author | nipkow |
Wed, 12 Feb 1997 18:54:39 +0100 | |
changeset 2609 | 4370e5f0fa3f |
parent 2031 | 03a843f0f447 |
child 3336 | 29ddef80bd49 |
permissions | -rw-r--r-- |
1465 | 1 |
(* Title: HOL/ex/puzzle.ML |
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ID: $Id$ |
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Author: Tobias Nipkow |
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Copyright 1993 TU Muenchen |
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For puzzle.thy. A question from "Bundeswettbewerb Mathematik" |
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Proof due to Herbert Ehler |
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*) |
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(*specialized form of induction needed below*) |
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val prems = goal Nat.thy "[| P(0); !!n. P(Suc(n)) |] ==> !n.P(n)"; |
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by (EVERY1 [rtac (nat_induct RS allI), resolve_tac prems, resolve_tac prems]); |
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qed "nat_exh"; |
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goal Puzzle.thy "! n. k=f(n) --> n <= f(n)"; |
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by (res_inst_tac [("n","k")] less_induct 1); |
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by (rtac nat_exh 1); |
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by (Simp_tac 1); |
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by (rtac impI 1); |
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by (rtac classical 1); |
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by (dtac not_leE 1); |
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by (subgoal_tac "f(na) <= f(f(na))" 1); |
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2017
dd3e2a91aeca
Proofs made more robust to work in presence of le_refl
paulson
parents:
1820
diff
changeset
|
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by (fast_tac (!claset addIs [Puzzle.f_ax]) 2); |
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by (rtac lessD 1); |
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by (fast_tac (!claset delrules [order_refl] |
2017
dd3e2a91aeca
Proofs made more robust to work in presence of le_refl
paulson
parents:
1820
diff
changeset
|
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addIs [Puzzle.f_ax, le_less_trans]) 1); |
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val lemma = result() RS spec RS mp; |
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goal Puzzle.thy "n <= f(n)"; |
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by (fast_tac (!claset addIs [lemma]) 1); |
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qed "lemma1"; |
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goal Puzzle.thy "f(n) < f(Suc(n))"; |
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2017
dd3e2a91aeca
Proofs made more robust to work in presence of le_refl
paulson
parents:
1820
diff
changeset
|
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by (deepen_tac (!claset addIs [Puzzle.f_ax, le_less_trans, lemma1]) 0 1); |
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qed "lemma2"; |
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val prems = goal Puzzle.thy "(!!n.f(n) <= f(Suc(n))) ==> m<n --> f(m) <= f(n)"; |
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by (res_inst_tac[("n","n")]nat_induct 1); |
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by (Simp_tac 1); |
1673
d22110ddd0af
repaired critical proofs depending on the order inside non-confluent SimpSets
oheimb
parents:
1465
diff
changeset
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by (simp_tac (!simpset addsimps [less_Suc_eq]) 1); |
2017
dd3e2a91aeca
Proofs made more robust to work in presence of le_refl
paulson
parents:
1820
diff
changeset
|
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by (best_tac (!claset addIs (le_trans::prems)) 1); |
dd3e2a91aeca
Proofs made more robust to work in presence of le_refl
paulson
parents:
1820
diff
changeset
|
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qed_spec_mp "mono_lemma1"; |
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val [p1,p2] = goal Puzzle.thy |
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"[| !! n. f(n)<=f(Suc(n)); m<=n |] ==> f(m) <= f(n)"; |
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by (rtac (p2 RS le_imp_less_or_eq RS disjE) 1); |
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by (etac (p1 RS mono_lemma1) 1); |
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2017
dd3e2a91aeca
Proofs made more robust to work in presence of le_refl
paulson
parents:
1820
diff
changeset
|
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by (Fast_tac 1); |
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qed "mono_lemma"; |
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val prems = goal Puzzle.thy "m <= n ==> f(m) <= f(n)"; |
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by (fast_tac (!claset addIs ([mono_lemma,less_imp_le,lemma2]@prems)) 1); |
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qed "f_mono"; |
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goal Puzzle.thy "f(n) = n"; |
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by (rtac le_anti_sym 1); |
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by (rtac lemma1 2); |
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by (fast_tac (!claset addIs [Puzzle.f_ax,leI] addDs [leD,f_mono,lessD]) 1); |
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result(); |