author | paulson |
Mon, 25 Jun 2001 15:35:59 +0200 | |
changeset 11377 | 0f16ad464c62 |
parent 11024 | 23bf8d787b04 |
child 11586 | d8a7f6318457 |
permissions | -rw-r--r-- |
11024 | 1 |
(* Title: HOL/ex/NatSum.ML |
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ID: $Id$ |
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Author: Tobias Nipkow |
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Copyright 1994 TU Muenchen |
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Summing natural numbers, squares, cubes, etc. |
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Originally demonstrated permutative rewriting, but add_ac is no longer |
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needed thanks to new simprocs. |
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Thanks to Sloane's On-Line Encyclopedia of Integer Sequences, |
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http://www.research.att.com/~njas/sequences/ |
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*) |
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header {* Summing natural numbers *} |
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theory NatSum = Main: |
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declare lessThan_Suc [simp] atMost_Suc [simp] |
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declare add_mult_distrib [simp] add_mult_distrib2 [simp] |
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declare diff_mult_distrib [simp] diff_mult_distrib2 [simp] |
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text {* |
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\medskip The sum of the first @{term n} odd numbers equals @{term n} |
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squared. |
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*} |
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lemma sum_of_odds: "setsum (\<lambda>i. Suc (i + i)) (lessThan n) = n * n" |
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apply (induct n) |
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apply auto |
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done |
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text {* |
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\medskip The sum of the first @{text n} odd squares. |
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*} |
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lemma sum_of_odd_squares: |
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"#3 * setsum (\<lambda>i. Suc (i + i) * Suc (i + i)) (lessThan n) = n * (#4 * n * n - #1)" |
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apply (induct n) |
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txt {* This removes the @{term "-#1"} from the inductive step *} |
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apply (case_tac [2] n) |
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apply auto |
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done |
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text {* |
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\medskip The sum of the first @{term n} odd cubes |
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*} |
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lemma sum_of_odd_cubes: |
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"setsum (\<lambda>i. Suc (i + i) * Suc (i + i) * Suc (i + i)) (lessThan n) = |
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n * n * (#2 * n * n - #1)" |
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apply (induct "n") |
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txt {* This removes the @{term "-#1"} from the inductive step *} |
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apply (case_tac [2] "n") |
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apply auto |
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done |
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text {* |
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\medskip The sum of the first @{term n} positive integers equals |
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@{text "n (n + 1) / 2"}.*} |
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lemma sum_of_naturals: "#2 * setsum id (atMost n) = n * Suc n" |
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apply (induct n) |
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apply auto |
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done |
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lemma sum_of_squares: "#6 * setsum (\<lambda>i. i * i) (atMost n) = n * Suc n * Suc (#2 * n)" |
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apply (induct n) |
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apply auto |
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done |
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lemma sum_of_cubes: "#4 * setsum (\<lambda>i. i * i * i) (atMost n) = n * n * Suc n * Suc n" |
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apply (induct n) |
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apply auto |
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done |
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text {* |
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\medskip Sum of fourth powers: two versions. |
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*} |
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lemma sum_of_fourth_powers: |
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"#30 * setsum (\<lambda>i. i * i * i * i) (atMost n) = |
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n * Suc n * Suc (#2 * n) * (#3 * n * n + #3 * n - #1)" |
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apply (induct n) |
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apply auto |
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txt {* Eliminates the subtraction *} |
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apply (case_tac n) |
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apply simp_all |
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done |
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text {* |
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Alternative proof, with a change of variables and much more |
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subtraction, performed using the integers. *} |
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declare |
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zmult_int [symmetric, simp] |
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zadd_zmult_distrib [simp] |
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zadd_zmult_distrib2 [simp] |
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zdiff_zmult_distrib [simp] |
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zdiff_zmult_distrib2 [simp] |
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lemma int_sum_of_fourth_powers: |
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"#30 * int (setsum (\<lambda>i. i * i * i * i) (lessThan m)) = |
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int m * (int m - #1) * (int (#2 * m) - #1) * |
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(int (#3 * m * m) - int (#3 * m) - #1)" |
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apply (induct m) |
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apply simp_all |
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done |
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text {* |
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\medskip Sums of geometric series: 2, 3 and the general case *} |
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lemma sum_of_2_powers: "setsum (\<lambda>i. #2^i) (lessThan n) = #2^n - 1" |
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apply (induct n) |
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0f16ad464c62
Simprocs for type "nat" no longer introduce numerals unless they are already
paulson
parents:
11024
diff
changeset
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apply (auto split: nat_diff_split) |
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done |
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lemma sum_of_3_powers: "#2 * setsum (\<lambda>i. #3^i) (lessThan n) = #3^n - 1" |
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apply (induct n) |
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apply auto |
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done |
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lemma sum_of_powers: "0 < k ==> (k - 1) * setsum (\<lambda>i. k^i) (lessThan n) = k^n - 1" |
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apply (induct n) |
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apply auto |
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done |
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end |