex_zero_not_eq_one"; val complex_inverse = thm"complex_inverse"; -val COMPLEX_INVERSE_ZERO = thm"COMPLEX_INVERSE_ZERO"; -val COMPLEX_DIVISION_BY_ZERO = thm"COMPLEX_DIVISION_BY_ZERO"; -val complex_mult_inv_left = thm"complex_mult_inv_left"; -val complex_mult_inv_right = thm"complex_mult_inv_right"; -val inj_complex_of_real = thm"inj_complex_of_real"; val complex_of_real_one = thm"complex_of_real_one"; val complex_of_real_zero = thm"complex_of_real_zero"; val complex_of_real_eq_iff = thm"complex_of_real_eq_iff"; @@ -1580,7 +1062,6 @@ val complex_mod_complex_of_real = thm"complex_mod_complex_of_real"; val complex_of_real_abs = thm"complex_of_real_abs"; val complex_cnj = thm"complex_cnj"; -val inj_cnj = thm"inj_cnj"; val complex_cnj_cancel_iff = thm"complex_cnj_cancel_iff"; val complex_cnj_cnj = thm"complex_cnj_cnj"; val complex_cnj_complex_of_real = thm"complex_cnj_complex_of_real"; @@ -1686,7 +1167,6 @@ val rcis_zero_arg = thm"rcis_zero_arg"; val complex_of_real_minus_one = thm"complex_of_real_minus_one"; val complex_i_mult_minus = thm"complex_i_mult_minus"; -val complex_i_mult_minus2 = thm"complex_i_mult_minus2"; val cis_real_of_nat_Suc_mult = thm"cis_real_of_nat_Suc_mult"; val DeMoivre = thm"DeMoivre"; val DeMoivre2 = thm"DeMoivre2"; @@ -1706,9 +1186,6 @@ val complex_Re_mult_eq = thm"complex_Re_mult_eq"; val complex_Im_mult_eq = thm"complex_Im_mult_eq"; val complex_expi_Ex = thm"complex_expi_Ex"; - -val complex_add_ac = thms"complex_add_ac"; -val complex_mult_ac = thms"complex_mult_ac"; *} end diff -r 51ddf8963c95 -r 67a628beb981 src/HOL/Complex/ComplexArith0.ML --- a/src/HOL/Complex/ComplexArith0.ML Tue Feb 03 10:19:21 2004 +0100 +++ b/src/HOL/Complex/ComplexArith0.ML Tue Feb 03 11:06:36 2004 +0100 @@ -19,7 +19,7 @@ val trans_tac = Real_Numeral_Simprocs.trans_tac val norm_tac = ALLGOALS (simp_tac (HOL_ss addsimps complex_minus_from_mult_simps @ mult_1s)) THEN ALLGOALS (simp_tac (HOL_ss addsimps bin_simps@complex_mult_minus_simps)) - THEN ALLGOALS (simp_tac (HOL_ss addsimps complex_mult_ac)) + THEN ALLGOALS (simp_tac (HOL_ss addsimps mult_ac)) val numeral_simp_tac = ALLGOALS (simp_tac (HOL_ss addsimps rel_complex_number_of@bin_simps)) val simplify_meta_eq = simplify_meta_eq end @@ -105,7 +105,7 @@ val dest_coeff = dest_coeff val find_first = find_first [] val trans_tac = Real_Numeral_Simprocs.trans_tac - val norm_tac = ALLGOALS (simp_tac (HOL_ss addsimps mult_1s@complex_mult_ac)) + val norm_tac = ALLGOALS (simp_tac (HOL_ss addsimps mult_1s@mult_ac)) end; @@ -173,13 +173,13 @@ Addsimps [complex_minus1_divide]; Goal "(x + - a = (0::complex)) = (x=a)"; -by (simp_tac (simpset() addsimps [complex_diff_eq_eq,symmetric complex_diff_def]) 1); +by (simp_tac (simpset() addsimps [diff_eq_eq,symmetric complex_diff_def]) 1); qed "complex_add_minus_iff"; Addsimps [complex_add_minus_iff]; Goal "(x+y = (0::complex)) = (y = -x)"; by Auto_tac; -by (dtac (sym RS (complex_diff_eq_eq RS iffD2)) 1); +by (dtac (sym RS (diff_eq_eq RS iffD2)) 1); by Auto_tac; qed "complex_add_eq_0_iff"; AddIffs [complex_add_eq_0_iff]; diff -r 51ddf8963c95 -r 67a628beb981 src/HOL/Complex/ComplexBin.ML --- a/src/HOL/Complex/ComplexBin.ML Tue Feb 03 10:19:21 2004 +0100 +++ b/src/HOL/Complex/ComplexBin.ML Tue Feb 03 11:06:36 2004 +0100 @@ -60,11 +60,11 @@ (*For specialist use: NOT as default simprules*) Goal "2 * z = (z+z::complex)"; -by (simp_tac (simpset () addsimps [lemma, complex_add_mult_distrib]) 1); +by (simp_tac (simpset () addsimps [lemma, left_distrib]) 1); qed "complex_mult_2"; Goal "z * 2 = (z+z::complex)"; -by (stac complex_mult_commute 1 THEN rtac complex_mult_2 1); +by (stac mult_commute 1 THEN rtac complex_mult_2 1); qed "complex_mult_2_right"; (** Equals (=) **) @@ -88,7 +88,7 @@ qed "complex_mult_minus1"; Goal "z * -1 = -(z::complex)"; -by (stac complex_mult_commute 1 THEN rtac complex_mult_minus1 1); +by (stac mult_commute 1 THEN rtac complex_mult_minus1 1); qed "complex_mult_minus1_right"; Addsimps [complex_mult_minus1,complex_mult_minus1_right]; @@ -111,7 +111,7 @@ qed "complex_add_number_of_left"; Goal "number_of v *(number_of w * z) = (number_of(bin_mult v w) * z::complex)"; -by (simp_tac (simpset() addsimps [complex_mult_assoc RS sym]) 1); +by (simp_tac (simpset() addsimps [mult_assoc RS sym]) 1); qed "complex_mult_number_of_left"; Goalw [complex_diff_def] @@ -121,7 +121,7 @@ Goal "number_of v + (c - number_of w) = \ \ number_of (bin_add v (bin_minus w)) + (c::complex)"; -by (auto_tac (claset(),simpset() addsimps [complex_diff_def]@ complex_add_ac)); +by (auto_tac (claset(),simpset() addsimps [complex_diff_def]@ add_ac)); qed "complex_add_number_of_diff2"; Addsimps [complex_add_number_of_left, complex_mult_number_of_left, @@ -133,40 +133,10 @@ (** Combining of literal coefficients in sums of products **) Goal "(x = y) = (x-y = (0::complex))"; -by (simp_tac (simpset() addsimps [complex_diff_eq_eq]) 1); +by (simp_tac (simpset() addsimps [diff_eq_eq]) 1); qed "complex_eq_iff_diff_eq_0"; -(** For combine_numerals **) -Goal "i*u + (j*u + k) = (i+j)*u + (k::complex)"; -by (asm_simp_tac (simpset() addsimps [complex_add_mult_distrib] - @ complex_add_ac) 1); -qed "left_complex_add_mult_distrib"; - -(** For cancel_numerals **) - -Goal "((x::complex) = u + v) = (x - (u + v) = 0)"; -by (auto_tac (claset(),simpset() addsimps [complex_diff_eq_eq])); -qed "complex_eq_add_diff_eq_0"; - -Goal "((x::complex) = n) = (x - n = 0)"; -by (auto_tac (claset(),simpset() addsimps [complex_diff_eq_eq])); -qed "complex_eq_diff_eq_0"; - -val complex_rel_iff_rel_0_rls = [complex_eq_diff_eq_0,complex_eq_add_diff_eq_0]; - -Goal "!!i::complex. (i*u + m = j*u + n) = ((i-j)*u + m = n)"; -by (auto_tac (claset(), simpset() addsimps [complex_add_mult_distrib, - complex_diff_def] @ complex_add_ac)); -by (asm_simp_tac (simpset() addsimps [complex_add_assoc RS sym]) 1); -by (simp_tac (simpset() addsimps [complex_add_assoc]) 1); -qed "complex_eq_add_iff1"; - -Goal "!!i::complex. (i*u + m = j*u + n) = (m = (j-i)*u + n)"; -by (simp_tac (simpset() addsimps [ complex_eq_add_iff1]) 1); -by (auto_tac (claset(), simpset() addsimps [complex_diff_def, - complex_add_mult_distrib]@ complex_add_ac)); -qed "complex_eq_add_iff2"; structure Complex_Numeral_Simprocs = struct @@ -276,29 +246,26 @@ bin_pred_1, bin_pred_0, bin_pred_Pls, bin_pred_Min]; (*To let us treat subtraction as addition*) -val diff_simps = [complex_diff_def, complex_minus_add_distrib, - complex_minus_minus]; +val diff_simps = [complex_diff_def, minus_add_distrib, minus_minus]; (* push the unary minus down: - x * y = x * - y *) val complex_minus_mult_eq_1_to_2 = - [complex_minus_mult_eq1 RS sym, complex_minus_mult_eq2] MRS trans + [minus_mult_left RS sym, minus_mult_right] MRS trans |> standard; (*to extract again any uncancelled minuses*) val complex_minus_from_mult_simps = - [complex_minus_minus, complex_minus_mult_eq1 RS sym, - complex_minus_mult_eq2 RS sym]; + [minus_minus, minus_mult_left RS sym, minus_mult_right RS sym]; (*combine unary minus with numeric literals, however nested within a product*) val complex_mult_minus_simps = - [complex_mult_assoc, complex_minus_mult_eq1, complex_minus_mult_eq_1_to_2]; + [mult_assoc, minus_mult_left, complex_minus_mult_eq_1_to_2]; (*Final simplification: cancel + and * *) val simplify_meta_eq = Int_Numeral_Simprocs.simplify_meta_eq - [complex_add_zero_left, complex_add_zero_right, - complex_mult_zero_left, complex_mult_zero_right, complex_mult_one_left, - complex_mult_one_right]; + [add_zero_left, add_zero_right, + mult_zero_left, mult_zero_right, mult_1, mult_1_right]; val prep_simproc = Real_Numeral_Simprocs.prep_simproc; @@ -313,11 +280,11 @@ val trans_tac = Real_Numeral_Simprocs.trans_tac val norm_tac = ALLGOALS (simp_tac (HOL_ss addsimps add_0s@mult_1s@diff_simps@ - complex_minus_simps@complex_add_ac)) + complex_minus_simps@add_ac)) THEN ALLGOALS (simp_tac (HOL_ss addsimps non_add_bin_simps@complex_mult_minus_simps)) THEN ALLGOALS (simp_tac (HOL_ss addsimps complex_minus_from_mult_simps@ - complex_add_ac@complex_mult_ac)) + add_ac@mult_ac)) val numeral_simp_tac = ALLGOALS (simp_tac (HOL_ss addsimps add_0s@bin_simps)) val simplify_meta_eq = simplify_meta_eq end; @@ -328,8 +295,8 @@ val prove_conv = Bin_Simprocs.prove_conv val mk_bal = HOLogic.mk_eq val dest_bal = HOLogic.dest_bin "op =" complexT - val bal_add1 = complex_eq_add_iff1 RS trans - val bal_add2 = complex_eq_add_iff2 RS trans + val bal_add1 = eq_add_iff1 RS trans + val bal_add2 = eq_add_iff2 RS trans ); @@ -348,15 +315,15 @@ val dest_sum = dest_sum val mk_coeff = mk_coeff val dest_coeff = dest_coeff 1 - val left_distrib = left_complex_add_mult_distrib RS trans + val left_distrib = combine_common_factor RS trans val prove_conv = Bin_Simprocs.prove_conv_nohyps val trans_tac = Real_Numeral_Simprocs.trans_tac val norm_tac = ALLGOALS (simp_tac (HOL_ss addsimps add_0s@mult_1s@diff_simps@ - complex_minus_simps@complex_add_ac)) + complex_minus_simps@add_ac)) THEN ALLGOALS (simp_tac (HOL_ss addsimps non_add_bin_simps@complex_mult_minus_simps)) THEN ALLGOALS (simp_tac (HOL_ss addsimps complex_minus_from_mult_simps@ - complex_add_ac@complex_mult_ac)) + add_ac@mult_ac)) val numeral_simp_tac = ALLGOALS (simp_tac (HOL_ss addsimps add_0s@bin_simps)) val simplify_meta_eq = simplify_meta_eq @@ -470,7 +437,7 @@ val sg_ref = Sign.self_ref (Theory.sign_of (the_context ())) val T = Complex_Numeral_Simprocs.complexT val plus = Const ("op *", [T,T] ---> T) - val add_ac = complex_mult_ac + val add_ac = mult_ac end; structure Complex_Times_Assoc = Assoc_Fold (Complex_Times_Assoc_Data); @@ -479,9 +446,6 @@ Addsimps [complex_of_real_zero_iff]; -(*Simplification of x-y = 0 *) - -AddIffs [complex_eq_iff_diff_eq_0 RS sym]; (*** Real and imaginary stuff ***) diff -r 51ddf8963c95 -r 67a628beb981 src/HOL/Complex/NSCA.ML --- a/src/HOL/Complex/NSCA.ML Tue Feb 03 10:19:21 2004 +0100 +++ b/src/HOL/Complex/NSCA.ML Tue Feb 03 11:06:36 2004 +0100 @@ -4,6 +4,9 @@ Description : Infinite, infinitesimal complex number etc! *) +val complex_induct = thm"complex.induct"; + + (*--------------------------------------------------------------------------------------*) (* Closure laws for members of (embedded) set standard complex SComplex *) (* -------------------------------------------------------------------------------------*) @@ -785,8 +788,8 @@ by (dres_inst_tac [("x","m")] spec 1); by (Ultra_tac 1); by (rename_tac "Z x" 1); -by (res_inst_tac [("z","X x")] eq_Abs_complex 1); -by (res_inst_tac [("z","Y x")] eq_Abs_complex 1); +by (case_tac "X x" 1); +by (case_tac "Y x" 1); by (auto_tac (claset(),simpset() addsimps [complex_minus,complex_add, complex_mod] delsimps [realpow_Suc])); by (rtac order_le_less_trans 1 THEN assume_tac 2); @@ -806,8 +809,8 @@ by (dres_inst_tac [("x","m")] spec 1); by (Ultra_tac 1); by (rename_tac "Z x" 1); -by (res_inst_tac [("z","X x")] eq_Abs_complex 1); -by (res_inst_tac [("z","Y x")] eq_Abs_complex 1); +by (case_tac "X x" 1); +by (case_tac "Y x" 1); by (auto_tac (claset(),simpset() addsimps [complex_minus,complex_add, complex_mod] delsimps [realpow_Suc])); by (rtac order_le_less_trans 1 THEN assume_tac 2); @@ -836,12 +839,13 @@ by (TRYALL(Force_tac)); by (ultra_tac (claset(),HOL_ss) 1); by (dtac sym 1 THEN dtac sym 1); -by (res_inst_tac [("z","X x")] eq_Abs_complex 1); -by (res_inst_tac [("z","Y x")] eq_Abs_complex 1); +by (case_tac "X x" 1); +by (case_tac "Y x" 1); by (auto_tac (claset(), HOL_ss addsimps [complex_minus,complex_add, complex_mod, snd_conv, fst_conv,numeral_2_eq_2])); -by (subgoal_tac "sqrt (abs(xa + - xb) ^ 2 + abs(y + - ya) ^ 2) < u" 1); +by (rename_tac "a b c d" 1); +by (subgoal_tac "sqrt (abs(a + - c) ^ 2 + abs(b + - d) ^ 2) < u" 1); by (rtac lemma_sqrt_hcomplex_capprox 2); by Auto_tac; by (asm_full_simp_tac (simpset() addsimps [power2_eq_square]) 1); @@ -868,7 +872,7 @@ by (rtac bexI 1 THEN rtac lemma_hyprel_refl 2); by (res_inst_tac [("x","u")] exI 1 THEN Auto_tac); by (Ultra_tac 1); -by (dtac sym 1 THEN res_inst_tac [("z","X x")] eq_Abs_complex 1); +by (dtac sym 1 THEN case_tac "X x" 1); by (auto_tac (claset(), simpset() addsimps [complex_mod,numeral_2_eq_2] delsimps [realpow_Suc])); by (rtac ccontr 1 THEN dtac (linorder_not_less RS iffD1) 1); @@ -884,7 +888,7 @@ by (rtac bexI 1 THEN rtac lemma_hyprel_refl 2); by (res_inst_tac [("x","u")] exI 1 THEN Auto_tac); by (Ultra_tac 1); -by (dtac sym 1 THEN res_inst_tac [("z","X x")] eq_Abs_complex 1); +by (dtac sym 1 THEN case_tac "X x" 1); by (auto_tac (claset(),simpset() addsimps [complex_mod] delsimps [realpow_Suc])); by (rtac ccontr 1 THEN dtac (linorder_not_less RS iffD1) 1); by (dtac order_less_le_trans 1 THEN assume_tac 1); @@ -901,7 +905,7 @@ by (rtac bexI 1 THEN rtac lemma_hyprel_refl 2); by (res_inst_tac [("x","2*(u + v)")] exI 1); by (Ultra_tac 1); -by (dtac sym 1 THEN res_inst_tac [("z","X x")] eq_Abs_complex 1); +by (dtac sym 1 THEN case_tac "X x" 1); by (auto_tac (claset(),simpset() addsimps [complex_mod,numeral_2_eq_2] delsimps [realpow_Suc])); by (subgoal_tac "0 < u" 1 THEN arith_tac 2); by (subgoal_tac "0 < v" 1 THEN arith_tac 2); @@ -941,7 +945,7 @@ hcomplex_of_complex_def,SReal_def,hypreal_of_real_def])); by (res_inst_tac [("x","complex_of_real r + ii * complex_of_real ra")] exI 1); by (Ultra_tac 1); -by (res_inst_tac [("z","X x")] eq_Abs_complex 1); +by (case_tac "X x" 1); by (auto_tac (claset(),simpset() addsimps [complex_of_real_def,i_def, complex_add,complex_mult])); qed "Reals_Re_Im_SComplex"; diff -r 51ddf8963c95 -r 67a628beb981 src/HOL/Complex/NSComplex.thy --- a/src/HOL/Complex/NSComplex.thy Tue Feb 03 10:19:21 2004 +0100 +++ b/src/HOL/Complex/NSComplex.thy Tue Feb 03 11:06:36 2004 +0100 @@ -372,7 +372,7 @@ apply (rule_tac z = "z1" in eq_Abs_hcomplex) apply (rule_tac z = "z2" in eq_Abs_hcomplex) apply (rule_tac z = "w" in eq_Abs_hcomplex) -apply (auto simp add: hcomplex_mult hcomplex_add complex_add_mult_distrib) +apply (auto simp add: hcomplex_mult hcomplex_add left_distrib) done lemma hcomplex_zero_not_eq_one: "(0::hcomplex) \ (1::hcomplex)" @@ -400,7 +400,7 @@ apply (auto simp add: hcomplex_inverse hcomplex_mult) apply (ultra) apply (rule ccontr) -apply (drule complex_mult_inv_left) +apply (drule left_inverse) apply auto done @@ -744,24 +744,6 @@ done declare hcnj_one [simp] - -(* MOVE to NSComplexBin -Goal "z + hcnj z = - hcomplex_of_hypreal (2 * hRe(z))" -by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1); -by (auto_tac (claset(),HOL_ss addsimps [hRe,hcnj,hcomplex_add, - hypreal_mult,hcomplex_of_hypreal,complex_add_cnj])); -qed "hcomplex_add_hcnj"; - -Goal "z - hcnj z = \ -\ hcomplex_of_hypreal (hypreal_of_real 2 * hIm(z)) * iii"; -by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1); -by (auto_tac (claset(),simpset() addsimps [hIm,hcnj,hcomplex_diff, - hypreal_of_real_def,hypreal_mult,hcomplex_of_hypreal, - complex_diff_cnj,iii_def,hcomplex_mult])); -qed "hcomplex_diff_hcnj"; -*) - lemma hcomplex_hcnj_zero: "hcnj 0 = 0" apply (unfold hcomplex_zero_def) diff -r 51ddf8963c95 -r 67a628beb981 src/HOL/Complex/NSComplexBin.ML --- a/src/HOL/Complex/NSComplexBin.ML Tue Feb 03 10:19:21 2004 +0100 +++ b/src/HOL/Complex/NSComplexBin.ML Tue Feb 03 11:06:36 2004 +0100 @@ -161,49 +161,6 @@ (**** Simprocs for numeric literals ****) -(** Combining of literal coefficients in sums of products **) - -Goal "(x = y) = (x-y = (0::hcomplex))"; -by (simp_tac (simpset() addsimps [hcomplex_diff_eq_eq]) 1); -qed "hcomplex_eq_iff_diff_eq_0"; - -(** For combine_numerals **) - -Goal "i*u + (j*u + k) = (i+j)*u + (k::hcomplex)"; -by (asm_simp_tac (simpset() addsimps [hcomplex_add_mult_distrib] - @ add_ac) 1); -qed "left_hcomplex_add_mult_distrib"; - -(** For cancel_numerals **) - -Goal "((x::hcomplex) = u + v) = (x - (u + v) = 0)"; -by (auto_tac (claset(),simpset() addsimps [hcomplex_diff_eq_eq])); -qed "hcomplex_eq_add_diff_eq_0"; - -Goal "((x::hcomplex) = n) = (x - n = 0)"; -by (auto_tac (claset(),simpset() addsimps [hcomplex_diff_eq_eq])); -qed "hcomplex_eq_diff_eq_0"; - -val hcomplex_rel_iff_rel_0_rls = [hcomplex_eq_diff_eq_0,hcomplex_eq_add_diff_eq_0]; - -Goal "!!i::hcomplex. (i*u + m = j*u + n) = ((i-j)*u + m = n)"; -by (auto_tac (claset(), simpset() addsimps [hcomplex_add_mult_distrib, - hcomplex_diff_def] @ add_ac)); -by (asm_simp_tac (simpset() addsimps [hcomplex_add_assoc RS sym]) 1); -by (simp_tac (simpset() addsimps [hcomplex_add_assoc]) 1); -qed "hcomplex_eq_add_iff1"; - -Goal "!!i::hcomplex. (i*u + m = j*u + n) = (m = (j-i)*u + n)"; -by (res_inst_tac [("z","i")] eq_Abs_hcomplex 1); -by (res_inst_tac [("z","j")] eq_Abs_hcomplex 1); -by (res_inst_tac [("z","u")] eq_Abs_hcomplex 1); -by (res_inst_tac [("z","m")] eq_Abs_hcomplex 1); -by (res_inst_tac [("z","n")] eq_Abs_hcomplex 1); -by (auto_tac (claset(), simpset() addsimps [hcomplex_diff,hcomplex_add, - hcomplex_mult,complex_eq_add_iff2])); -qed "hcomplex_eq_add_iff2"; - - structure HComplex_Numeral_Simprocs = struct @@ -358,8 +315,8 @@ val prove_conv = Bin_Simprocs.prove_conv val mk_bal = HOLogic.mk_eq val dest_bal = HOLogic.dest_bin "op =" hcomplexT - val bal_add1 = hcomplex_eq_add_iff1 RS trans - val bal_add2 = hcomplex_eq_add_iff2 RS trans + val bal_add1 = eq_add_iff1 RS trans + val bal_add2 = eq_add_iff2 RS trans ); @@ -378,7 +335,7 @@ val dest_sum = dest_sum val mk_coeff = mk_coeff val dest_coeff = dest_coeff 1 - val left_distrib = left_hcomplex_add_mult_distrib RS trans + val left_distrib = combine_common_factor RS trans val prove_conv = Bin_Simprocs.prove_conv_nohyps val trans_tac = Real_Numeral_Simprocs.trans_tac val norm_tac = @@ -507,9 +464,6 @@ Addsimps [hcomplex_of_complex_zero_iff]; -(*Simplification of x-y = 0 *) - -AddIffs [hcomplex_eq_iff_diff_eq_0 RS sym]; (** extra thms **) diff -r 51ddf8963c95 -r 67a628beb981 src/HOL/Complex/ex/BinEx.thy --- a/src/HOL/Complex/ex/BinEx.thy Tue Feb 03 10:19:21 2004 +0100 +++ b/src/HOL/Complex/ex/BinEx.thy Tue Feb 03 11:06:36 2004 +0100 @@ -381,8 +381,7 @@ text{*Multiplication requires distributive laws. Perhaps versions instantiated to literal constants should be added to the simpset.*} -lemmas distrib = complex_add_mult_distrib complex_add_mult_distrib2 - complex_diff_mult_distrib complex_diff_mult_distrib2 +lemmas distrib = left_distrib right_distrib left_diff_distrib right_diff_distrib lemma "(1 + ii) * (1 - ii) = 2" by (simp add: distrib) @@ -393,10 +392,8 @@ lemma "(-84 + 255*ii) + (51 * 255*ii) = -84 + 13260 * ii" by (simp add: distrib) -text{*No inequalities: we have no ordering on the complex numbers.*} +text{*No inequalities or linear arithmetic: the complex numbers are unordered!*} text{*No powers (not supported yet)*} -text{*No linear arithmetic*} - end diff -r 51ddf8963c95 -r 67a628beb981 src/HOL/IsaMakefile --- a/src/HOL/IsaMakefile Tue Feb 03 10:19:21 2004 +0100 +++ b/src/HOL/IsaMakefile Tue Feb 03 11:06:36 2004 +0100 @@ -138,7 +138,7 @@ $(OUT)/HOL-Complex: $(OUT)/HOL Complex/ROOT.ML\ Library/Zorn.thy\ - Real/Complex_Numbers.thy Real/Lubs.thy Real/rat_arith.ML Real/RatArith.thy\ + Real/Lubs.thy Real/rat_arith.ML Real/RatArith.thy\ Real/Rational.thy Real/PReal.thy Real/RComplete.thy \ Real/ROOT.ML Real/Real.thy \ Real/RealArith.thy Real/real_arith.ML Real/RealDef.thy \