author | paulson |
Thu, 04 Jan 2001 10:23:01 +0100 | |
changeset 10778 | 2c6605049646 |
parent 10751 | a81ea5d3dd41 |
child 10797 | 028d22926a41 |
permissions | -rw-r--r-- |
10751 | 1 |
(* Title : HSeries.ML |
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Author : Jacques D. Fleuriot |
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Copyright : 1998 University of Cambridge |
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Description : Finite summation and infinite series |
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for hyperreals |
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*) |
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Goalw [sumhr_def] |
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"sumhr(M,N,f) = \ |
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\ Abs_hypreal(UN X:Rep_hypnat(M). UN Y: Rep_hypnat(N). \ |
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\ hyprel ^^{%n::nat. sumr (X n) (Y n) f})"; |
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by (Auto_tac); |
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qed "sumhr_iff"; |
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Goalw [sumhr_def] |
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"sumhr(Abs_hypnat(hypnatrel^^{%n. M n}), \ |
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\ Abs_hypnat(hypnatrel^^{%n. N n}), f) = \ |
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\ Abs_hypreal(hyprel ^^ {%n. sumr (M n) (N n) f})"; |
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by (res_inst_tac [("f","Abs_hypreal")] arg_cong 1); |
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by (Auto_tac THEN Ultra_tac 1); |
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qed "sumhr"; |
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(*------------------------------------------------------- |
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lcp's suggestion: exploit pattern matching |
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facilities and use as definition instead (to do) |
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-------------------------------------------------------*) |
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Goalw [sumhr_def] |
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"sumhr p = (%(M,N,f). Abs_hypreal(UN X:Rep_hypnat(M). \ |
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\ UN Y: Rep_hypnat(N). \ |
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\ hyprel ^^{%n::nat. sumr (X n) (Y n) f})) p"; |
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by (res_inst_tac [("p","p")] PairE 1); |
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by (res_inst_tac [("p","y")] PairE 1); |
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by (Auto_tac); |
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qed "sumhr_iff2"; |
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(* Theorem corresponding to base case in def of sumr *) |
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Goalw [hypnat_zero_def] |
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"sumhr (m,0,f) = #0"; |
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by (res_inst_tac [("z","m")] eq_Abs_hypnat 1); |
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by (auto_tac (claset(), |
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simpset() addsimps [sumhr, symmetric hypreal_zero_def])); |
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qed "sumhr_zero"; |
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Addsimps [sumhr_zero]; |
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(* Theorem corresponding to recursive case in def of sumr *) |
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Goalw [hypnat_one_def] |
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"sumhr(m,n+1hn,f) = (if n + 1hn <= m then #0 \ |
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\ else sumhr(m,n,f) + (*fNat* f) n)"; |
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by (simp_tac (HOL_ss addsimps |
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[zero_eq_numeral_0 RS sym, hypreal_zero_def]) 1); |
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by (res_inst_tac [("z","m")] eq_Abs_hypnat 1); |
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by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
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by (auto_tac (claset(), |
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simpset() addsimps [sumhr, hypnat_add,hypnat_le,starfunNat,hypreal_add])); |
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by (ALLGOALS(Ultra_tac)); |
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qed "sumhr_if"; |
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Goalw [hypnat_one_def] "sumhr (n + 1hn, n, f) = #0"; |
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by (simp_tac (HOL_ss addsimps |
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[zero_eq_numeral_0 RS sym, hypreal_zero_def]) 1); |
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by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
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by (auto_tac (claset(), |
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simpset() addsimps [sumhr, hypnat_add])); |
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qed "sumhr_Suc_zero"; |
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Addsimps [sumhr_Suc_zero]; |
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Goal "sumhr (n,n,f) = #0"; |
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by (simp_tac (HOL_ss addsimps |
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[zero_eq_numeral_0 RS sym, hypreal_zero_def]) 1); |
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by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
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by (auto_tac (claset(), simpset() addsimps [sumhr])); |
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qed "sumhr_eq_bounds"; |
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Addsimps [sumhr_eq_bounds]; |
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Goalw [hypnat_one_def] |
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"sumhr (m,m + 1hn,f) = (*fNat* f) m"; |
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by (res_inst_tac [("z","m")] eq_Abs_hypnat 1); |
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by (auto_tac (claset(), |
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simpset() addsimps [sumhr, hypnat_add,starfunNat])); |
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qed "sumhr_Suc"; |
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Addsimps [sumhr_Suc]; |
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Goal "sumhr(m+k,k,f) = #0"; |
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by (simp_tac (HOL_ss addsimps |
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[zero_eq_numeral_0 RS sym, hypreal_zero_def]) 1); |
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by (res_inst_tac [("z","m")] eq_Abs_hypnat 1); |
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by (res_inst_tac [("z","k")] eq_Abs_hypnat 1); |
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by (auto_tac (claset(), |
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simpset() addsimps [sumhr, hypnat_add])); |
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qed "sumhr_add_lbound_zero"; |
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Addsimps [sumhr_add_lbound_zero]; |
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Goal "sumhr (m,n,f) + sumhr(m,n,g) = sumhr(m,n,%i. f i + g i)"; |
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by (res_inst_tac [("z","m")] eq_Abs_hypnat 1); |
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by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
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by (auto_tac (claset(), |
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simpset() addsimps [sumhr, hypreal_add,sumr_add])); |
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qed "sumhr_add"; |
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Goalw [hypreal_of_real_def] |
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"hypreal_of_real r * sumhr(m,n,f) = sumhr(m,n,%n. r * f n)"; |
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by (res_inst_tac [("z","m")] eq_Abs_hypnat 1); |
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by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
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by (auto_tac (claset(), |
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simpset() addsimps [sumhr, hypreal_mult,sumr_mult])); |
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qed "sumhr_mult"; |
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Goalw [hypnat_zero_def] |
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"n < p ==> sumhr (0,n,f) + sumhr (n,p,f) = sumhr (0,p,f)"; |
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by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
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by (res_inst_tac [("z","p")] eq_Abs_hypnat 1); |
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by (auto_tac (claset() addSEs [FreeUltrafilterNat_subset], |
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simpset() addsimps [sumhr,hypreal_add,hypnat_less, sumr_split_add])); |
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qed "sumhr_split_add"; |
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(*FIXME delete*) |
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Goal "n < p ==> sumhr (0, p, f) + - sumhr (0, n, f) = sumhr (n,p,f)"; |
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by (dres_inst_tac [("f1","f")] (sumhr_split_add RS sym) 1); |
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by (Asm_simp_tac 1); |
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qed "sumhr_split_add_minus"; |
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Goal "n < p ==> sumhr (0, p, f) - sumhr (0, n, f) = sumhr (n,p,f)"; |
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by (dres_inst_tac [("f1","f")] (sumhr_split_add RS sym) 1); |
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by (Asm_simp_tac 1); |
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qed "sumhr_split_diff"; |
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Goal "abs(sumhr(m,n,f)) <= sumhr(m,n,%i. abs(f i))"; |
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by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
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by (res_inst_tac [("z","m")] eq_Abs_hypnat 1); |
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by (auto_tac (claset(), |
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simpset() addsimps [sumhr, hypreal_le,hypreal_hrabs,sumr_rabs])); |
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qed "sumhr_hrabs"; |
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(* other general version also needed *) |
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Goalw [hypnat_of_nat_def] |
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"(ALL r. m <= r & r < n --> f r = g r) --> \ |
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\ sumhr(hypnat_of_nat m, hypnat_of_nat n, f) = \ |
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\ sumhr(hypnat_of_nat m, hypnat_of_nat n, g)"; |
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by (Step_tac 1 THEN dtac sumr_fun_eq 1); |
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by (auto_tac (claset(), simpset() addsimps [sumhr])); |
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qed "sumhr_fun_hypnat_eq"; |
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Goalw [hypnat_zero_def,hypreal_of_real_def] |
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"sumhr(0,n,%i. r) = hypreal_of_hypnat n*hypreal_of_real r"; |
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by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
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by (asm_simp_tac |
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(simpset() addsimps [sumhr, hypreal_of_hypnat,hypreal_mult]) 1); |
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qed "sumhr_const"; |
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Goalw [hypnat_zero_def,hypreal_of_real_def] |
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"sumhr(0,n,f) + -(hypreal_of_hypnat n*hypreal_of_real r) = \ |
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\ sumhr(0,n,%i. f i + -r)"; |
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by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
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by (asm_simp_tac (simpset() addsimps [sumhr, |
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hypreal_of_hypnat,hypreal_mult,hypreal_add, |
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hypreal_minus,sumr_add RS sym]) 1); |
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qed "sumhr_add_mult_const"; |
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Goal "n < m ==> sumhr (m,n,f) = #0"; |
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by (simp_tac (HOL_ss addsimps |
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[zero_eq_numeral_0 RS sym, hypreal_zero_def]) 1); |
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by (res_inst_tac [("z","m")] eq_Abs_hypnat 1); |
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by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
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by (auto_tac (claset() addEs [FreeUltrafilterNat_subset], |
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simpset() addsimps [sumhr,hypnat_less])); |
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qed "sumhr_less_bounds_zero"; |
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Addsimps [sumhr_less_bounds_zero]; |
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Goal "sumhr(m, n, %i. - f i) = - sumhr(m, n, f)"; |
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by (res_inst_tac [("z","m")] eq_Abs_hypnat 1); |
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by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
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by (auto_tac (claset(), simpset() addsimps [sumhr, hypreal_minus,sumr_minus])); |
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qed "sumhr_minus"; |
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Goalw [hypnat_of_nat_def] |
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"sumhr(m+hypnat_of_nat k,n+hypnat_of_nat k,f) = sumhr(m,n,%i. f(i + k))"; |
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by (res_inst_tac [("z","m")] eq_Abs_hypnat 1); |
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by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
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by (auto_tac (claset(), |
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simpset() addsimps [sumhr, hypnat_add,sumr_shift_bounds])); |
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qed "sumhr_shift_bounds"; |
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(*------------------------------------------------------------------ |
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Theorems about NS sums - infinite sums are obtained |
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by summing to some infinite hypernatural (such as whn) |
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-----------------------------------------------------------------*) |
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Goalw [hypnat_omega_def,hypnat_zero_def] |
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"sumhr(0,whn,%i. #1) = hypreal_of_hypnat whn"; |
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by (auto_tac (claset(), |
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simpset() addsimps [sumhr, hypreal_of_hypnat])); |
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qed "sumhr_hypreal_of_hypnat_omega"; |
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Goalw [hypnat_omega_def,hypnat_zero_def,omega_def] |
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"sumhr(0, whn, %i. #1) = whr - #1"; |
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by (simp_tac (HOL_ss addsimps |
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[one_eq_numeral_1 RS sym, hypreal_one_def]) 1); |
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by (auto_tac (claset(), |
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2c6605049646
more tidying, especially to remove real_of_posnat
paulson
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simpset() addsimps [sumhr, hypreal_diff, real_of_nat_Suc])); |
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qed "sumhr_hypreal_omega_minus_one"; |
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Goalw [hypnat_zero_def, hypnat_omega_def] |
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"sumhr(0, whn + whn, %i. (-#1) ^ (i+1)) = #0"; |
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by (simp_tac (HOL_ss addsimps |
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[zero_eq_numeral_0 RS sym, hypreal_zero_def]) 1); |
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by (simp_tac (simpset() addsimps [sumhr,hypnat_add] delsimps [realpow_Suc]) 1); |
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qed "sumhr_minus_one_realpow_zero"; |
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Addsimps [sumhr_minus_one_realpow_zero]; |
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Goalw [hypnat_of_nat_def,hypreal_of_real_def] |
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"(ALL n. m <= Suc n --> f n = r) & m <= na \ |
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\ ==> sumhr(hypnat_of_nat m,hypnat_of_nat na,f) = \ |
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\ (hypreal_of_nat (na - m) * hypreal_of_real r)"; |
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by (auto_tac (claset() addSDs [sumr_interval_const], |
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simpset() addsimps [sumhr,hypreal_of_nat_def, |
2c6605049646
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hypreal_of_real_def, hypreal_mult])); |
10751 | 216 |
qed "sumhr_interval_const"; |
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Goalw [hypnat_zero_def] |
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"(*fNat* (%n. sumr 0 n f)) N = sumhr(0,N,f)"; |
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by (res_inst_tac [("z","N")] eq_Abs_hypnat 1); |
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by (asm_full_simp_tac (simpset() addsimps [starfunNat,sumhr]) 1); |
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qed "starfunNat_sumr"; |
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Goal "sumhr (0, M, f) @= sumhr (0, N, f) \ |
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\ ==> abs (sumhr (M, N, f)) @= #0"; |
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by (cut_inst_tac [("x","M"),("y","N")] hypnat_linear 1); |
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by (auto_tac (claset(), simpset() addsimps [inf_close_refl])); |
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by (dtac (inf_close_sym RS (inf_close_minus_iff RS iffD1)) 1); |
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by (auto_tac (claset() addDs [inf_close_hrabs], |
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simpset() addsimps [sumhr_split_add_minus])); |
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qed "sumhr_hrabs_inf_close"; |
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Addsimps [sumhr_hrabs_inf_close]; |
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(*---------------------------------------------------------------- |
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infinite sums: Standard and NS theorems |
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----------------------------------------------------------------*) |
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Goalw [sums_def,NSsums_def] "(f sums l) = (f NSsums l)"; |
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by (simp_tac (simpset() addsimps [LIMSEQ_NSLIMSEQ_iff]) 1); |
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qed "sums_NSsums_iff"; |
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Goalw [summable_def,NSsummable_def] |
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"(summable f) = (NSsummable f)"; |
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by (simp_tac (simpset() addsimps [sums_NSsums_iff]) 1); |
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qed "summable_NSsummable_iff"; |
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Goalw [suminf_def,NSsuminf_def] |
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"(suminf f) = (NSsuminf f)"; |
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by (simp_tac (simpset() addsimps [sums_NSsums_iff]) 1); |
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qed "suminf_NSsuminf_iff"; |
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Goalw [NSsums_def,NSsummable_def] |
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"f NSsums l ==> NSsummable f"; |
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by (Blast_tac 1); |
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qed "NSsums_NSsummable"; |
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Goalw [NSsummable_def,NSsuminf_def] |
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"NSsummable f ==> f NSsums (NSsuminf f)"; |
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by (blast_tac (claset() addIs [someI2]) 1); |
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qed "NSsummable_NSsums"; |
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260 |
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Goal "f NSsums s ==> (s = NSsuminf f)"; |
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by (asm_full_simp_tac |
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(simpset() addsimps [suminf_NSsuminf_iff RS sym,sums_NSsums_iff, |
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sums_unique]) 1); |
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qed "NSsums_unique"; |
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266 |
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Goal "ALL m. n <= Suc m --> f(m) = #0 ==> f NSsums (sumr 0 n f)"; |
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by (asm_simp_tac (simpset() addsimps [sums_NSsums_iff RS sym, series_zero]) 1); |
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qed "NSseries_zero"; |
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270 |
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Goal "NSsummable f = \ |
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\ (ALL M: HNatInfinite. ALL N: HNatInfinite. abs (sumhr(M,N,f)) @= #0)"; |
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by (auto_tac (claset(), |
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simpset() addsimps [summable_NSsummable_iff RS sym, |
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summable_convergent_sumr_iff, convergent_NSconvergent_iff, |
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NSCauchy_NSconvergent_iff RS sym, NSCauchy_def, |
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starfunNat_sumr])); |
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by (cut_inst_tac [("x","M"),("y","N")] hypnat_linear 1); |
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by (auto_tac (claset(), simpset() addsimps [inf_close_refl])); |
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by (rtac ((inf_close_minus_iff RS iffD2) RS inf_close_sym) 1); |
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by (rtac (inf_close_minus_iff RS iffD2) 2); |
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282 |
by (auto_tac (claset() addDs [inf_close_hrabs_zero_cancel], |
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simpset() addsimps [sumhr_split_add_minus])); |
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qed "NSsummable_NSCauchy"; |
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285 |
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286 |
(*------------------------------------------------------------------- |
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287 |
Terms of a convergent series tend to zero |
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288 |
-------------------------------------------------------------------*) |
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289 |
Goalw [NSLIMSEQ_def] "NSsummable f ==> f ----NS> #0"; |
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by (auto_tac (claset(), simpset() addsimps [NSsummable_NSCauchy])); |
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by (dtac bspec 1 THEN Auto_tac); |
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by (dres_inst_tac [("x","N + 1hn")] bspec 1); |
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by (auto_tac (claset() addIs [HNatInfinite_add_one, |
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294 |
inf_close_hrabs_zero_cancel], |
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simpset() addsimps [rename_numerals hypreal_of_real_zero])); |
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296 |
qed "NSsummable_NSLIMSEQ_zero"; |
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297 |
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298 |
(* Easy to prove stsandard case now *) |
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Goal "summable f ==> f ----> #0"; |
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300 |
by (auto_tac (claset(), |
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simpset() addsimps [summable_NSsummable_iff, |
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LIMSEQ_NSLIMSEQ_iff, NSsummable_NSLIMSEQ_zero])); |
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303 |
qed "summable_LIMSEQ_zero"; |
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304 |
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305 |
(*------------------------------------------------------------------- |
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306 |
NS Comparison test |
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307 |
-------------------------------------------------------------------*) |
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308 |
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309 |
Goal "[| EX N. ALL n. N <= n --> abs(f n) <= g n; \ |
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310 |
\ NSsummable g \ |
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311 |
\ |] ==> NSsummable f"; |
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by (auto_tac (claset() addIs [summable_comparison_test], |
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313 |
simpset() addsimps [summable_NSsummable_iff RS sym])); |
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314 |
qed "NSsummable_comparison_test"; |
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315 |
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316 |
Goal "[| EX N. ALL n. N <= n --> abs(f n) <= g n; \ |
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\ NSsummable g \ |
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318 |
\ |] ==> NSsummable (%k. abs (f k))"; |
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by (rtac NSsummable_comparison_test 1); |
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320 |
by (auto_tac (claset(), simpset() addsimps [abs_idempotent])); |
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321 |
qed "NSsummable_rabs_comparison_test"; |