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theory Pigeonhole(* Title: HOL/Extraction/Pigeonhole.thy ID: $Id: Pigeonhole.thy,v 1.4 2005/09/23 14:05:42 nipkow Exp $ 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: "!!n::nat. m = n ∨ m ≠ n" apply (induct m) apply (case_tac n) apply (case_tac [3] n) apply (simp only: nat.simps, iprover?)+ done text {* We can decide whether an array @{term "f"} of length @{term "l"} contains an element @{term "x"}. *} lemma search: "(∃j<(l::nat). (x::nat) = f j) ∨ ¬ (∃j<l. x = f j)" proof (induct l) case 0 have "¬ (∃j<0. x = f j)" proof assume "∃j<0. x = f j" then obtain j where j: "j < (0::nat)" by iprover thus "False" by simp qed thus ?case .. next case (Suc l) thus ?case proof assume "∃j<l. x = f j" then obtain j where j: "j < l" and eq: "x = f j" by iprover from j have "j < Suc l" by simp with eq show ?case by iprover next assume nex: "¬ (∃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 iprover next assume neq: "x ≠ f l" have "¬ (∃j<Suc l. x = f j)" proof assume "∃j<Suc l. x = f j" then obtain j where j: "j < Suc l" and eq: "x = f j" by iprover show False proof cases assume "j = l" with eq have "x = f l" by simp with neq show False .. next assume "j ≠ l" with j have "j < l" by simp with nex and eq show False by iprover qed qed thus ?case .. qed qed qed text {* This proof yields a polynomial program. *} theorem pigeonhole: "!!f. (!!i. i ≤ Suc n ==> f i ≤ n) ==> ∃i j. i ≤ Suc n ∧ j < i ∧ f i = f j" proof (induct n) case 0 hence "Suc 0 ≤ Suc 0 ∧ 0 < Suc 0 ∧ f (Suc 0) = f 0" by simp thus ?case by iprover next case (Suc n) { fix k have "k ≤ Suc (Suc n) ==> (!!i j. Suc k ≤ i ==> i ≤ Suc (Suc n) ==> j < i ==> f i ≠ f j) ==> (∃i j. i ≤ k ∧ j < i ∧ f i = f j)" proof (induct k) case 0 let ?f = "λi. if f i = Suc n then f (Suc (Suc n)) else f i" have "¬ (∃i j. i ≤ Suc n ∧ j < i ∧ ?f i = ?f j)" proof assume "∃i j. i ≤ Suc n ∧ j < i ∧ ?f i = ?f j" then obtain i j where i: "i ≤ Suc n" and j: "j < i" and f: "?f i = ?f j" by iprover from j have i_nz: "Suc 0 ≤ i" by simp from i have iSSn: "i ≤ Suc (Suc n)" by simp have S0SSn: "Suc 0 ≤ 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 ≠ 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)) ≠ f j" by (rule 0) moreover assume "f j ≠ Suc n" with fi and f have "f (Suc (Suc n)) = f j" by simp ultimately show False .. qed next assume fi: "f i ≠ 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)) ≠ 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 ≠ f j" by (rule 0) moreover assume "f j ≠ Suc n" with fi and f have "f i = f j" by simp ultimately show False .. qed qed qed moreover have "!!i. i ≤ Suc n ==> ?f i ≤ n" proof - fix i assume "i ≤ Suc n" hence i: "i < Suc (Suc n)" by simp have "f (Suc (Suc n)) ≠ f i" by (rule 0) (simp_all add: i) moreover have "f (Suc (Suc n)) ≤ Suc n" by (rule Suc) simp moreover from i have "i ≤ Suc (Suc n)" by simp hence "f i ≤ Suc n" by (rule Suc) ultimately show "?thesis i" by simp qed hence "∃i j. i ≤ Suc n ∧ j < i ∧ ?f i = ?f j" by (rule Suc) ultimately show ?case .. next case (Suc k) from search show ?case proof assume "∃j<Suc k. f (Suc k) = f j" thus ?case by (iprover intro: le_refl) next assume nex: "¬ (∃j<Suc k. f (Suc k) = f j)" have "∃i j. i ≤ k ∧ j < i ∧ f i = f j" proof (rule Suc) from Suc show "k ≤ Suc (Suc n)" by simp fix i j assume k: "Suc k ≤ i" and i: "i ≤ Suc (Suc n)" and j: "j < i" show "f i ≠ 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 iprover qed next assume "i ≠ Suc k" with k have "Suc (Suc k) ≤ i" by simp thus ?thesis using i and j by (rule Suc) qed qed thus ?thesis by (iprover 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: "!!f. (!!i. i ≤ Suc n ==> f i ≤ n) ==> ∃i j. i ≤ Suc n ∧ j < i ∧ f i = f j" proof (induct n) case 0 have "Suc 0 ≤ Suc 0" .. moreover have "0 < Suc 0" .. moreover from 0 have "f (Suc 0) = f 0" by simp ultimately show ?case by iprover next case (Suc n) from search show ?case proof assume "∃j < Suc (Suc n). f (Suc (Suc n)) = f j" thus ?case by (iprover intro: le_refl) next assume "¬ (∃j < Suc (Suc n). f (Suc (Suc n)) = f j)" hence nex: "∀j < Suc (Suc n). f (Suc (Suc n)) ≠ f j" by iprover let ?f = "λi. if f i = Suc n then f (Suc (Suc n)) else f i" have "!!i. i ≤ Suc n ==> ?f i ≤ n" proof - fix i assume i: "i ≤ Suc n" show "?thesis i" proof (cases "f i = Suc n") case True from i and nex have "f (Suc (Suc n)) ≠ f i" by simp with True have "f (Suc (Suc n)) ≠ Suc n" by simp moreover from Suc have "f (Suc (Suc n)) ≤ Suc n" by simp ultimately have "f (Suc (Suc n)) ≤ n" by simp with True show ?thesis by simp next case False from Suc and i have "f i ≤ Suc n" by simp with False show ?thesis by simp qed qed hence "∃i j. i ≤ Suc n ∧ j < i ∧ ?f i = ?f j" by (rule Suc) then obtain i j where i: "i ≤ Suc n" and ji: "j < i" and f: "?f i = ?f j" by iprover 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 ≠ Suc n" moreover from i ji nex have "f (Suc (Suc n)) ≠ 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)) ≠ f i" by simp ultimately show ?thesis using False f by simp next assume "f j ≠ Suc n" with False f show ?thesis by simp qed qed moreover from i have "i ≤ Suc (Suc n)" by simp ultimately show ?thesis using ji by iprover 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 × nat" ("{* (0::nat, 0::nat) *}") code_module PH contains test = "λn. pigeonhole n (λm. m - 1)" test' = "λn. pigeonhole_slow n (λm. m - 1)" sel = "op !" ML "timeit (fn () => PH.test 10)" ML "timeit (fn () => PH.test' 10)" ML "timeit (fn () => PH.test 20)" ML "timeit (fn () => PH.test' 20)" ML "timeit (fn () => PH.test 25)" ML "timeit (fn () => PH.test' 25)" ML "timeit (fn () => PH.test 500)" ML "PH.pigeonhole 8 (PH.sel [0,1,2,3,4,5,6,3,7,8])" end
lemma nat_eq_dec:
m = n ∨ m ≠ n
lemma search:
(∃j<l. x = f j) ∨ ¬ (∃j<l. x = f j)
theorem pigeonhole:
(!!i. i ≤ Suc n ==> f i ≤ n) ==> ∃i j. i ≤ Suc n ∧ j < i ∧ f i = f j
theorem pigeonhole_slow:
(!!i. i ≤ Suc n ==> f i ≤ n) ==> ∃i j. i ≤ Suc n ∧ j < i ∧ f i = f j