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src/HOL/Extraction/Pigeonhole.thy

author | berghofe |

Sat, 06 Aug 2005 18:06:56 +0200 | |

changeset 17027 | 8bbe57116d13 |

parent 17024 | ae4a8446df16 |

child 17145 | e623e57b0f44 |

permissions | -rw-r--r-- |

Tuned comment.

(* Title: HOL/Extraction/Pigeonhole.thy ID: $Id$ Author: Stefan Berghofer, TU Muenchen *) header {* The pigeonhole principle *} theory Pigeonhole imports EfficientNat begin text {* We formalize two proofs of the pigeonhole principle, which lead to extracted programs of quite different complexity. The original formalization of these proofs in {\sc Nuprl} is due to Aleksey Nogin \cite{Nogin-ENTCS-2000}. We need decidability of equality on natural numbers: *} lemma nat_eq_dec: "\<And>n::nat. m = n \<or> m \<noteq> n" apply (induct m) apply (case_tac n) apply (case_tac [3] n) apply (simp only: nat.simps, rules?)+ done text {* We can decide whether an array @{term "f"} of length @{term "l"} contains an element @{term "x"}. *} lemma search: "(\<exists>j<(l::nat). (x::nat) = f j) \<or> \<not> (\<exists>j<l. x = f j)" proof (induct l) case 0 have "\<not> (\<exists>j<0. x = f j)" proof assume "\<exists>j<0. x = f j" then obtain j where j: "j < (0::nat)" by rules thus "False" by simp qed thus ?case .. next case (Suc l) thus ?case proof assume "\<exists>j<l. x = f j" then obtain j where j: "j < l" and eq: "x = f j" by rules from j have "j < Suc l" by simp with eq show ?case by rules next assume nex: "\<not> (\<exists>j<l. x = f j)" from nat_eq_dec show ?case proof assume eq: "x = f l" have "l < Suc l" by simp with eq show ?case by rules next assume neq: "x \<noteq> f l" have "\<not> (\<exists>j<Suc l. x = f j)" proof assume "\<exists>j<Suc l. x = f j" then obtain j where j: "j < Suc l" and eq: "x = f j" by rules show False proof cases assume "j = l" with eq have "x = f l" by simp with neq show False .. next assume "j \<noteq> l" with j have "j < l" by simp with nex and eq show False by rules qed qed thus ?case .. qed qed qed text {* This proof yields a polynomial program. *} theorem pigeonhole: "\<And>f. (\<And>i. i \<le> Suc n \<Longrightarrow> f i \<le> n) \<Longrightarrow> \<exists>i j. i \<le> Suc n \<and> j < i \<and> f i = f j" proof (induct n) case 0 hence "Suc 0 \<le> Suc 0 \<and> 0 < Suc 0 \<and> f (Suc 0) = f 0" by simp thus ?case by rules next case (Suc n) { fix k have "k \<le> Suc (Suc n) \<Longrightarrow> (\<And>i j. Suc k \<le> i \<Longrightarrow> i \<le> Suc (Suc n) \<Longrightarrow> j < i \<Longrightarrow> f i \<noteq> f j) \<Longrightarrow> (\<exists>i j. i \<le> k \<and> j < i \<and> f i = f j)" proof (induct k) case 0 let ?f = "\<lambda>i. if f i = Suc n then f (Suc (Suc n)) else f i" have "\<not> (\<exists>i j. i \<le> Suc n \<and> j < i \<and> ?f i = ?f j)" proof assume "\<exists>i j. i \<le> Suc n \<and> j < i \<and> ?f i = ?f j" then obtain i j where i: "i \<le> Suc n" and j: "j < i" and f: "?f i = ?f j" by rules from j have i_nz: "Suc 0 \<le> i" by simp from i have iSSn: "i \<le> Suc (Suc n)" by simp have S0SSn: "Suc 0 \<le> Suc (Suc n)" by simp show False proof cases assume fi: "f i = Suc n" show False proof cases assume fj: "f j = Suc n" from i_nz and iSSn and j have "f i \<noteq> f j" by (rule 0) moreover from fi have "f i = f j" by (simp add: fj [symmetric]) ultimately show ?thesis .. next from i and j have "j < Suc (Suc n)" by simp with S0SSn and le_refl have "f (Suc (Suc n)) \<noteq> f j" by (rule 0) moreover assume "f j \<noteq> Suc n" with fi and f have "f (Suc (Suc n)) = f j" by simp ultimately show False .. qed next assume fi: "f i \<noteq> Suc n" show False proof cases from i have "i < Suc (Suc n)" by simp with S0SSn and le_refl have "f (Suc (Suc n)) \<noteq> f i" by (rule 0) moreover assume "f j = Suc n" with fi and f have "f (Suc (Suc n)) = f i" by simp ultimately show False .. next from i_nz and iSSn and j have "f i \<noteq> f j" by (rule 0) moreover assume "f j \<noteq> Suc n" with fi and f have "f i = f j" by simp ultimately show False .. qed qed qed moreover have "\<And>i. i \<le> Suc n \<Longrightarrow> ?f i \<le> n" proof - fix i assume "i \<le> Suc n" hence i: "i < Suc (Suc n)" by simp have "f (Suc (Suc n)) \<noteq> f i" by (rule 0) (simp_all add: i) moreover have "f (Suc (Suc n)) \<le> Suc n" by (rule Suc) simp moreover from i have "i \<le> Suc (Suc n)" by simp hence "f i \<le> Suc n" by (rule Suc) ultimately show "?thesis i" by simp qed hence "\<exists>i j. i \<le> Suc n \<and> j < i \<and> ?f i = ?f j" by (rule Suc) ultimately show ?case .. next case (Suc k) from search show ?case proof assume "\<exists>j<Suc k. f (Suc k) = f j" thus ?case by (rules intro: le_refl) next assume nex: "\<not> (\<exists>j<Suc k. f (Suc k) = f j)" have "\<exists>i j. i \<le> k \<and> j < i \<and> f i = f j" proof (rule Suc) from Suc show "k \<le> Suc (Suc n)" by simp fix i j assume k: "Suc k \<le> i" and i: "i \<le> Suc (Suc n)" and j: "j < i" show "f i \<noteq> f j" proof cases assume eq: "i = Suc k" show ?thesis proof assume "f i = f j" hence "f (Suc k) = f j" by (simp add: eq) with nex and j and eq show False by rules qed next assume "i \<noteq> Suc k" with k have "Suc (Suc k) \<le> i" by simp thus ?thesis using i and j by (rule Suc) qed qed thus ?thesis by (rules intro: le_SucI) qed qed } note r = this show ?case by (rule r) simp_all qed text {* The following proof, although quite elegant from a mathematical point of view, leads to an exponential program: *} theorem pigeonhole_slow: "\<And>f. (\<And>i. i \<le> Suc n \<Longrightarrow> f i \<le> n) \<Longrightarrow> \<exists>i j. i \<le> Suc n \<and> j < i \<and> f i = f j" proof (induct n) case 0 have "Suc 0 \<le> Suc 0" .. moreover have "0 < Suc 0" .. moreover from 0 have "f (Suc 0) = f 0" by simp ultimately show ?case by rules next case (Suc n) from search show ?case proof assume "\<exists>j < Suc (Suc n). f (Suc (Suc n)) = f j" thus ?case by (rules intro: le_refl) next assume "\<not> (\<exists>j < Suc (Suc n). f (Suc (Suc n)) = f j)" hence nex: "\<forall>j < Suc (Suc n). f (Suc (Suc n)) \<noteq> f j" by rules let ?f = "\<lambda>i. if f i = Suc n then f (Suc (Suc n)) else f i" have "\<And>i. i \<le> Suc n \<Longrightarrow> ?f i \<le> n" proof - fix i assume i: "i \<le> Suc n" show "?thesis i" proof (cases "f i = Suc n") case True from i and nex have "f (Suc (Suc n)) \<noteq> f i" by simp with True have "f (Suc (Suc n)) \<noteq> Suc n" by simp moreover from Suc have "f (Suc (Suc n)) \<le> Suc n" by simp ultimately have "f (Suc (Suc n)) \<le> n" by simp with True show ?thesis by simp next case False from Suc and i have "f i \<le> Suc n" by simp with False show ?thesis by simp qed qed hence "\<exists>i j. i \<le> Suc n \<and> j < i \<and> ?f i = ?f j" by (rule Suc) then obtain i j where i: "i \<le> Suc n" and ji: "j < i" and f: "?f i = ?f j" by rules have "f i = f j" proof (cases "f i = Suc n") case True show ?thesis proof (cases "f j = Suc n") assume "f j = Suc n" with True show ?thesis by simp next assume "f j \<noteq> Suc n" moreover from i ji nex have "f (Suc (Suc n)) \<noteq> f j" by simp ultimately show ?thesis using True f by simp qed next case False show ?thesis proof (cases "f j = Suc n") assume "f j = Suc n" moreover from i nex have "f (Suc (Suc n)) \<noteq> f i" by simp ultimately show ?thesis using False f by simp next assume "f j \<noteq> Suc n" with False f show ?thesis by simp qed qed moreover from i have "i \<le> Suc (Suc n)" by simp ultimately show ?thesis using ji by rules qed qed extract pigeonhole pigeonhole_slow text {* The programs extracted from the above proofs look as follows: @{thm [display] pigeonhole_def} @{thm [display] pigeonhole_slow_def} The program for searching for an element in an array is @{thm [display,eta_contract=false] search_def} The correctness statement for @{term "pigeonhole"} is @{thm [display] pigeonhole_correctness [no_vars]} In order to analyze the speed of the above programs, we generate ML code from them. *} consts_code arbitrary :: "nat \<times> nat" ("{* (0::nat, 0::nat) *}") generate_code test = "\<lambda>n. pigeonhole n (\<lambda>m. m - 1)" test' = "\<lambda>n. pigeonhole_slow n (\<lambda>m. m - 1)" sel = "op !" ML "timeit (fn () => test 10)" ML "timeit (fn () => test' 10)" ML "timeit (fn () => test 20)" ML "timeit (fn () => test' 20)" ML "timeit (fn () => test 25)" ML "timeit (fn () => test' 25)" ML "timeit (fn () => test 500)" ML "pigeonhole 8 (sel [0,1,2,3,4,5,6,3,7,8])" end