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(* 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 and cubes. Could be continued...
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*)
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val natsum_ss = arith_ss addsimps
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([NatSum.sum_0,NatSum.sum_Suc] @ add_ac);
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(*The sum of the first n positive integers equals n(n+1)/2.*)
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goal NatSum.thy "Suc(Suc(0))*sum (%i.i) (Suc n) = n*Suc(n)";
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by (simp_tac natsum_ss 1);
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by (nat_ind_tac "n" 1);
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by (simp_tac natsum_ss 1);
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by (asm_simp_tac natsum_ss 1);
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qed "sum_of_naturals";
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goal NatSum.thy
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"Suc(Suc(Suc(Suc(Suc(Suc(0))))))*sum (%i.i*i) (Suc n) = \
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\ n*Suc(n)*Suc(Suc(Suc(0))*n)";
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by (simp_tac natsum_ss 1);
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by (nat_ind_tac "n" 1);
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by (simp_tac natsum_ss 1);
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by (asm_simp_tac natsum_ss 1);
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qed "sum_of_squares";
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goal NatSum.thy
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"Suc(Suc(Suc(Suc(0))))*sum (%i.i*i*i) (Suc n) = n*n*Suc(n)*Suc(n)";
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by (simp_tac natsum_ss 1);
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by (nat_ind_tac "n" 1);
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by (simp_tac natsum_ss 1);
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by (asm_simp_tac natsum_ss 1);
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qed "sum_of_cubes";
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(*The sum of the first n odd numbers equals n squared.*)
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goal NatSum.thy "sum (%i.Suc(i+i)) n = n*n";
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by (nat_ind_tac "n" 1);
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by (simp_tac natsum_ss 1);
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by (asm_simp_tac natsum_ss 1);
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qed "sum_of_odds";
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