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(* Title: HOL/ex/Fib
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ID: $Id$
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Author: Lawrence C Paulson
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Copyright 1997 University of Cambridge
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Fibonacci numbers: proofs of laws taken from
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R. L. Graham, D. E. Knuth, O. Patashnik.
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Concrete Mathematics.
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(Addison-Wesley, 1989)
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*)
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(** The difficulty in these proofs is to ensure that the induction hypotheses
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are applied before the definition of "fib". Towards this end, the
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"fib" equations are not added to the simpset and are applied very
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selectively at first.
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**)
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bind_thm ("fib_Suc_Suc", hd(rev fib.rules));
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(*Concrete Mathematics, page 280*)
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goal thy "fib (Suc (n + k)) = fib(Suc k) * fib(Suc n) + fib k * fib n";
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by (res_inst_tac [("u","n")] fib.induct 1);
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(*Simplify the LHS just enough to apply the induction hypotheses*)
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by (asm_full_simp_tac
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(simpset() addsimps [read_instantiate[("x", "Suc(?m+?n)")] fib_Suc_Suc]) 3);
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by (ALLGOALS
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(asm_simp_tac (simpset() addsimps
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(fib.rules @ add_ac @ mult_ac @
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[add_mult_distrib, add_mult_distrib2]))));
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qed "fib_add";
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goal thy "fib (Suc n) ~= 0";
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by (res_inst_tac [("u","n")] fib.induct 1);
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by (ALLGOALS (asm_simp_tac (simpset() addsimps fib.rules)));
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qed "fib_Suc_neq_0";
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Addsimps [fib_Suc_neq_0];
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goal thy "0 < fib (Suc n)";
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by (res_inst_tac [("u","n")] fib.induct 1);
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by (ALLGOALS (asm_simp_tac (simpset() addsimps fib.rules)));
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qed "fib_Suc_gr_0";
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Addsimps [fib_Suc_gr_0];
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(*Concrete Mathematics, page 278: Cassini's identity*)
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goal thy "fib (Suc (Suc n)) * fib n = \
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\ (if n mod 2 = 0 then pred(fib(Suc n) * fib(Suc n)) \
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\ else Suc (fib(Suc n) * fib(Suc n)))";
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by (res_inst_tac [("u","n")] fib.induct 1);
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by (res_inst_tac [("P", "%z. ?ff(x) * z = ?kk(x)")] (fib_Suc_Suc RS ssubst) 3);
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by (stac (read_instantiate [("x", "Suc(Suc ?n)")] fib_Suc_Suc) 3);
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by (asm_simp_tac (simpset() addsimps [add_mult_distrib, add_mult_distrib2]) 3);
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by (stac (read_instantiate [("x", "Suc ?n")] fib_Suc_Suc) 3);
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by (ALLGOALS (*using fib.rules here results in a longer proof!*)
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(asm_simp_tac (simpset() addsimps [add_mult_distrib, add_mult_distrib2,
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mod_less, mod_Suc]
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addsplits [expand_if])));
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by (safe_tac (claset() addSDs [mod2_neq_0]));
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by (ALLGOALS
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(asm_full_simp_tac
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(simpset() addsimps (fib.rules @ add_ac @ mult_ac @
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[add_mult_distrib, add_mult_distrib2,
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mod_less, mod_Suc]))));
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qed "fib_Cassini";
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(** exercise: prove gcd(fib m, fib n) = fib(gcd(m,n)) **)
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