author | paulson |
Sun, 15 Feb 2004 10:46:37 +0100 | |
changeset 14387 | e96d5c42c4b0 |
parent 14373 | 67a628beb981 |
permissions | -rw-r--r-- |
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(* Title : CLim.ML |
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Author : Jacques D. Fleuriot |
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Copyright : 2001 University of Edinburgh |
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Description : A first theory of limits, continuity and |
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differentiation for complex functions |
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*) |
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(*FIXME: MOVE these two to Complex.thy*) |
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Goal "(x + - a = (0::complex)) = (x=a)"; |
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by (simp_tac (simpset() addsimps [diff_eq_eq,symmetric complex_diff_def]) 1); |
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qed "complex_add_minus_iff"; |
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Addsimps [complex_add_minus_iff]; |
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Goal "(x+y = (0::complex)) = (y = -x)"; |
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by Auto_tac; |
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by (dtac (sym RS (diff_eq_eq RS iffD2)) 1); |
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by Auto_tac; |
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qed "complex_add_eq_0_iff"; |
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AddIffs [complex_add_eq_0_iff]; |
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(*-----------------------------------------------------------------------*) |
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(* Limit of complex to complex function *) |
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(*-----------------------------------------------------------------------*) |
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Goalw [NSCLIM_def,NSCRLIM_def] |
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"f -- a --NSC> L ==> (%x. Re(f x)) -- a --NSCR> Re(L)"; |
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by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1); |
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by (auto_tac (claset(),simpset() addsimps [starfunC_approx_Re_Im_iff, |
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hRe_hcomplex_of_complex])); |
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qed "NSCLIM_NSCRLIM_Re"; |
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Goalw [NSCLIM_def,NSCRLIM_def] |
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"f -- a --NSC> L ==> (%x. Im(f x)) -- a --NSCR> Im(L)"; |
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by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1); |
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by (auto_tac (claset(),simpset() addsimps [starfunC_approx_Re_Im_iff, |
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hIm_hcomplex_of_complex])); |
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qed "NSCLIM_NSCRLIM_Im"; |
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Goalw [CLIM_def,NSCLIM_def,capprox_def] |
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"f -- x --C> L ==> f -- x --NSC> L"; |
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by Auto_tac; |
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by (res_inst_tac [("z","xa")] eq_Abs_hcomplex 1); |
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by (auto_tac (claset(),simpset() addsimps [hcomplex_of_complex_def, |
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starfunC,hcomplex_diff,CInfinitesimal_hcmod_iff,hcmod, |
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Infinitesimal_FreeUltrafilterNat_iff])); |
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by (rtac bexI 1 THEN rtac lemma_hyprel_refl 2); |
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by (Step_tac 1); |
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by (dres_inst_tac [("x","u")] spec 1 THEN Auto_tac); |
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by (dres_inst_tac [("x","s")] spec 1 THEN Auto_tac); |
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by (Ultra_tac 1); |
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by (dtac sym 1 THEN Auto_tac); |
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qed "CLIM_NSCLIM"; |
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Goal "(ALL t. P t) = (ALL X. P (Abs_hcomplex(hcomplexrel `` {X})))"; |
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by Auto_tac; |
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by (res_inst_tac [("z","t")] eq_Abs_hcomplex 1); |
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by Auto_tac; |
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qed "eq_Abs_hcomplex_ALL"; |
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Goal "ALL s. 0 < s --> (EX xa. xa ~= x & \ |
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\ cmod (xa - x) < s & r <= cmod (f xa - L)) \ |
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\ ==> ALL (n::nat). EX xa. xa ~= x & \ |
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\ cmod(xa - x) < inverse(real(Suc n)) & r <= cmod(f xa - L)"; |
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by (Clarify_tac 1); |
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by (cut_inst_tac [("n1","n")] |
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(real_of_nat_Suc_gt_zero RS positive_imp_inverse_positive) 1); |
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by Auto_tac; |
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val lemma_CLIM = result(); |
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(* not needed? *) |
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Goal "ALL x z. EX y. Q x z y ==> EX f. ALL x z. Q x z (f x z)"; |
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by (rtac choice 1 THEN Step_tac 1); |
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by (blast_tac (claset() addIs [choice]) 1); |
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qed "choice2"; |
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Goal "ALL s. 0 < s --> (EX xa. xa ~= x & \ |
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\ cmod (xa - x) < s & r <= cmod (f xa - L)) \ |
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\ ==> EX X. ALL (n::nat). X n ~= x & \ |
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\ cmod(X n - x) < inverse(real(Suc n)) & r <= cmod(f (X n) - L)"; |
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by (dtac lemma_CLIM 1); |
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by (dtac choice 1); |
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by (Blast_tac 1); |
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val lemma_skolemize_CLIM2 = result(); |
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Goal "ALL n. X n ~= x & \ |
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\ cmod (X n - x) < inverse (real(Suc n)) & \ |
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\ r <= cmod (f (X n) - L) ==> \ |
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\ ALL n. cmod (X n - x) < inverse (real(Suc n))"; |
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by (Auto_tac ); |
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val lemma_csimp = result(); |
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Goalw [CLIM_def,NSCLIM_def] |
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"f -- x --NSC> L ==> f -- x --C> L"; |
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by (auto_tac (claset(),simpset() addsimps [eq_Abs_hcomplex_ALL, |
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starfunC,CInfinitesimal_capprox_minus RS sym,hcomplex_diff, |
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CInfinitesimal_hcmod_iff,hcomplex_of_complex_def, |
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Infinitesimal_FreeUltrafilterNat_iff,hcmod])); |
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by (EVERY1[rtac ccontr, Asm_full_simp_tac]); |
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by (asm_full_simp_tac (simpset() addsimps [linorder_not_less]) 1); |
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by (dtac lemma_skolemize_CLIM2 1); |
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by (Step_tac 1); |
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by (dres_inst_tac [("x","X")] spec 1); |
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by Auto_tac; |
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by (dtac (lemma_csimp RS complex_seq_to_hcomplex_CInfinitesimal) 1); |
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by (asm_full_simp_tac (simpset() addsimps [CInfinitesimal_hcmod_iff, |
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hcomplex_of_complex_def,Infinitesimal_FreeUltrafilterNat_iff, |
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hcomplex_diff,hcmod]) 1); |
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by (Blast_tac 1); |
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by (dres_inst_tac [("x","r")] spec 1); |
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by (Clarify_tac 1); |
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by (dtac FreeUltrafilterNat_all 1); |
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by (Ultra_tac 1); |
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by (arith_tac 1); |
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qed "NSCLIM_CLIM"; |
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(**** First key result ****) |
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Goal "(f -- x --C> L) = (f -- x --NSC> L)"; |
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by (blast_tac (claset() addIs [CLIM_NSCLIM,NSCLIM_CLIM]) 1); |
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qed "CLIM_NSCLIM_iff"; |
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(*-----------------------------------------------------------------------*) |
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(* Limit of complex to real function *) |
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(*-----------------------------------------------------------------------*) |
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Goalw [CRLIM_def,NSCRLIM_def,capprox_def] |
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"f -- x --CR> L ==> f -- x --NSCR> L"; |
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by Auto_tac; |
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by (res_inst_tac [("z","xa")] eq_Abs_hcomplex 1); |
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by (auto_tac (claset(),simpset() addsimps [hcomplex_of_complex_def, |
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starfunCR,hcomplex_diff,CInfinitesimal_hcmod_iff,hcmod,hypreal_diff, |
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Infinitesimal_FreeUltrafilterNat_iff,Infinitesimal_approx_minus RS sym, |
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hypreal_of_real_def])); |
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by (rtac bexI 1 THEN rtac lemma_hyprel_refl 2); |
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by (Step_tac 1); |
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by (dres_inst_tac [("x","u")] spec 1 THEN Auto_tac); |
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by (dres_inst_tac [("x","s")] spec 1 THEN Auto_tac); |
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by (Ultra_tac 1); |
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by (dtac sym 1 THEN Auto_tac); |
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qed "CRLIM_NSCRLIM"; |
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Goal "ALL s. 0 < s --> (EX xa. xa ~= x & \ |
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\ cmod (xa - x) < s & r <= abs (f xa - L)) \ |
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\ ==> ALL (n::nat). EX xa. xa ~= x & \ |
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\ cmod(xa - x) < inverse(real(Suc n)) & r <= abs (f xa - L)"; |
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by (Clarify_tac 1); |
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by (cut_inst_tac [("n1","n")] |
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(real_of_nat_Suc_gt_zero RS positive_imp_inverse_positive) 1); |
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by Auto_tac; |
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val lemma_CRLIM = result(); |
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Goal "ALL s. 0 < s --> (EX xa. xa ~= x & \ |
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\ cmod (xa - x) < s & r <= abs (f xa - L)) \ |
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\ ==> EX X. ALL (n::nat). X n ~= x & \ |
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\ cmod(X n - x) < inverse(real(Suc n)) & r <= abs (f (X n) - L)"; |
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by (dtac lemma_CRLIM 1); |
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by (dtac choice 1); |
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by (Blast_tac 1); |
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val lemma_skolemize_CRLIM2 = result(); |
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Goal "ALL n. X n ~= x & \ |
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\ cmod (X n - x) < inverse (real(Suc n)) & \ |
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\ r <= abs (f (X n) - L) ==> \ |
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\ ALL n. cmod (X n - x) < inverse (real(Suc n))"; |
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by (Auto_tac ); |
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val lemma_crsimp = result(); |
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Goalw [CRLIM_def,NSCRLIM_def,capprox_def] |
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"f -- x --NSCR> L ==> f -- x --CR> L"; |
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by (auto_tac (claset(),simpset() addsimps [eq_Abs_hcomplex_ALL, |
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starfunCR,hcomplex_diff,hcomplex_of_complex_def,hypreal_diff, |
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CInfinitesimal_hcmod_iff,hcmod,Infinitesimal_approx_minus RS sym, |
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Infinitesimal_FreeUltrafilterNat_iff])); |
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by (EVERY1[rtac ccontr, Asm_full_simp_tac]); |
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by (asm_full_simp_tac (simpset() addsimps [linorder_not_less]) 1); |
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by (dtac lemma_skolemize_CRLIM2 1); |
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by (Step_tac 1); |
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by (dres_inst_tac [("x","X")] spec 1); |
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by Auto_tac; |
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by (dtac (lemma_crsimp RS complex_seq_to_hcomplex_CInfinitesimal) 1); |
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by (asm_full_simp_tac (simpset() addsimps [CInfinitesimal_hcmod_iff, |
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hcomplex_of_complex_def,Infinitesimal_FreeUltrafilterNat_iff, |
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hcomplex_diff,hcmod]) 1); |
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by (Blast_tac 1); |
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by (auto_tac (claset(),simpset() addsimps [hypreal_of_real_def, |
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hypreal_diff])); |
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by (dres_inst_tac [("x","r")] spec 1); |
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by (Clarify_tac 1); |
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by (dtac FreeUltrafilterNat_all 1); |
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by (Ultra_tac 1); |
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qed "NSCRLIM_CRLIM"; |
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(** second key result **) |
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Goal "(f -- x --CR> L) = (f -- x --NSCR> L)"; |
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by (blast_tac (claset() addIs [CRLIM_NSCRLIM,NSCRLIM_CRLIM]) 1); |
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qed "CRLIM_NSCRLIM_iff"; |
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(** get this result easily now **) |
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Goal "f -- a --C> L ==> (%x. Re(f x)) -- a --CR> Re(L)"; |
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by (auto_tac (claset() addDs [NSCLIM_NSCRLIM_Re],simpset() |
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addsimps [CLIM_NSCLIM_iff,CRLIM_NSCRLIM_iff RS sym])); |
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qed "CLIM_CRLIM_Re"; |
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Goal "f -- a --C> L ==> (%x. Im(f x)) -- a --CR> Im(L)"; |
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by (auto_tac (claset() addDs [NSCLIM_NSCRLIM_Im],simpset() |
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addsimps [CLIM_NSCLIM_iff,CRLIM_NSCRLIM_iff RS sym])); |
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qed "CLIM_CRLIM_Im"; |
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Goal "f -- a --C> L ==> (%x. cnj (f x)) -- a --C> cnj L"; |
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by (auto_tac (claset(),simpset() addsimps [CLIM_def, |
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complex_cnj_diff RS sym])); |
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qed "CLIM_cnj"; |
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Goal "((%x. cnj (f x)) -- a --C> cnj L) = (f -- a --C> L)"; |
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by (auto_tac (claset(),simpset() addsimps [CLIM_def, |
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complex_cnj_diff RS sym])); |
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qed "CLIM_cnj_iff"; |
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(*** NSLIM_add hence CLIM_add *) |
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Goalw [NSCLIM_def] |
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"[| f -- x --NSC> l; g -- x --NSC> m |] \ |
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\ ==> (%x. f(x) + g(x)) -- x --NSC> (l + m)"; |
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by (auto_tac (claset() addSIs [capprox_add], simpset())); |
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qed "NSCLIM_add"; |
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Goal "[| f -- x --C> l; g -- x --C> m |] \ |
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\ ==> (%x. f(x) + g(x)) -- x --C> (l + m)"; |
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by (asm_full_simp_tac (simpset() addsimps [CLIM_NSCLIM_iff, NSCLIM_add]) 1); |
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qed "CLIM_add"; |
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(*** NSLIM_mult hence CLIM_mult *) |
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Goalw [NSCLIM_def] |
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"[| f -- x --NSC> l; g -- x --NSC> m |] \ |
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\ ==> (%x. f(x) * g(x)) -- x --NSC> (l * m)"; |
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by (auto_tac (claset() addSIs [capprox_mult_CFinite], simpset())); |
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qed "NSCLIM_mult"; |
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Goal "[| f -- x --C> l; g -- x --C> m |] \ |
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\ ==> (%x. f(x) * g(x)) -- x --C> (l * m)"; |
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by (asm_full_simp_tac (simpset() addsimps [CLIM_NSCLIM_iff, NSCLIM_mult]) 1); |
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qed "CLIM_mult"; |
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(*** NSCLIM_const and CLIM_const ***) |
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Goalw [NSCLIM_def] "(%x. k) -- x --NSC> k"; |
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by Auto_tac; |
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qed "NSCLIM_const"; |
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Addsimps [NSCLIM_const]; |
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252 |
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Goalw [CLIM_def] "(%x. k) -- x --C> k"; |
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by Auto_tac; |
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qed "CLIM_const"; |
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Addsimps [CLIM_const]; |
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(*** NSCLIM_minus and CLIM_minus ***) |
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Goalw [NSCLIM_def] |
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"f -- a --NSC> L ==> (%x. -f(x)) -- a --NSC> -L"; |
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by Auto_tac; |
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qed "NSCLIM_minus"; |
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Goal "f -- a --C> L ==> (%x. -f(x)) -- a --C> -L"; |
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by (asm_full_simp_tac (simpset() addsimps [CLIM_NSCLIM_iff, NSCLIM_minus]) 1); |
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qed "CLIM_minus"; |
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269 |
(*** NSCLIM_diff hence CLIM_diff ***) |
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270 |
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271 |
Goalw [complex_diff_def] |
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"[| f -- x --NSC> l; g -- x --NSC> m |] \ |
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\ ==> (%x. f(x) - g(x)) -- x --NSC> (l - m)"; |
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274 |
by (auto_tac (claset(), simpset() addsimps [NSCLIM_add,NSCLIM_minus])); |
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qed "NSCLIM_diff"; |
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Goal "[| f -- x --C> l; g -- x --C> m |] \ |
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\ ==> (%x. f(x) - g(x)) -- x --C> (l - m)"; |
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by (asm_full_simp_tac (simpset() addsimps [CLIM_NSCLIM_iff, NSCLIM_diff]) 1); |
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qed "CLIM_diff"; |
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281 |
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282 |
(*** NSCLIM_inverse and hence CLIM_inverse *) |
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283 |
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284 |
Goalw [NSCLIM_def] |
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285 |
"[| f -- a --NSC> L; L ~= 0 |] \ |
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286 |
\ ==> (%x. inverse(f(x))) -- a --NSC> (inverse L)"; |
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by (Clarify_tac 1); |
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288 |
by (dtac spec 1); |
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289 |
by (auto_tac (claset(), |
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290 |
simpset() addsimps [hcomplex_of_complex_capprox_inverse])); |
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qed "NSCLIM_inverse"; |
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292 |
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293 |
Goal "[| f -- a --C> L; L ~= 0 |] \ |
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\ ==> (%x. inverse(f(x))) -- a --C> (inverse L)"; |
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by (asm_full_simp_tac (simpset() addsimps [CLIM_NSCLIM_iff, NSCLIM_inverse]) 1); |
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qed "CLIM_inverse"; |
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(*** NSCLIM_zero, CLIM_zero, etc. ***) |
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Goal "f -- a --NSC> l ==> (%x. f(x) - l) -- a --NSC> 0"; |
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by (res_inst_tac [("a1","l")] (right_minus RS subst) 1); |
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by (rewtac complex_diff_def); |
303 |
by (rtac NSCLIM_add 1 THEN Auto_tac); |
|
304 |
qed "NSCLIM_zero"; |
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305 |
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306 |
Goal "f -- a --C> l ==> (%x. f(x) - l) -- a --C> 0"; |
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307 |
by (asm_full_simp_tac (simpset() addsimps [CLIM_NSCLIM_iff, NSCLIM_zero]) 1); |
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308 |
qed "CLIM_zero"; |
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309 |
||
310 |
Goal "(%x. f(x) - l) -- x --NSC> 0 ==> f -- x --NSC> l"; |
|
311 |
by (dres_inst_tac [("g","%x. l"),("m","l")] NSCLIM_add 1); |
|
312 |
by Auto_tac; |
|
313 |
qed "NSCLIM_zero_cancel"; |
|
314 |
||
315 |
Goal "(%x. f(x) - l) -- x --C> 0 ==> f -- x --C> l"; |
|
316 |
by (dres_inst_tac [("g","%x. l"),("m","l")] CLIM_add 1); |
|
317 |
by Auto_tac; |
|
318 |
qed "CLIM_zero_cancel"; |
|
319 |
||
320 |
(*** NSCLIM_not zero and hence CLIM_not_zero ***) |
|
321 |
||
322 |
(*not in simpset?*) |
|
323 |
Addsimps [hypreal_epsilon_not_zero]; |
|
324 |
||
325 |
Goalw [NSCLIM_def] "k ~= 0 ==> ~ ((%x. k) -- x --NSC> 0)"; |
|
326 |
by (auto_tac (claset(),simpset() delsimps [hcomplex_of_complex_zero])); |
|
327 |
by (res_inst_tac [("x","hcomplex_of_complex x + hcomplex_of_hypreal epsilon")] exI 1); |
|
328 |
by (auto_tac (claset() addIs [CInfinitesimal_add_capprox_self RS capprox_sym],simpset() |
|
329 |
delsimps [hcomplex_of_complex_zero])); |
|
330 |
qed "NSCLIM_not_zero"; |
|
331 |
||
332 |
(* [| k ~= 0; (%x. k) -- x --NSC> 0 |] ==> R *) |
|
333 |
bind_thm("NSCLIM_not_zeroE", NSCLIM_not_zero RS notE); |
|
334 |
||
335 |
Goal "k ~= 0 ==> ~ ((%x. k) -- x --C> 0)"; |
|
336 |
by (asm_full_simp_tac (simpset() addsimps [CLIM_NSCLIM_iff, NSCLIM_not_zero]) 1); |
|
337 |
qed "CLIM_not_zero"; |
|
338 |
||
339 |
(*** NSCLIM_const hence CLIM_const ***) |
|
340 |
||
341 |
Goal "(%x. k) -- x --NSC> L ==> k = L"; |
|
342 |
by (rtac ccontr 1); |
|
343 |
by (dtac NSCLIM_zero 1); |
|
344 |
by (rtac NSCLIM_not_zeroE 1 THEN assume_tac 2); |
|
345 |
by Auto_tac; |
|
346 |
qed "NSCLIM_const_eq"; |
|
347 |
||
348 |
Goal "(%x. k) -- x --C> L ==> k = L"; |
|
349 |
by (asm_full_simp_tac (simpset() addsimps [CLIM_NSCLIM_iff,NSCLIM_const_eq]) 1); |
|
350 |
qed "CLIM_const_eq"; |
|
351 |
||
352 |
(*** NSCLIM and hence CLIM are unique ***) |
|
353 |
||
354 |
Goal "[| f -- x --NSC> L; f -- x --NSC> M |] ==> L = M"; |
|
355 |
by (dtac NSCLIM_minus 1); |
|
356 |
by (dtac NSCLIM_add 1 THEN assume_tac 1); |
|
357 |
by (auto_tac (claset() addSDs [NSCLIM_const_eq RS sym], simpset())); |
|
358 |
qed "NSCLIM_unique"; |
|
359 |
||
360 |
Goal "[| f -- x --C> L; f -- x --C> M |] ==> L = M"; |
|
361 |
by (asm_full_simp_tac (simpset() addsimps [CLIM_NSCLIM_iff, NSCLIM_unique]) 1); |
|
362 |
qed "CLIM_unique"; |
|
363 |
||
364 |
(*** NSCLIM_mult_zero and CLIM_mult_zero ***) |
|
365 |
||
366 |
Goal "[| f -- x --NSC> 0; g -- x --NSC> 0 |] \ |
|
367 |
\ ==> (%x. f(x)*g(x)) -- x --NSC> 0"; |
|
368 |
by (dtac NSCLIM_mult 1 THEN Auto_tac); |
|
369 |
qed "NSCLIM_mult_zero"; |
|
370 |
||
371 |
Goal "[| f -- x --C> 0; g -- x --C> 0 |] \ |
|
372 |
\ ==> (%x. f(x)*g(x)) -- x --C> 0"; |
|
373 |
by (dtac CLIM_mult 1 THEN Auto_tac); |
|
374 |
qed "CLIM_mult_zero"; |
|
375 |
||
376 |
(*** NSCLIM_self hence CLIM_self ***) |
|
377 |
||
378 |
Goalw [NSCLIM_def] "(%x. x) -- a --NSC> a"; |
|
379 |
by (auto_tac (claset() addIs [starfunC_Idfun_capprox],simpset())); |
|
380 |
qed "NSCLIM_self"; |
|
381 |
||
382 |
Goal "(%x. x) -- a --C> a"; |
|
383 |
by (simp_tac (simpset() addsimps [CLIM_NSCLIM_iff,NSCLIM_self]) 1); |
|
384 |
qed "CLIM_self"; |
|
385 |
||
386 |
(** another equivalence result **) |
|
387 |
Goalw [NSCLIM_def,NSCRLIM_def] |
|
388 |
"(f -- x --NSC> L) = ((%y. cmod(f y - L)) -- x --NSCR> 0)"; |
|
389 |
by (auto_tac (claset(),simpset() addsimps [CInfinitesimal_capprox_minus |
|
390 |
RS sym,CInfinitesimal_hcmod_iff])); |
|
391 |
by (ALLGOALS(dtac spec) THEN Auto_tac); |
|
392 |
by (ALLGOALS(res_inst_tac [("z","xa")] eq_Abs_hcomplex)); |
|
393 |
by (auto_tac (claset(),simpset() addsimps [hcomplex_diff, |
|
394 |
starfunC,starfunCR,hcomplex_of_complex_def,hcmod,mem_infmal_iff])); |
|
395 |
qed "NSCLIM_NSCRLIM_iff"; |
|
396 |
||
397 |
(** much, much easier standard proof **) |
|
398 |
Goalw [CLIM_def,CRLIM_def] |
|
399 |
"(f -- x --C> L) = ((%y. cmod(f y - L)) -- x --CR> 0)"; |
|
400 |
by Auto_tac; |
|
401 |
qed "CLIM_CRLIM_iff"; |
|
402 |
||
403 |
(* so this is nicer nonstandard proof *) |
|
404 |
Goal "(f -- x --NSC> L) = ((%y. cmod(f y - L)) -- x --NSCR> 0)"; |
|
405 |
by (auto_tac (claset(),simpset() addsimps [CRLIM_NSCRLIM_iff RS sym, |
|
406 |
CLIM_CRLIM_iff,CLIM_NSCLIM_iff RS sym])); |
|
407 |
qed "NSCLIM_NSCRLIM_iff2"; |
|
408 |
||
409 |
Goal "(f -- a --NSC> L) = ((%x. Re(f x)) -- a --NSCR> Re(L) & \ |
|
410 |
\ (%x. Im(f x)) -- a --NSCR> Im(L))"; |
|
411 |
by (auto_tac (claset() addIs [NSCLIM_NSCRLIM_Re,NSCLIM_NSCRLIM_Im],simpset())); |
|
412 |
by (auto_tac (claset(),simpset() addsimps [NSCLIM_def,NSCRLIM_def])); |
|
413 |
by (REPEAT(dtac spec 1) THEN Auto_tac); |
|
414 |
by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1); |
|
415 |
by (auto_tac (claset(),simpset() addsimps [capprox_approx_iff,starfunC, |
|
416 |
hcomplex_of_complex_def,starfunCR,hypreal_of_real_def])); |
|
417 |
qed "NSCLIM_NSCRLIM_Re_Im_iff"; |
|
418 |
||
419 |
Goal "(f -- a --C> L) = ((%x. Re(f x)) -- a --CR> Re(L) & \ |
|
420 |
\ (%x. Im(f x)) -- a --CR> Im(L))"; |
|
421 |
by (auto_tac (claset(),simpset() addsimps [CLIM_NSCLIM_iff,CRLIM_NSCRLIM_iff, |
|
422 |
NSCLIM_NSCRLIM_Re_Im_iff])); |
|
423 |
qed "CLIM_CRLIM_Re_Im_iff"; |
|
424 |
||
425 |
||
14335 | 426 |
(*-----------------------------------------------------------------------*) |
13957 | 427 |
(* Continuity *) |
14335 | 428 |
(*-----------------------------------------------------------------------*) |
13957 | 429 |
|
430 |
Goalw [isNSContc_def] |
|
431 |
"[| isNSContc f a; y @c= hcomplex_of_complex a |] \ |
|
432 |
\ ==> ( *fc* f) y @c= hcomplex_of_complex (f a)"; |
|
433 |
by (Blast_tac 1); |
|
434 |
qed "isNSContcD"; |
|
435 |
||
436 |
Goalw [isNSContc_def,NSCLIM_def] |
|
437 |
"isNSContc f a ==> f -- a --NSC> (f a) "; |
|
438 |
by (Blast_tac 1); |
|
439 |
qed "isNSContc_NSCLIM"; |
|
440 |
||
441 |
Goalw [isNSContc_def,NSCLIM_def] |
|
442 |
"f -- a --NSC> (f a) ==> isNSContc f a"; |
|
443 |
by Auto_tac; |
|
444 |
by (res_inst_tac [("Q","y = hcomplex_of_complex a")] |
|
445 |
(excluded_middle RS disjE) 1); |
|
446 |
by Auto_tac; |
|
447 |
qed "NSCLIM_isNSContc"; |
|
448 |
||
449 |
(*--------------------------------------------------*) |
|
450 |
(* NS continuity can be defined using NS Limit in *) |
|
451 |
(* similar fashion to standard def of continuity *) |
|
452 |
(* -------------------------------------------------*) |
|
453 |
||
454 |
Goal "(isNSContc f a) = (f -- a --NSC> (f a))"; |
|
455 |
by (blast_tac (claset() addIs [isNSContc_NSCLIM,NSCLIM_isNSContc]) 1); |
|
456 |
qed "isNSContc_NSCLIM_iff"; |
|
457 |
||
458 |
Goal "(isNSContc f a) = (f -- a --C> (f a))"; |
|
459 |
by (asm_full_simp_tac (simpset() addsimps |
|
460 |
[CLIM_NSCLIM_iff,isNSContc_NSCLIM_iff]) 1); |
|
461 |
qed "isNSContc_CLIM_iff"; |
|
462 |
||
463 |
(*** key result for continuity ***) |
|
464 |
Goalw [isContc_def] "(isNSContc f a) = (isContc f a)"; |
|
465 |
by (rtac isNSContc_CLIM_iff 1); |
|
466 |
qed "isNSContc_isContc_iff"; |
|
467 |
||
468 |
Goal "isContc f a ==> isNSContc f a"; |
|
469 |
by (etac (isNSContc_isContc_iff RS iffD2) 1); |
|
470 |
qed "isContc_isNSContc"; |
|
471 |
||
472 |
Goal "isNSContc f a ==> isContc f a"; |
|
473 |
by (etac (isNSContc_isContc_iff RS iffD1) 1); |
|
474 |
qed "isNSContc_isContc"; |
|
475 |
||
476 |
(*--------------------------------------------------*) |
|
477 |
(* Alternative definition of continuity *) |
|
478 |
(* -------------------------------------------------*) |
|
479 |
||
480 |
Goalw [NSCLIM_def] |
|
481 |
"(f -- a --NSC> L) = ((%h. f(a + h)) -- 0 --NSC> L)"; |
|
482 |
by Auto_tac; |
|
483 |
by (dres_inst_tac [("x","hcomplex_of_complex a + x")] spec 1); |
|
484 |
by (dres_inst_tac [("x","- hcomplex_of_complex a + x")] spec 2); |
|
14320 | 485 |
by Safe_tac; |
13957 | 486 |
by (Asm_full_simp_tac 1); |
487 |
by (rtac ((mem_cinfmal_iff RS iffD2) RS |
|
488 |
(CInfinitesimal_add_capprox_self RS capprox_sym)) 1); |
|
489 |
by (rtac (capprox_minus_iff2 RS iffD1) 4); |
|
14320 | 490 |
by (asm_full_simp_tac (simpset() addsimps compare_rls@[hcomplex_add_commute]) 3); |
13957 | 491 |
by (res_inst_tac [("z","x")] eq_Abs_hcomplex 2); |
492 |
by (res_inst_tac [("z","x")] eq_Abs_hcomplex 4); |
|
493 |
by (auto_tac (claset(), |
|
494 |
simpset() addsimps [starfunC, hcomplex_of_complex_def, |
|
495 |
hcomplex_minus, hcomplex_add])); |
|
496 |
qed "NSCLIM_h_iff"; |
|
497 |
||
498 |
Goal "(f -- a --NSC> f a) = ((%h. f(a + h)) -- 0 --NSC> f a)"; |
|
499 |
by (rtac NSCLIM_h_iff 1); |
|
500 |
qed "NSCLIM_isContc_iff"; |
|
501 |
||
502 |
Goal "(f -- a --C> f a) = ((%h. f(a + h)) -- 0 --C> f(a))"; |
|
503 |
by (simp_tac (simpset() addsimps [CLIM_NSCLIM_iff, NSCLIM_isContc_iff]) 1); |
|
504 |
qed "CLIM_isContc_iff"; |
|
505 |
||
506 |
Goalw [isContc_def] "(isContc f x) = ((%h. f(x + h)) -- 0 --C> f(x))"; |
|
507 |
by (simp_tac (simpset() addsimps [CLIM_isContc_iff]) 1); |
|
508 |
qed "isContc_iff"; |
|
509 |
||
510 |
Goal "[| isContc f a; isContc g a |] ==> isContc (%x. f(x) + g(x)) a"; |
|
511 |
by (auto_tac (claset() addIs [capprox_add], |
|
512 |
simpset() addsimps [isNSContc_isContc_iff RS sym, isNSContc_def])); |
|
513 |
qed "isContc_add"; |
|
514 |
||
515 |
Goal "[| isContc f a; isContc g a |] ==> isContc (%x. f(x) * g(x)) a"; |
|
516 |
by (auto_tac (claset() addSIs [starfunC_mult_CFinite_capprox], |
|
517 |
simpset() delsimps [starfunC_mult RS sym] |
|
518 |
addsimps [isNSContc_isContc_iff RS sym, isNSContc_def])); |
|
519 |
qed "isContc_mult"; |
|
520 |
||
521 |
(*** more theorems: note simple proofs ***) |
|
522 |
||
523 |
Goal "[| isContc f a; isContc g (f a) |] \ |
|
524 |
\ ==> isContc (g o f) a"; |
|
525 |
by (auto_tac (claset(),simpset() addsimps [isNSContc_isContc_iff RS sym, |
|
526 |
isNSContc_def,starfunC_o RS sym])); |
|
527 |
qed "isContc_o"; |
|
528 |
||
529 |
Goal "[| isContc f a; isContc g (f a) |] \ |
|
530 |
\ ==> isContc (%x. g (f x)) a"; |
|
531 |
by (auto_tac (claset() addDs [isContc_o],simpset() addsimps [o_def])); |
|
532 |
qed "isContc_o2"; |
|
533 |
||
534 |
Goalw [isNSContc_def] "isNSContc f a ==> isNSContc (%x. - f x) a"; |
|
535 |
by Auto_tac; |
|
536 |
qed "isNSContc_minus"; |
|
537 |
||
538 |
Goal "isContc f a ==> isContc (%x. - f x) a"; |
|
539 |
by (auto_tac (claset(),simpset() addsimps [isNSContc_isContc_iff RS sym, |
|
540 |
isNSContc_minus])); |
|
541 |
qed "isContc_minus"; |
|
542 |
||
543 |
Goalw [isContc_def] |
|
544 |
"[| isContc f x; f x ~= 0 |] ==> isContc (%x. inverse (f x)) x"; |
|
545 |
by (blast_tac (claset() addIs [CLIM_inverse]) 1); |
|
546 |
qed "isContc_inverse"; |
|
547 |
||
548 |
Goal "[| isNSContc f x; f x ~= 0 |] ==> isNSContc (%x. inverse (f x)) x"; |
|
549 |
by (auto_tac (claset() addIs [isContc_inverse],simpset() addsimps |
|
550 |
[isNSContc_isContc_iff])); |
|
551 |
qed "isNSContc_inverse"; |
|
552 |
||
553 |
Goalw [complex_diff_def] |
|
554 |
"[| isContc f a; isContc g a |] ==> isContc (%x. f(x) - g(x)) a"; |
|
555 |
by (auto_tac (claset() addIs [isContc_add,isContc_minus],simpset())); |
|
556 |
qed "isContc_diff"; |
|
557 |
||
558 |
Goalw [isContc_def] "isContc (%x. k) a"; |
|
559 |
by (Simp_tac 1); |
|
560 |
qed "isContc_const"; |
|
561 |
Addsimps [isContc_const]; |
|
562 |
||
563 |
Goalw [isNSContc_def] "isNSContc (%x. k) a"; |
|
564 |
by (Simp_tac 1); |
|
565 |
qed "isNSContc_const"; |
|
566 |
Addsimps [isNSContc_const]; |
|
567 |
||
568 |
||
14335 | 569 |
(*-----------------------------------------------------------------------*) |
13957 | 570 |
(* functions from complex to reals *) |
14335 | 571 |
(* ----------------------------------------------------------------------*) |
13957 | 572 |
|
573 |
Goalw [isNSContCR_def] |
|
574 |
"[| isNSContCR f a; y @c= hcomplex_of_complex a |] \ |
|
575 |
\ ==> ( *fcR* f) y @= hypreal_of_real (f a)"; |
|
576 |
by (Blast_tac 1); |
|
577 |
qed "isNSContCRD"; |
|
578 |
||
579 |
Goalw [isNSContCR_def,NSCRLIM_def] |
|
580 |
"isNSContCR f a ==> f -- a --NSCR> (f a) "; |
|
581 |
by (Blast_tac 1); |
|
582 |
qed "isNSContCR_NSCRLIM"; |
|
583 |
||
584 |
Goalw [isNSContCR_def,NSCRLIM_def] |
|
585 |
"f -- a --NSCR> (f a) ==> isNSContCR f a"; |
|
586 |
by Auto_tac; |
|
587 |
by (res_inst_tac [("Q","y = hcomplex_of_complex a")] |
|
588 |
(excluded_middle RS disjE) 1); |
|
589 |
by Auto_tac; |
|
590 |
qed "NSCRLIM_isNSContCR"; |
|
591 |
||
592 |
Goal "(isNSContCR f a) = (f -- a --NSCR> (f a))"; |
|
593 |
by (blast_tac (claset() addIs [isNSContCR_NSCRLIM,NSCRLIM_isNSContCR]) 1); |
|
594 |
qed "isNSContCR_NSCRLIM_iff"; |
|
595 |
||
596 |
Goal "(isNSContCR f a) = (f -- a --CR> (f a))"; |
|
597 |
by (asm_full_simp_tac (simpset() addsimps |
|
598 |
[CRLIM_NSCRLIM_iff,isNSContCR_NSCRLIM_iff]) 1); |
|
599 |
qed "isNSContCR_CRLIM_iff"; |
|
600 |
||
601 |
(*** another key result for continuity ***) |
|
602 |
Goalw [isContCR_def] "(isNSContCR f a) = (isContCR f a)"; |
|
603 |
by (rtac isNSContCR_CRLIM_iff 1); |
|
604 |
qed "isNSContCR_isContCR_iff"; |
|
605 |
||
606 |
Goal "isContCR f a ==> isNSContCR f a"; |
|
607 |
by (etac (isNSContCR_isContCR_iff RS iffD2) 1); |
|
608 |
qed "isContCR_isNSContCR"; |
|
609 |
||
610 |
Goal "isNSContCR f a ==> isContCR f a"; |
|
611 |
by (etac (isNSContCR_isContCR_iff RS iffD1) 1); |
|
612 |
qed "isNSContCR_isContCR"; |
|
613 |
||
614 |
Goalw [isNSContCR_def] "isNSContCR cmod (a)"; |
|
615 |
by (auto_tac (claset() addIs [capprox_hcmod_approx], |
|
616 |
simpset() addsimps [starfunCR_cmod,hcmod_hcomplex_of_complex |
|
617 |
RS sym])); |
|
618 |
qed "isNSContCR_cmod"; |
|
619 |
Addsimps [isNSContCR_cmod]; |
|
620 |
||
621 |
Goal "isContCR cmod (a)"; |
|
622 |
by (auto_tac (claset(),simpset() addsimps [isNSContCR_isContCR_iff RS sym])); |
|
623 |
qed "isContCR_cmod"; |
|
624 |
Addsimps [isContCR_cmod]; |
|
625 |
||
626 |
Goalw [isContc_def,isContCR_def] |
|
627 |
"isContc f a ==> isContCR (%x. Re (f x)) a"; |
|
628 |
by (etac CLIM_CRLIM_Re 1); |
|
629 |
qed "isContc_isContCR_Re"; |
|
630 |
||
631 |
Goalw [isContc_def,isContCR_def] |
|
632 |
"isContc f a ==> isContCR (%x. Im (f x)) a"; |
|
633 |
by (etac CLIM_CRLIM_Im 1); |
|
634 |
qed "isContc_isContCR_Im"; |
|
635 |
||
14335 | 636 |
(*-----------------------------------------------------------------------*) |
13957 | 637 |
(* Derivatives *) |
14335 | 638 |
(*-----------------------------------------------------------------------*) |
13957 | 639 |
|
640 |
Goalw [cderiv_def] |
|
641 |
"(CDERIV f x :> D) = ((%h. (f(x + h) - f(x))/h) -- 0 --C> D)"; |
|
642 |
by (Blast_tac 1); |
|
643 |
qed "CDERIV_iff"; |
|
644 |
||
645 |
Goalw [cderiv_def] |
|
646 |
"(CDERIV f x :> D) = ((%h. (f(x + h) - f(x))/h) -- 0 --NSC> D)"; |
|
647 |
by (simp_tac (simpset() addsimps [CLIM_NSCLIM_iff]) 1); |
|
648 |
qed "CDERIV_NSC_iff"; |
|
649 |
||
650 |
Goalw [cderiv_def] |
|
651 |
"CDERIV f x :> D \ |
|
652 |
\ ==> (%h. (f(x + h) - f(x))/h) -- 0 --C> D"; |
|
653 |
by (Blast_tac 1); |
|
654 |
qed "CDERIVD"; |
|
655 |
||
656 |
Goalw [cderiv_def] |
|
657 |
"CDERIV f x :> D ==> (%h. (f(x + h) - f(x))/h) -- 0 --NSC> D"; |
|
658 |
by (asm_full_simp_tac (simpset() addsimps [CLIM_NSCLIM_iff]) 1); |
|
659 |
qed "NSC_DERIVD"; |
|
660 |
||
661 |
(*** Uniqueness ***) |
|
662 |
||
663 |
Goalw [cderiv_def] |
|
664 |
"[| CDERIV f x :> D; CDERIV f x :> E |] ==> D = E"; |
|
665 |
by (blast_tac (claset() addIs [CLIM_unique]) 1); |
|
666 |
qed "CDERIV_unique"; |
|
667 |
||
668 |
(*** uniqueness: a nonstandard proof ***) |
|
669 |
Goalw [nscderiv_def] |
|
670 |
"[| NSCDERIV f x :> D; NSCDERIV f x :> E |] ==> D = E"; |
|
671 |
by (auto_tac (claset() addSDs [inst "x" "hcomplex_of_hypreal epsilon" bspec] |
|
672 |
addSIs [inj_hcomplex_of_complex RS injD] |
|
673 |
addDs [capprox_trans3], |
|
674 |
simpset())); |
|
675 |
qed "NSCDeriv_unique"; |
|
676 |
||
677 |
||
14335 | 678 |
(*-----------------------------------------------------------------------*) |
13957 | 679 |
(* Differentiability *) |
14335 | 680 |
(*-----------------------------------------------------------------------*) |
13957 | 681 |
|
682 |
Goalw [cdifferentiable_def] |
|
683 |
"f cdifferentiable x ==> EX D. CDERIV f x :> D"; |
|
684 |
by (assume_tac 1); |
|
685 |
qed "cdifferentiableD"; |
|
686 |
||
687 |
Goalw [cdifferentiable_def] |
|
688 |
"CDERIV f x :> D ==> f cdifferentiable x"; |
|
689 |
by (Blast_tac 1); |
|
690 |
qed "cdifferentiableI"; |
|
691 |
||
692 |
Goalw [NSCdifferentiable_def] |
|
693 |
"f NSCdifferentiable x ==> EX D. NSCDERIV f x :> D"; |
|
694 |
by (assume_tac 1); |
|
695 |
qed "NSCdifferentiableD"; |
|
696 |
||
697 |
Goalw [NSCdifferentiable_def] |
|
698 |
"NSCDERIV f x :> D ==> f NSCdifferentiable x"; |
|
699 |
by (Blast_tac 1); |
|
700 |
qed "NSCdifferentiableI"; |
|
701 |
||
702 |
||
14335 | 703 |
(*-----------------------------------------------------------------------*) |
13957 | 704 |
(* Alternative definition for differentiability *) |
14335 | 705 |
(*-----------------------------------------------------------------------*) |
13957 | 706 |
|
707 |
Goalw [CLIM_def] |
|
708 |
"((%h. (f(a + h) - f(a))/h) -- 0 --C> D) = \ |
|
709 |
\ ((%x. (f(x) - f(a)) / (x - a)) -- a --C> D)"; |
|
710 |
by (Step_tac 1); |
|
711 |
by (ALLGOALS(dtac spec)); |
|
712 |
by (Step_tac 1); |
|
713 |
by (Blast_tac 1 THEN Blast_tac 2); |
|
714 |
by (ALLGOALS(res_inst_tac [("x","s")] exI)); |
|
715 |
by (Step_tac 1); |
|
716 |
by (dres_inst_tac [("x","x - a")] spec 1); |
|
717 |
by (dres_inst_tac [("x","x + a")] spec 2); |
|
14373 | 718 |
by (auto_tac (claset(), simpset() addsimps add_ac)); |
13957 | 719 |
qed "CDERIV_CLIM_iff"; |
720 |
||
721 |
Goalw [cderiv_def] "(CDERIV f x :> D) = \ |
|
722 |
\ ((%z. (f(z) - f(x)) / (z - x)) -- x --C> D)"; |
|
723 |
by (simp_tac (simpset() addsimps [CDERIV_CLIM_iff]) 1); |
|
724 |
qed "CDERIV_iff2"; |
|
725 |
||
726 |
||
14335 | 727 |
(*-----------------------------------------------------------------------*) |
13957 | 728 |
(* Equivalence of NS and standard defs of differentiation *) |
14335 | 729 |
(*-----------------------------------------------------------------------*) |
13957 | 730 |
|
731 |
(*** first equivalence ***) |
|
732 |
Goalw [nscderiv_def,NSCLIM_def] |
|
733 |
"(NSCDERIV f x :> D) = ((%h. (f(x + h) - f(x))/h) -- 0 --NSC> D)"; |
|
734 |
by Auto_tac; |
|
735 |
by (dres_inst_tac [("x","xa")] bspec 1); |
|
736 |
by (rtac ccontr 3); |
|
737 |
by (dres_inst_tac [("x","h")] spec 3); |
|
738 |
by (auto_tac (claset(), |
|
739 |
simpset() addsimps [mem_cinfmal_iff, starfunC_lambda_cancel])); |
|
740 |
qed "NSCDERIV_NSCLIM_iff"; |
|
741 |
||
742 |
(*** 2nd equivalence ***) |
|
743 |
Goal "(NSCDERIV f x :> D) = \ |
|
744 |
\ ((%z. (f(z) - f(x)) / (z - x)) -- x --NSC> D)"; |
|
745 |
by (full_simp_tac (simpset() addsimps |
|
746 |
[NSCDERIV_NSCLIM_iff, CDERIV_CLIM_iff, CLIM_NSCLIM_iff RS sym]) 1); |
|
747 |
qed "NSCDERIV_NSCLIM_iff2"; |
|
748 |
||
749 |
Goal "(NSCDERIV f x :> D) = \ |
|
750 |
\ (ALL xa. xa ~= hcomplex_of_complex x & xa @c= hcomplex_of_complex x --> \ |
|
751 |
\ ( *fc* (%z. (f z - f x) / (z - x))) xa @c= hcomplex_of_complex D)"; |
|
752 |
by (auto_tac (claset(), simpset() addsimps [NSCDERIV_NSCLIM_iff2, NSCLIM_def])); |
|
753 |
qed "NSCDERIV_iff2"; |
|
754 |
||
755 |
Goalw [cderiv_def] "(NSCDERIV f x :> D) = (CDERIV f x :> D)"; |
|
756 |
by (simp_tac (simpset() addsimps [NSCDERIV_NSCLIM_iff,CLIM_NSCLIM_iff]) 1); |
|
757 |
qed "NSCDERIV_CDERIV_iff"; |
|
758 |
||
759 |
Goalw [nscderiv_def] |
|
760 |
"NSCDERIV f x :> D ==> isNSContc f x"; |
|
761 |
by (auto_tac (claset(),simpset() addsimps [isNSContc_NSCLIM_iff, |
|
762 |
NSCLIM_def,hcomplex_diff_def])); |
|
763 |
by (dtac (capprox_minus_iff RS iffD1) 1); |
|
14320 | 764 |
by (subgoal_tac "xa + - (hcomplex_of_complex x) ~= 0" 1); |
765 |
by (asm_full_simp_tac (simpset() addsimps compare_rls) 2); |
|
13957 | 766 |
by (dres_inst_tac [("x","- hcomplex_of_complex x + xa")] bspec 1); |
767 |
by (asm_full_simp_tac (simpset() addsimps [hcomplex_add_assoc RS sym]) 2); |
|
768 |
by (auto_tac (claset(),simpset() addsimps |
|
769 |
[mem_cinfmal_iff RS sym,hcomplex_add_commute])); |
|
770 |
by (dres_inst_tac [("c","xa + - hcomplex_of_complex x")] capprox_mult1 1); |
|
771 |
by (auto_tac (claset() addIs [CInfinitesimal_subset_CFinite |
|
14373 | 772 |
RS subsetD],simpset() addsimps [mult_assoc])); |
13957 | 773 |
by (dres_inst_tac [("x3","D")] (CFinite_hcomplex_of_complex RSN |
774 |
(2,CInfinitesimal_CFinite_mult) RS (mem_cinfmal_iff RS iffD1)) 1); |
|
14373 | 775 |
by (blast_tac (claset() addIs [capprox_trans,mult_commute RS subst, |
13957 | 776 |
(capprox_minus_iff RS iffD2)]) 1); |
777 |
qed "NSCDERIV_isNSContc"; |
|
778 |
||
779 |
Goal "CDERIV f x :> D ==> isContc f x"; |
|
780 |
by (asm_full_simp_tac (simpset() addsimps [NSCDERIV_CDERIV_iff RS sym, |
|
781 |
isNSContc_isContc_iff RS sym,NSCDERIV_isNSContc]) 1); |
|
782 |
qed "CDERIV_isContc"; |
|
783 |
||
14335 | 784 |
(*-----------------------------------------------------------------------*) |
13957 | 785 |
(* Differentiation rules for combinations of functions follow from clear, *) |
786 |
(* straightforard, algebraic manipulations *) |
|
14335 | 787 |
(*-----------------------------------------------------------------------*) |
13957 | 788 |
|
789 |
(* use simple constant nslimit theorem *) |
|
790 |
Goal "(NSCDERIV (%x. k) x :> 0)"; |
|
791 |
by (simp_tac (simpset() addsimps [NSCDERIV_NSCLIM_iff]) 1); |
|
792 |
qed "NSCDERIV_const"; |
|
793 |
Addsimps [NSCDERIV_const]; |
|
794 |
||
795 |
Goal "(CDERIV (%x. k) x :> 0)"; |
|
796 |
by (simp_tac (simpset() addsimps [NSCDERIV_CDERIV_iff RS sym]) 1); |
|
797 |
qed "CDERIV_const"; |
|
798 |
Addsimps [CDERIV_const]; |
|
799 |
||
800 |
Goal "[| NSCDERIV f x :> Da; NSCDERIV g x :> Db |] \ |
|
801 |
\ ==> NSCDERIV (%x. f x + g x) x :> Da + Db"; |
|
802 |
by (asm_full_simp_tac (simpset() addsimps [NSCDERIV_NSCLIM_iff, |
|
803 |
NSCLIM_def]) 1 THEN REPEAT(Step_tac 1)); |
|
804 |
by (auto_tac (claset(), |
|
805 |
simpset() addsimps [hcomplex_add_divide_distrib,hcomplex_diff_def])); |
|
806 |
by (dres_inst_tac [("b","hcomplex_of_complex Da"), |
|
807 |
("d","hcomplex_of_complex Db")] capprox_add 1); |
|
14335 | 808 |
by (auto_tac (claset(), simpset() addsimps add_ac)); |
13957 | 809 |
qed "NSCDERIV_add"; |
810 |
||
811 |
Goal "[| CDERIV f x :> Da; CDERIV g x :> Db |] \ |
|
812 |
\ ==> CDERIV (%x. f x + g x) x :> Da + Db"; |
|
813 |
by (asm_full_simp_tac (simpset() addsimps [NSCDERIV_add, |
|
814 |
NSCDERIV_CDERIV_iff RS sym]) 1); |
|
815 |
qed "CDERIV_add"; |
|
816 |
||
817 |
(*** lemmas for multiplication ***) |
|
818 |
||
819 |
Goal "((a::hcomplex)*b) - (c*d) = (b*(a - c)) + (c*(b - d))"; |
|
14335 | 820 |
by (simp_tac (simpset() addsimps [right_diff_distrib]) 1); |
13957 | 821 |
val lemma_nscderiv1 = result(); |
822 |
||
823 |
Goal "[| (x + y) / z = hcomplex_of_complex D + yb; z ~= 0; \ |
|
824 |
\ z : CInfinitesimal; yb : CInfinitesimal |] \ |
|
825 |
\ ==> x + y @c= 0"; |
|
826 |
by (forw_inst_tac [("c1","z")] (hcomplex_mult_right_cancel RS iffD2) 1 |
|
827 |
THEN assume_tac 1); |
|
828 |
by (thin_tac "(x + y) / z = hcomplex_of_complex D + yb" 1); |
|
829 |
by (auto_tac (claset() addSIs [CInfinitesimal_CFinite_mult2, CFinite_add], |
|
830 |
simpset() addsimps [mem_cinfmal_iff RS sym])); |
|
831 |
by (etac (CInfinitesimal_subset_CFinite RS subsetD) 1); |
|
832 |
val lemma_nscderiv2 = result(); |
|
833 |
||
834 |
Goal "[| NSCDERIV f x :> Da; NSCDERIV g x :> Db |] \ |
|
835 |
\ ==> NSCDERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))"; |
|
836 |
by (asm_full_simp_tac (simpset() addsimps [NSCDERIV_NSCLIM_iff, NSCLIM_def]) 1 |
|
837 |
THEN REPEAT(Step_tac 1)); |
|
838 |
by (auto_tac (claset(), |
|
839 |
simpset() addsimps [starfunC_lambda_cancel, lemma_nscderiv1, |
|
840 |
hcomplex_of_complex_zero])); |
|
841 |
by (simp_tac (simpset() addsimps [hcomplex_add_divide_distrib]) 1); |
|
842 |
by (REPEAT(dtac (bex_CInfinitesimal_iff2 RS iffD2) 1)); |
|
843 |
by (auto_tac (claset(), |
|
14318 | 844 |
simpset() delsimps [times_divide_eq_right] |
845 |
addsimps [times_divide_eq_right RS sym])); |
|
13957 | 846 |
by (rewtac hcomplex_diff_def); |
847 |
by (dres_inst_tac [("D","Db")] lemma_nscderiv2 1); |
|
848 |
by (dtac (capprox_minus_iff RS iffD2 RS (bex_CInfinitesimal_iff2 RS iffD2)) 4); |
|
849 |
by (auto_tac (claset() addSIs [capprox_add_mono1], |
|
14373 | 850 |
simpset() addsimps [left_distrib, right_distrib, mult_commute, add_assoc])); |
851 |
by (res_inst_tac [("b1","hcomplex_of_complex Db * hcomplex_of_complex (f x)")] |
|
852 |
(add_commute RS subst) 1); |
|
13957 | 853 |
by (auto_tac (claset() addSIs [CInfinitesimal_add_capprox_self2 RS capprox_sym, |
854 |
CInfinitesimal_add, CInfinitesimal_mult, |
|
855 |
CInfinitesimal_hcomplex_of_complex_mult, |
|
856 |
CInfinitesimal_hcomplex_of_complex_mult2], |
|
857 |
simpset() addsimps [hcomplex_add_assoc RS sym])); |
|
858 |
qed "NSCDERIV_mult"; |
|
859 |
||
860 |
Goal "[| CDERIV f x :> Da; CDERIV g x :> Db |] \ |
|
861 |
\ ==> CDERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))"; |
|
862 |
by (asm_full_simp_tac (simpset() addsimps [NSCDERIV_mult, |
|
863 |
NSCDERIV_CDERIV_iff RS sym]) 1); |
|
864 |
qed "CDERIV_mult"; |
|
865 |
||
866 |
Goal "NSCDERIV f x :> D ==> NSCDERIV (%x. c * f x) x :> c*D"; |
|
867 |
by (asm_full_simp_tac |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14335
diff
changeset
|
868 |
(simpset() addsimps [times_divide_eq_right RS sym, NSCDERIV_NSCLIM_iff, |
14373 | 869 |
minus_mult_right, right_distrib RS sym, |
13957 | 870 |
complex_diff_def] |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14335
diff
changeset
|
871 |
delsimps [times_divide_eq_right, minus_mult_right RS sym]) 1); |
13957 | 872 |
by (etac (NSCLIM_const RS NSCLIM_mult) 1); |
873 |
qed "NSCDERIV_cmult"; |
|
874 |
||
875 |
Goal "CDERIV f x :> D ==> CDERIV (%x. c * f x) x :> c*D"; |
|
876 |
by (auto_tac (claset(),simpset() addsimps [NSCDERIV_cmult,NSCDERIV_CDERIV_iff |
|
877 |
RS sym])); |
|
878 |
qed "CDERIV_cmult"; |
|
879 |
||
880 |
Goal "NSCDERIV f x :> D ==> NSCDERIV (%x. -(f x)) x :> -D"; |
|
881 |
by (asm_full_simp_tac (simpset() addsimps [NSCDERIV_NSCLIM_iff,complex_diff_def]) 1); |
|
14373 | 882 |
by (res_inst_tac [("t","f x")] (minus_minus RS subst) 1); |
883 |
by (asm_simp_tac (simpset() addsimps [minus_add_distrib RS sym] |
|
14335 | 884 |
delsimps [minus_add_distrib, minus_minus] |
885 |
||
886 |
) 1); |
|
13957 | 887 |
by (etac NSCLIM_minus 1); |
888 |
qed "NSCDERIV_minus"; |
|
889 |
||
890 |
Goal "CDERIV f x :> D ==> CDERIV (%x. -(f x)) x :> -D"; |
|
891 |
by (asm_full_simp_tac (simpset() addsimps [NSCDERIV_minus,NSCDERIV_CDERIV_iff RS sym]) 1); |
|
892 |
qed "CDERIV_minus"; |
|
893 |
||
894 |
Goal "[| NSCDERIV f x :> Da; NSCDERIV g x :> Db |] \ |
|
895 |
\ ==> NSCDERIV (%x. f x + -g x) x :> Da + -Db"; |
|
896 |
by (blast_tac (claset() addDs [NSCDERIV_add,NSCDERIV_minus]) 1); |
|
897 |
qed "NSCDERIV_add_minus"; |
|
898 |
||
899 |
Goal "[| CDERIV f x :> Da; CDERIV g x :> Db |] \ |
|
900 |
\ ==> CDERIV (%x. f x + -g x) x :> Da + -Db"; |
|
901 |
by (blast_tac (claset() addDs [CDERIV_add,CDERIV_minus]) 1); |
|
902 |
qed "CDERIV_add_minus"; |
|
903 |
||
904 |
Goalw [complex_diff_def] |
|
905 |
"[| NSCDERIV f x :> Da; NSCDERIV g x :> Db |] \ |
|
906 |
\ ==> NSCDERIV (%x. f x - g x) x :> Da - Db"; |
|
907 |
by (blast_tac (claset() addIs [NSCDERIV_add_minus]) 1); |
|
908 |
qed "NSCDERIV_diff"; |
|
909 |
||
910 |
Goalw [complex_diff_def] |
|
911 |
"[| CDERIV f x :> Da; CDERIV g x :> Db |] \ |
|
912 |
\ ==> CDERIV (%x. f x - g x) x :> Da - Db"; |
|
913 |
by (blast_tac (claset() addIs [CDERIV_add_minus]) 1); |
|
914 |
qed "CDERIV_diff"; |
|
915 |
||
916 |
||
917 |
(*--------------------------------------------------*) |
|
918 |
(* Chain rule *) |
|
919 |
(*--------------------------------------------------*) |
|
920 |
||
921 |
(* lemmas *) |
|
922 |
Goalw [nscderiv_def] |
|
923 |
"[| NSCDERIV g x :> D; \ |
|
924 |
\ ( *fc* g) (hcomplex_of_complex(x) + xa) = hcomplex_of_complex(g x);\ |
|
925 |
\ xa : CInfinitesimal; xa ~= 0 \ |
|
926 |
\ |] ==> D = 0"; |
|
927 |
by (dtac bspec 1); |
|
928 |
by Auto_tac; |
|
929 |
qed "NSCDERIV_zero"; |
|
930 |
||
931 |
Goalw [nscderiv_def] |
|
932 |
"[| NSCDERIV f x :> D; h: CInfinitesimal; h ~= 0 |] \ |
|
933 |
\ ==> ( *fc* f) (hcomplex_of_complex(x) + h) - hcomplex_of_complex(f x) @c= 0"; |
|
934 |
by (asm_full_simp_tac (simpset() addsimps [mem_cinfmal_iff RS sym]) 1); |
|
935 |
by (rtac CInfinitesimal_ratio 1); |
|
936 |
by (rtac capprox_hcomplex_of_complex_CFinite 3); |
|
937 |
by Auto_tac; |
|
938 |
qed "NSCDERIV_capprox"; |
|
939 |
||
940 |
||
941 |
(*--------------------------------------------------*) |
|
942 |
(* from one version of differentiability *) |
|
943 |
(* *) |
|
944 |
(* f(x) - f(a) *) |
|
945 |
(* --------------- @= Db *) |
|
946 |
(* x - a *) |
|
947 |
(* -------------------------------------------------*) |
|
948 |
||
949 |
Goal "[| NSCDERIV f (g x) :> Da; \ |
|
950 |
\ ( *fc* g) (hcomplex_of_complex(x) + xa) ~= hcomplex_of_complex (g x); \ |
|
951 |
\ ( *fc* g) (hcomplex_of_complex(x) + xa) @c= hcomplex_of_complex (g x) \ |
|
952 |
\ |] ==> (( *fc* f) (( *fc* g) (hcomplex_of_complex(x) + xa)) \ |
|
953 |
\ - hcomplex_of_complex (f (g x))) \ |
|
954 |
\ / (( *fc* g) (hcomplex_of_complex(x) + xa) - hcomplex_of_complex (g x)) \ |
|
955 |
\ @c= hcomplex_of_complex (Da)"; |
|
956 |
by (auto_tac (claset(),simpset() addsimps [NSCDERIV_NSCLIM_iff2, NSCLIM_def])); |
|
957 |
qed "NSCDERIVD1"; |
|
958 |
||
959 |
(*--------------------------------------------------*) |
|
960 |
(* from other version of differentiability *) |
|
961 |
(* *) |
|
962 |
(* f(x + h) - f(x) *) |
|
963 |
(* ----------------- @= Db *) |
|
964 |
(* h *) |
|
965 |
(*--------------------------------------------------*) |
|
966 |
||
967 |
Goal "[| NSCDERIV g x :> Db; xa: CInfinitesimal; xa ~= 0 |] \ |
|
968 |
\ ==> (( *fc* g) (hcomplex_of_complex(x) + xa) - hcomplex_of_complex(g x)) / xa \ |
|
969 |
\ @c= hcomplex_of_complex (Db)"; |
|
970 |
by (auto_tac (claset(), |
|
971 |
simpset() addsimps [NSCDERIV_NSCLIM_iff, NSCLIM_def, |
|
972 |
mem_cinfmal_iff, starfunC_lambda_cancel])); |
|
973 |
qed "NSCDERIVD2"; |
|
974 |
||
975 |
Goal "(z::hcomplex) ~= 0 ==> x*y = (x*inverse(z))*(z*y)"; |
|
976 |
by Auto_tac; |
|
977 |
qed "lemma_complex_chain"; |
|
978 |
||
979 |
(*** chain rule ***) |
|
980 |
||
981 |
Goal "[| NSCDERIV f (g x) :> Da; NSCDERIV g x :> Db |] \ |
|
982 |
\ ==> NSCDERIV (f o g) x :> Da * Db"; |
|
983 |
by (asm_simp_tac (simpset() addsimps [NSCDERIV_NSCLIM_iff, |
|
984 |
NSCLIM_def,mem_cinfmal_iff RS sym]) 1 THEN Step_tac 1); |
|
985 |
by (forw_inst_tac [("f","g")] NSCDERIV_capprox 1); |
|
986 |
by (auto_tac (claset(), |
|
987 |
simpset() addsimps [starfunC_lambda_cancel2, starfunC_o RS sym])); |
|
988 |
by (case_tac "( *fc* g) (hcomplex_of_complex(x) + xa) = hcomplex_of_complex (g x)" 1); |
|
989 |
by (dres_inst_tac [("g","g")] NSCDERIV_zero 1); |
|
990 |
by (auto_tac (claset(),simpset() addsimps [hcomplex_divide_def])); |
|
991 |
by (res_inst_tac [("z1","( *fc* g) (hcomplex_of_complex(x) + xa) - hcomplex_of_complex (g x)"), |
|
992 |
("y1","inverse xa")] (lemma_complex_chain RS ssubst) 1); |
|
993 |
by (Asm_simp_tac 1); |
|
994 |
by (rtac capprox_mult_hcomplex_of_complex 1); |
|
995 |
by (fold_tac [hcomplex_divide_def]); |
|
996 |
by (blast_tac (claset() addIs [NSCDERIVD2]) 2); |
|
997 |
by (auto_tac (claset() addSIs [NSCDERIVD1] addIs [capprox_minus_iff RS iffD2], |
|
998 |
simpset() addsimps [symmetric hcomplex_diff_def])); |
|
999 |
qed "NSCDERIV_chain"; |
|
1000 |
||
1001 |
(* standard version *) |
|
1002 |
Goal "[| CDERIV f (g x) :> Da; CDERIV g x :> Db |] \ |
|
1003 |
\ ==> CDERIV (f o g) x :> Da * Db"; |
|
1004 |
by (asm_full_simp_tac (simpset() addsimps [NSCDERIV_CDERIV_iff RS sym, |
|
1005 |
NSCDERIV_chain]) 1); |
|
1006 |
qed "CDERIV_chain"; |
|
1007 |
||
1008 |
Goal "[| CDERIV f (g x) :> Da; CDERIV g x :> Db |] \ |
|
1009 |
\ ==> CDERIV (%x. f (g x)) x :> Da * Db"; |
|
1010 |
by (auto_tac (claset() addDs [CDERIV_chain], simpset() addsimps [o_def])); |
|
1011 |
qed "CDERIV_chain2"; |
|
1012 |
||
14335 | 1013 |
(*-----------------------------------------------------------------------*) |
13957 | 1014 |
(* Differentiation of natural number powers *) |
14335 | 1015 |
(*-----------------------------------------------------------------------*) |
13957 | 1016 |
|
1017 |
Goal "NSCDERIV (%x. x) x :> 1"; |
|
1018 |
by (auto_tac (claset(), |
|
1019 |
simpset() addsimps [NSCDERIV_NSCLIM_iff,NSCLIM_def])); |
|
1020 |
qed "NSCDERIV_Id"; |
|
1021 |
Addsimps [NSCDERIV_Id]; |
|
1022 |
||
1023 |
Goal "CDERIV (%x. x) x :> 1"; |
|
1024 |
by (simp_tac (simpset() addsimps [NSCDERIV_CDERIV_iff RS sym]) 1); |
|
1025 |
qed "CDERIV_Id"; |
|
1026 |
Addsimps [CDERIV_Id]; |
|
1027 |
||
1028 |
bind_thm ("isContc_Id", CDERIV_Id RS CDERIV_isContc); |
|
1029 |
||
1030 |
(*derivative of linear multiplication*) |
|
1031 |
Goal "CDERIV (op * c) x :> c"; |
|
1032 |
by (cut_inst_tac [("c","c"),("x","x")] (CDERIV_Id RS CDERIV_cmult) 1); |
|
1033 |
by (Asm_full_simp_tac 1); |
|
1034 |
qed "CDERIV_cmult_Id"; |
|
1035 |
Addsimps [CDERIV_cmult_Id]; |
|
1036 |
||
1037 |
Goal "NSCDERIV (op * c) x :> c"; |
|
1038 |
by (simp_tac (simpset() addsimps [NSCDERIV_CDERIV_iff]) 1); |
|
1039 |
qed "NSCDERIV_cmult_Id"; |
|
1040 |
Addsimps [NSCDERIV_cmult_Id]; |
|
1041 |
||
1042 |
Goal "CDERIV (%x. x ^ n) x :> (complex_of_real (real n)) * (x ^ (n - 1))"; |
|
1043 |
by (induct_tac "n" 1); |
|
1044 |
by (dtac (CDERIV_Id RS CDERIV_mult) 2); |
|
1045 |
by (auto_tac (claset(), |
|
1046 |
simpset() addsimps [complex_of_real_add RS sym, |
|
14373 | 1047 |
left_distrib,real_of_nat_Suc] |
14335 | 1048 |
delsimps [complex_of_real_add])); |
13957 | 1049 |
by (case_tac "n" 1); |
1050 |
by (auto_tac (claset(), |
|
14373 | 1051 |
simpset() addsimps [mult_assoc, add_commute])); |
1052 |
by (auto_tac (claset(),simpset() addsimps [mult_commute])); |
|
13957 | 1053 |
qed "CDERIV_pow"; |
1054 |
Addsimps [CDERIV_pow,simplify (simpset()) CDERIV_pow]; |
|
1055 |
||
1056 |
(* NS version *) |
|
1057 |
Goal "NSCDERIV (%x. x ^ n) x :> complex_of_real (real n) * (x ^ (n - 1))"; |
|
1058 |
by (simp_tac (simpset() addsimps [NSCDERIV_CDERIV_iff]) 1); |
|
1059 |
qed "NSCDERIV_pow"; |
|
1060 |
||
14320 | 1061 |
Goal "[|CDERIV f x :> D; D = E|] ==> CDERIV f x :> E"; |
13957 | 1062 |
by Auto_tac; |
1063 |
qed "lemma_CDERIV_subst"; |
|
1064 |
||
1065 |
(*used once, in NSCDERIV_inverse*) |
|
1066 |
Goal "[| h: CInfinitesimal; x ~= 0 |] ==> hcomplex_of_complex x + h ~= 0"; |
|
14320 | 1067 |
by (Clarify_tac 1); |
1068 |
by (dtac (thm"equals_zero_I") 1); |
|
13957 | 1069 |
by Auto_tac; |
1070 |
qed "CInfinitesimal_add_not_zero"; |
|
1071 |
||
1072 |
(*Can't get rid of x ~= 0 because it isn't continuous at zero*) |
|
1073 |
||
1074 |
Goalw [nscderiv_def] |
|
1075 |
"x ~= 0 ==> NSCDERIV (%x. inverse(x)) x :> (- (inverse x ^ 2))"; |
|
1076 |
by (rtac ballI 1 THEN Asm_full_simp_tac 1 THEN Step_tac 1); |
|
1077 |
by (forward_tac [CInfinitesimal_add_not_zero] 1); |
|
14323 | 1078 |
by (asm_full_simp_tac (simpset() addsimps [hcomplex_add_commute,numeral_2_eq_2]) 2); |
13957 | 1079 |
by (auto_tac (claset(), |
1080 |
simpset() addsimps [starfunC_inverse_inverse,hcomplex_diff_def] |
|
14335 | 1081 |
delsimps [minus_mult_left RS sym, minus_mult_right RS sym])); |
13957 | 1082 |
by (asm_simp_tac |
14320 | 1083 |
(simpset() addsimps [inverse_add, |
14335 | 1084 |
inverse_mult_distrib RS sym, inverse_minus_eq RS sym] |
1085 |
@ add_ac @ mult_ac |
|
14331 | 1086 |
delsimps [inverse_minus_eq, |
14335 | 1087 |
inverse_mult_distrib, minus_mult_right RS sym, minus_mult_left RS sym] ) 1); |
1088 |
by (asm_simp_tac (HOL_ss addsimps [mult_assoc RS sym, right_distrib]) 1); |
|
13957 | 1089 |
by (res_inst_tac [("y"," inverse(- hcomplex_of_complex x * hcomplex_of_complex x)")] |
1090 |
capprox_trans 1); |
|
1091 |
by (rtac inverse_add_CInfinitesimal_capprox2 1); |
|
1092 |
by (auto_tac (claset() addSDs [hcomplex_of_complex_CFinite_diff_CInfinitesimal] addIs [CFinite_mult], |
|
14335 | 1093 |
simpset() addsimps [inverse_minus_eq RS sym])); |
13957 | 1094 |
by (rtac CInfinitesimal_CFinite_mult2 1); |
1095 |
by Auto_tac; |
|
1096 |
qed "NSCDERIV_inverse"; |
|
1097 |
||
1098 |
Goal "x ~= 0 ==> CDERIV (%x. inverse(x)) x :> (-(inverse x ^ 2))"; |
|
1099 |
by (asm_simp_tac (simpset() addsimps [NSCDERIV_inverse, |
|
1100 |
NSCDERIV_CDERIV_iff RS sym] delsimps [complexpow_Suc]) 1); |
|
1101 |
qed "CDERIV_inverse"; |
|
1102 |
||
1103 |
||
14335 | 1104 |
(*-----------------------------------------------------------------------*) |
13957 | 1105 |
(* Derivative of inverse *) |
14335 | 1106 |
(*-----------------------------------------------------------------------*) |
13957 | 1107 |
|
1108 |
Goal "[| CDERIV f x :> d; f(x) ~= 0 |] \ |
|
1109 |
\ ==> CDERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ 2)))"; |
|
14373 | 1110 |
by (rtac (mult_commute RS subst) 1); |
1111 |
by (asm_simp_tac (simpset() addsimps [minus_mult_left, |
|
1112 |
power_inverse] delsimps [complexpow_Suc, minus_mult_left RS sym]) 1); |
|
13957 | 1113 |
by (fold_goals_tac [o_def]); |
1114 |
by (blast_tac (claset() addSIs [CDERIV_chain,CDERIV_inverse]) 1); |
|
1115 |
qed "CDERIV_inverse_fun"; |
|
1116 |
||
1117 |
Goal "[| NSCDERIV f x :> d; f(x) ~= 0 |] \ |
|
1118 |
\ ==> NSCDERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ 2)))"; |
|
1119 |
by (asm_full_simp_tac (simpset() addsimps [NSCDERIV_CDERIV_iff, |
|
1120 |
CDERIV_inverse_fun] delsimps [complexpow_Suc]) 1); |
|
1121 |
qed "NSCDERIV_inverse_fun"; |
|
1122 |
||
14335 | 1123 |
(*-----------------------------------------------------------------------*) |
13957 | 1124 |
(* Derivative of quotient *) |
14335 | 1125 |
(*-----------------------------------------------------------------------*) |
13957 | 1126 |
|
1127 |
||
1128 |
Goal "x ~= (0::complex) \\<Longrightarrow> (x * inverse(x) ^ 2) = inverse x"; |
|
14323 | 1129 |
by (auto_tac (claset(),simpset() addsimps [numeral_2_eq_2])); |
13957 | 1130 |
qed "lemma_complex_mult_inverse_squared"; |
1131 |
Addsimps [lemma_complex_mult_inverse_squared]; |
|
1132 |
||
1133 |
Goalw [complex_diff_def] |
|
1134 |
"[| CDERIV f x :> d; CDERIV g x :> e; g(x) ~= 0 |] \ |
|
1135 |
\ ==> CDERIV (%y. f(y) / (g y)) x :> (d*g(x) - (e*f(x))) / (g(x) ^ 2)"; |
|
1136 |
by (dres_inst_tac [("f","g")] CDERIV_inverse_fun 1); |
|
1137 |
by (dtac CDERIV_mult 2); |
|
1138 |
by (REPEAT(assume_tac 1)); |
|
1139 |
by (asm_full_simp_tac |
|
14373 | 1140 |
(simpset() addsimps [complex_divide_def, right_distrib, |
1141 |
power_inverse,minus_mult_left] @ mult_ac |
|
14335 | 1142 |
delsimps [complexpow_Suc, minus_mult_right RS sym, minus_mult_left RS sym]) 1); |
13957 | 1143 |
qed "CDERIV_quotient"; |
1144 |
||
1145 |
Goal "[| NSCDERIV f x :> d; NSCDERIV g x :> e; g(x) ~= 0 |] \ |
|
1146 |
\ ==> NSCDERIV (%y. f(y) / (g y)) x :> (d*g(x) - (e*f(x))) / (g(x) ^ 2)"; |
|
1147 |
by (asm_full_simp_tac (simpset() addsimps [NSCDERIV_CDERIV_iff, |
|
1148 |
CDERIV_quotient] delsimps [complexpow_Suc]) 1); |
|
1149 |
qed "NSCDERIV_quotient"; |
|
1150 |
||
1151 |
||
14335 | 1152 |
(*-----------------------------------------------------------------------*) |
13957 | 1153 |
(* Caratheodory formulation of derivative at a point: standard proof *) |
14335 | 1154 |
(*-----------------------------------------------------------------------*) |
13957 | 1155 |
|
1156 |
||
1157 |
Goalw [CLIM_def] |
|
1158 |
"[| ALL x. x ~= a --> (f x = g x) |] \ |
|
1159 |
\ ==> (f -- a --C> l) = (g -- a --C> l)"; |
|
1160 |
by (auto_tac (claset(), simpset() addsimps [complex_add_minus_iff])); |
|
1161 |
qed "CLIM_equal"; |
|
1162 |
||
1163 |
Goal "[| (%x. f(x) + -g(x)) -- a --C> 0; \ |
|
1164 |
\ g -- a --C> l |] \ |
|
1165 |
\ ==> f -- a --C> l"; |
|
1166 |
by (dtac CLIM_add 1 THEN assume_tac 1); |
|
1167 |
by (auto_tac (claset(), simpset() addsimps [complex_add_assoc])); |
|
1168 |
qed "CLIM_trans"; |
|
1169 |
||
1170 |
Goal "(CDERIV f x :> l) = \ |
|
1171 |
\ (EX g. (ALL z. f z - f x = g z * (z - x)) & isContc g x & g x = l)"; |
|
1172 |
by (Step_tac 1); |
|
1173 |
by (res_inst_tac |
|
1174 |
[("x","%z. if z = x then l else (f(z) - f(x)) / (z - x)")] exI 1); |
|
14373 | 1175 |
by (auto_tac (claset(),simpset() addsimps [mult_assoc, |
13957 | 1176 |
CLAIM "z ~= x ==> z - x ~= (0::complex)"])); |
1177 |
by (auto_tac (claset(),simpset() addsimps [isContc_iff,CDERIV_iff])); |
|
1178 |
by (ALLGOALS(rtac (CLIM_equal RS iffD1))); |
|
1179 |
by Auto_tac; |
|
1180 |
qed "CARAT_CDERIV"; |
|
1181 |
||
1182 |
Goal "NSCDERIV f x :> l ==> \ |
|
1183 |
\ EX g. (ALL z. f z - f x = g z * (z - x)) & isNSContc g x & g x = l"; |
|
1184 |
by (auto_tac (claset(),simpset() addsimps [NSCDERIV_CDERIV_iff, |
|
1185 |
isNSContc_isContc_iff,CARAT_CDERIV])); |
|
1186 |
qed "CARAT_NSCDERIV"; |
|
1187 |
||
1188 |
(* How about a NS proof? *) |
|
1189 |
Goal "(ALL z. f z - f x = g z * (z - x)) & isNSContc g x & g x = l \ |
|
1190 |
\ ==> NSCDERIV f x :> l"; |
|
1191 |
by (auto_tac (claset(), |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14373
diff
changeset
|
1192 |
simpset() delsimprocs field_cancel_factor |
13957 | 1193 |
addsimps [NSCDERIV_iff2])); |
1194 |
by (asm_full_simp_tac (simpset() addsimps [isNSContc_def]) 1); |
|
1195 |
qed "CARAT_CDERIVD"; |
|
1196 |