# HG changeset patch # User paulson # Date 1390999897 0 # Node ID 608c157d743dee1871a48d28cc9c85a851d1e292 # Parent 39bcdf19dd14c5a483d15b95e2d2bc0263347653 Replacing the theory Library/Binomial by Number_Theory/Binomial diff -r 39bcdf19dd14 -r 608c157d743d src/Doc/ROOT --- a/src/Doc/ROOT Mon Jan 27 17:13:33 2014 +0000 +++ b/src/Doc/ROOT Wed Jan 29 12:51:37 2014 +0000 @@ -355,7 +355,7 @@ "Rules/Blast" "Rules/Force" theories [pretty_margin = 64, thy_output_indent = 5] - "Rules/Primes" + "Rules/TPrimes" "Rules/Forward" "Rules/Tacticals" "Rules/find2" diff -r 39bcdf19dd14 -r 608c157d743d src/Doc/Tutorial/Rules/Forward.thy --- a/src/Doc/Tutorial/Rules/Forward.thy Mon Jan 27 17:13:33 2014 +0000 +++ b/src/Doc/Tutorial/Rules/Forward.thy Wed Jan 29 12:51:37 2014 +0000 @@ -1,4 +1,4 @@ -theory Forward imports Primes begin +theory Forward imports TPrimes begin text{*\noindent Forward proof material: of, OF, THEN, simplify, rule_format. @@ -166,7 +166,7 @@ example of "insert" *} -lemma relprime_dvd_mult: +lemma relprime_dvd_mult: "\ gcd k n = 1; k dvd m*n \ \ k dvd m" apply (insert gcd_mult_distrib2 [of m k n]) txt{*@{subgoals[display,indent=0,margin=65]}*} diff -r 39bcdf19dd14 -r 608c157d743d src/Doc/Tutorial/Rules/Primes.thy --- a/src/Doc/Tutorial/Rules/Primes.thy Mon Jan 27 17:13:33 2014 +0000 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,155 +0,0 @@ -(* EXTRACT from HOL/ex/Primes.thy*) - -(*Euclid's algorithm - This material now appears AFTER that of Forward.thy *) -theory Primes imports Main begin - -fun gcd :: "nat \ nat \ nat" where - "gcd m n = (if n=0 then m else gcd n (m mod n))" - - -text {*Now in Basic.thy! -@{thm[display]"dvd_def"} -\rulename{dvd_def} -*}; - - -(*** Euclid's Algorithm ***) - -lemma gcd_0 [simp]: "gcd m 0 = m" -apply (simp); -done - -lemma gcd_non_0 [simp]: "0 gcd m n = gcd n (m mod n)" -apply (simp) -done; - -declare gcd.simps [simp del]; - -(*gcd(m,n) divides m and n. The conjunctions don't seem provable separately*) -lemma gcd_dvd_both: "(gcd m n dvd m) \ (gcd m n dvd n)" -apply (induct_tac m n rule: gcd.induct) - --{* @{subgoals[display,indent=0,margin=65]} *} -apply (case_tac "n=0") -txt{*subgoals after the case tac -@{subgoals[display,indent=0,margin=65]} -*}; -apply (simp_all) - --{* @{subgoals[display,indent=0,margin=65]} *} -by (blast dest: dvd_mod_imp_dvd) - - - -text {* -@{thm[display] dvd_mod_imp_dvd} -\rulename{dvd_mod_imp_dvd} - -@{thm[display] dvd_trans} -\rulename{dvd_trans} -*} - -lemmas gcd_dvd1 [iff] = gcd_dvd_both [THEN conjunct1] -lemmas gcd_dvd2 [iff] = gcd_dvd_both [THEN conjunct2]; - - -text {* -\begin{quote} -@{thm[display] gcd_dvd1} -\rulename{gcd_dvd1} - -@{thm[display] gcd_dvd2} -\rulename{gcd_dvd2} -\end{quote} -*}; - -(*Maximality: for all m,n,k naturals, - if k divides m and k divides n then k divides gcd(m,n)*) -lemma gcd_greatest [rule_format]: - "k dvd m \ k dvd n \ k dvd gcd m n" -apply (induct_tac m n rule: gcd.induct) -apply (case_tac "n=0") -txt{*subgoals after the case tac -@{subgoals[display,indent=0,margin=65]} -*}; -apply (simp_all add: dvd_mod) -done - -text {* -@{thm[display] dvd_mod} -\rulename{dvd_mod} -*} - -(*just checking the claim that case_tac "n" works too*) -lemma "k dvd m \ k dvd n \ k dvd gcd m n" -apply (induct_tac m n rule: gcd.induct) -apply (case_tac "n") -apply (simp_all add: dvd_mod) -done - - -theorem gcd_greatest_iff [iff]: - "(k dvd gcd m n) = (k dvd m \ k dvd n)" -by (blast intro!: gcd_greatest intro: dvd_trans) - - -(**** The material below was omitted from the book ****) - -definition is_gcd :: "[nat,nat,nat] \ bool" where (*gcd as a relation*) - "is_gcd p m n == p dvd m \ p dvd n \ - (ALL d. d dvd m \ d dvd n \ d dvd p)" - -(*Function gcd yields the Greatest Common Divisor*) -lemma is_gcd: "is_gcd (gcd m n) m n" -apply (simp add: is_gcd_def gcd_greatest); -done - -(*uniqueness of GCDs*) -lemma is_gcd_unique: "\ is_gcd m a b; is_gcd n a b \ \ m=n" -apply (simp add: is_gcd_def); -apply (blast intro: dvd_antisym) -done - - -text {* -@{thm[display] dvd_antisym} -\rulename{dvd_antisym} - -\begin{isabelle} -proof\ (prove):\ step\ 1\isanewline -\isanewline -goal\ (lemma\ is_gcd_unique):\isanewline -\isasymlbrakk is_gcd\ m\ a\ b;\ is_gcd\ n\ a\ b\isasymrbrakk \ \isasymLongrightarrow \ m\ =\ n\isanewline -\ 1.\ \isasymlbrakk m\ dvd\ a\ \isasymand \ m\ dvd\ b\ \isasymand \ (\isasymforall d.\ d\ dvd\ a\ \isasymand \ d\ dvd\ b\ \isasymlongrightarrow \ d\ dvd\ m);\isanewline -\ \ \ \ \ \ \ n\ dvd\ a\ \isasymand \ n\ dvd\ b\ \isasymand \ (\isasymforall d.\ d\ dvd\ a\ \isasymand \ d\ dvd\ b\ \isasymlongrightarrow \ d\ dvd\ n)\isasymrbrakk \isanewline -\ \ \ \ \isasymLongrightarrow \ m\ =\ n -\end{isabelle} -*}; - -lemma gcd_assoc: "gcd (gcd k m) n = gcd k (gcd m n)" - apply (rule is_gcd_unique) - apply (rule is_gcd) - apply (simp add: is_gcd_def); - apply (blast intro: dvd_trans); - done - -text{* -\begin{isabelle} -proof\ (prove):\ step\ 3\isanewline -\isanewline -goal\ (lemma\ gcd_assoc):\isanewline -gcd\ (gcd\ (k,\ m),\ n)\ =\ gcd\ (k,\ gcd\ (m,\ n))\isanewline -\ 1.\ gcd\ (k,\ gcd\ (m,\ n))\ dvd\ k\ \isasymand \isanewline -\ \ \ \ gcd\ (k,\ gcd\ (m,\ n))\ dvd\ m\ \isasymand \ gcd\ (k,\ gcd\ (m,\ n))\ dvd\ n -\end{isabelle} -*} - - -lemma gcd_dvd_gcd_mult: "gcd m n dvd gcd (k*m) n" - apply (auto intro: dvd_trans [of _ m]) - done - -(*This is half of the proof (by dvd_antisym) of*) -lemma gcd_mult_cancel: "gcd k n = 1 \ gcd (k*m) n = gcd m n" - oops - -end diff -r 39bcdf19dd14 -r 608c157d743d src/Doc/Tutorial/Rules/TPrimes.thy --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/Doc/Tutorial/Rules/TPrimes.thy Wed Jan 29 12:51:37 2014 +0000 @@ -0,0 +1,155 @@ +(* EXTRACT from HOL/ex/Primes.thy*) + +(*Euclid's algorithm + This material now appears AFTER that of Forward.thy *) +theory TPrimes imports Main begin + +fun gcd :: "nat \ nat \ nat" where + "gcd m n = (if n=0 then m else gcd n (m mod n))" + + +text {*Now in Basic.thy! +@{thm[display]"dvd_def"} +\rulename{dvd_def} +*}; + + +(*** Euclid's Algorithm ***) + +lemma gcd_0 [simp]: "gcd m 0 = m" +apply (simp); +done + +lemma gcd_non_0 [simp]: "0 gcd m n = gcd n (m mod n)" +apply (simp) +done; + +declare gcd.simps [simp del]; + +(*gcd(m,n) divides m and n. The conjunctions don't seem provable separately*) +lemma gcd_dvd_both: "(gcd m n dvd m) \ (gcd m n dvd n)" +apply (induct_tac m n rule: gcd.induct) + --{* @{subgoals[display,indent=0,margin=65]} *} +apply (case_tac "n=0") +txt{*subgoals after the case tac +@{subgoals[display,indent=0,margin=65]} +*}; +apply (simp_all) + --{* @{subgoals[display,indent=0,margin=65]} *} +by (blast dest: dvd_mod_imp_dvd) + + + +text {* +@{thm[display] dvd_mod_imp_dvd} +\rulename{dvd_mod_imp_dvd} + +@{thm[display] dvd_trans} +\rulename{dvd_trans} +*} + +lemmas gcd_dvd1 [iff] = gcd_dvd_both [THEN conjunct1] +lemmas gcd_dvd2 [iff] = gcd_dvd_both [THEN conjunct2]; + + +text {* +\begin{quote} +@{thm[display] gcd_dvd1} +\rulename{gcd_dvd1} + +@{thm[display] gcd_dvd2} +\rulename{gcd_dvd2} +\end{quote} +*}; + +(*Maximality: for all m,n,k naturals, + if k divides m and k divides n then k divides gcd(m,n)*) +lemma gcd_greatest [rule_format]: + "k dvd m \ k dvd n \ k dvd gcd m n" +apply (induct_tac m n rule: gcd.induct) +apply (case_tac "n=0") +txt{*subgoals after the case tac +@{subgoals[display,indent=0,margin=65]} +*}; +apply (simp_all add: dvd_mod) +done + +text {* +@{thm[display] dvd_mod} +\rulename{dvd_mod} +*} + +(*just checking the claim that case_tac "n" works too*) +lemma "k dvd m \ k dvd n \ k dvd gcd m n" +apply (induct_tac m n rule: gcd.induct) +apply (case_tac "n") +apply (simp_all add: dvd_mod) +done + + +theorem gcd_greatest_iff [iff]: + "(k dvd gcd m n) = (k dvd m \ k dvd n)" +by (blast intro!: gcd_greatest intro: dvd_trans) + + +(**** The material below was omitted from the book ****) + +definition is_gcd :: "[nat,nat,nat] \ bool" where (*gcd as a relation*) + "is_gcd p m n == p dvd m \ p dvd n \ + (ALL d. d dvd m \ d dvd n \ d dvd p)" + +(*Function gcd yields the Greatest Common Divisor*) +lemma is_gcd: "is_gcd (gcd m n) m n" +apply (simp add: is_gcd_def gcd_greatest); +done + +(*uniqueness of GCDs*) +lemma is_gcd_unique: "\ is_gcd m a b; is_gcd n a b \ \ m=n" +apply (simp add: is_gcd_def); +apply (blast intro: dvd_antisym) +done + + +text {* +@{thm[display] dvd_antisym} +\rulename{dvd_antisym} + +\begin{isabelle} +proof\ (prove):\ step\ 1\isanewline +\isanewline +goal\ (lemma\ is_gcd_unique):\isanewline +\isasymlbrakk is_gcd\ m\ a\ b;\ is_gcd\ n\ a\ b\isasymrbrakk \ \isasymLongrightarrow \ m\ =\ n\isanewline +\ 1.\ \isasymlbrakk m\ dvd\ a\ \isasymand \ m\ dvd\ b\ \isasymand \ (\isasymforall d.\ d\ dvd\ a\ \isasymand \ d\ dvd\ b\ \isasymlongrightarrow \ d\ dvd\ m);\isanewline +\ \ \ \ \ \ \ n\ dvd\ a\ \isasymand \ n\ dvd\ b\ \isasymand \ (\isasymforall d.\ d\ dvd\ a\ \isasymand \ d\ dvd\ b\ \isasymlongrightarrow \ d\ dvd\ n)\isasymrbrakk \isanewline +\ \ \ \ \isasymLongrightarrow \ m\ =\ n +\end{isabelle} +*}; + +lemma gcd_assoc: "gcd (gcd k m) n = gcd k (gcd m n)" + apply (rule is_gcd_unique) + apply (rule is_gcd) + apply (simp add: is_gcd_def); + apply (blast intro: dvd_trans); + done + +text{* +\begin{isabelle} +proof\ (prove):\ step\ 3\isanewline +\isanewline +goal\ (lemma\ gcd_assoc):\isanewline +gcd\ (gcd\ (k,\ m),\ n)\ =\ gcd\ (k,\ gcd\ (m,\ n))\isanewline +\ 1.\ gcd\ (k,\ gcd\ (m,\ n))\ dvd\ k\ \isasymand \isanewline +\ \ \ \ gcd\ (k,\ gcd\ (m,\ n))\ dvd\ m\ \isasymand \ gcd\ (k,\ gcd\ (m,\ n))\ dvd\ n +\end{isabelle} +*} + + +lemma gcd_dvd_gcd_mult: "gcd m n dvd gcd (k*m) n" + apply (auto intro: dvd_trans [of _ m]) + done + +(*This is half of the proof (by dvd_antisym) of*) +lemma gcd_mult_cancel: "gcd k n = 1 \ gcd (k*m) n = gcd m n" + oops + +end diff -r 39bcdf19dd14 -r 608c157d743d src/Doc/Tutorial/Sets/Examples.thy --- a/src/Doc/Tutorial/Sets/Examples.thy Mon Jan 27 17:13:33 2014 +0000 +++ b/src/Doc/Tutorial/Sets/Examples.thy Wed Jan 29 12:51:37 2014 +0000 @@ -1,4 +1,4 @@ -theory Examples imports Main "~~/src/HOL/Library/Binomial" begin +theory Examples imports "~~/src/HOL/Number_Theory/Binomial" begin declare [[eta_contract = false]] @@ -44,7 +44,7 @@ \rulename{Diff_disjoint} *} - + lemma "A \ -A = UNIV" by blast @@ -142,7 +142,7 @@ lemma "{x. x \ A} = A" by blast - + text{* @{thm[display] Collect_mem_eq[no_vars]} \rulename{Collect_mem_eq} @@ -157,7 +157,7 @@ definition prime :: "nat set" where "prime == {p. 1

m=1 | m=p)}" -lemma "{p*q | p q. p\prime \ q\prime} = +lemma "{p*q | p q. p\prime \ q\prime} = {z. \p q. z = p*q \ p\prime \ q\prime}" by (rule refl) diff -r 39bcdf19dd14 -r 608c157d743d src/HOL/Library/Binomial.thy --- a/src/HOL/Library/Binomial.thy Mon Jan 27 17:13:33 2014 +0000 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,606 +0,0 @@ -(* Title: HOL/Library/Binomial.thy - Author: Lawrence C Paulson, Amine Chaieb - Copyright 1997 University of Cambridge -*) - -header {* Binomial Coefficients *} - -theory Binomial -imports Complex_Main -begin - -text {* This development is based on the work of Andy Gordon and - Florian Kammueller. *} - -subsection {* Basic definitions and lemmas *} - -primrec binomial :: "nat \ nat \ nat" (infixl "choose" 65) -where - "0 choose k = (if k = 0 then 1 else 0)" -| "Suc n choose k = (if k = 0 then 1 else (n choose (k - 1)) + (n choose k))" - -lemma binomial_n_0 [simp]: "(n choose 0) = 1" - by (cases n) simp_all - -lemma binomial_0_Suc [simp]: "(0 choose Suc k) = 0" - by simp - -lemma binomial_Suc_Suc [simp]: "(Suc n choose Suc k) = (n choose k) + (n choose Suc k)" - by simp - -lemma choose_reduce_nat: - "0 < (n::nat) \ 0 < k \ - (n choose k) = ((n - 1) choose k) + ((n - 1) choose (k - 1))" - by (metis Suc_diff_1 binomial.simps(2) nat_add_commute neq0_conv) - -lemma binomial_eq_0: "n < k \ n choose k = 0" - by (induct n arbitrary: k) auto - -declare binomial.simps [simp del] - -lemma binomial_n_n [simp]: "n choose n = 1" - by (induct n) (simp_all add: binomial_eq_0) - -lemma binomial_Suc_n [simp]: "Suc n choose n = Suc n" - by (induct n) simp_all - -lemma binomial_1 [simp]: "n choose Suc 0 = n" - by (induct n) simp_all - -lemma zero_less_binomial: "k \ n \ n choose k > 0" - by (induct n k rule: diff_induct) simp_all - -lemma binomial_eq_0_iff [simp]: "n choose k = 0 \ n < k" - by (metis binomial_eq_0 less_numeral_extra(3) not_less zero_less_binomial) - -lemma zero_less_binomial_iff [simp]: "n choose k > 0 \ k \ n" - by (metis binomial_eq_0_iff not_less0 not_less zero_less_binomial) - -(*Might be more useful if re-oriented*) -lemma Suc_times_binomial_eq: - "k \ n \ Suc n * (n choose k) = (Suc n choose Suc k) * Suc k" - apply (induct n arbitrary: k) - apply (simp add: binomial.simps) - apply (case_tac k) - apply (auto simp add: add_mult_distrib add_mult_distrib2 le_Suc_eq binomial_eq_0) - done - -text{*This is the well-known version, but it's harder to use because of the - need to reason about division.*} -lemma binomial_Suc_Suc_eq_times: - "k \ n \ (Suc n choose Suc k) = (Suc n * (n choose k)) div Suc k" - by (simp add: Suc_times_binomial_eq del: mult_Suc mult_Suc_right) - -text{*Another version, with -1 instead of Suc.*} -lemma times_binomial_minus1_eq: - "k \ n \ 0 < k \ (n choose k) * k = n * ((n - 1) choose (k - 1))" - using Suc_times_binomial_eq [where n = "n - 1" and k = "k - 1"] - by (auto split add: nat_diff_split) - - -subsection {* Combinatorial theorems involving @{text "choose"} *} - -text {*By Florian Kamm\"uller, tidied by LCP.*} - -lemma card_s_0_eq_empty: "finite A \ card {B. B \ A & card B = 0} = 1" - by (simp cong add: conj_cong add: finite_subset [THEN card_0_eq]) - -lemma choose_deconstruct: "finite M \ x \ M \ - {s. s \ insert x M \ card s = Suc k} = - {s. s \ M \ card s = Suc k} \ {s. \t. t \ M \ card t = k \ s = insert x t}" - apply safe - apply (auto intro: finite_subset [THEN card_insert_disjoint]) - by (metis (full_types) Diff_insert_absorb Set.set_insert Zero_neq_Suc card_Diff_singleton_if - card_eq_0_iff diff_Suc_1 in_mono subset_insert_iff) - -lemma finite_bex_subset [simp]: - assumes "finite B" - and "\A. A \ B \ finite {x. P x A}" - shows "finite {x. \A \ B. P x A}" - by (metis (no_types) assms finite_Collect_bounded_ex finite_Collect_subsets) - -text{*There are as many subsets of @{term A} having cardinality @{term k} - as there are sets obtained from the former by inserting a fixed element - @{term x} into each.*} -lemma constr_bij: - "finite A \ x \ A \ - card {B. \C. C \ A \ card C = k \ B = insert x C} = - card {B. B \ A & card(B) = k}" - apply (rule card_bij_eq [where f = "\s. s - {x}" and g = "insert x"]) - apply (auto elim!: equalityE simp add: inj_on_def) - apply (metis card_Diff_singleton_if finite_subset in_mono) - done - -text {* - Main theorem: combinatorial statement about number of subsets of a set. -*} - -theorem n_subsets: "finite A \ card {B. B \ A \ card B = k} = (card A choose k)" -proof (induct k arbitrary: A) - case 0 then show ?case by (simp add: card_s_0_eq_empty) -next - case (Suc k) - show ?case using `finite A` - proof (induct A) - case empty show ?case by (simp add: card_s_0_eq_empty) - next - case (insert x A) - then show ?case using Suc.hyps - apply (simp add: card_s_0_eq_empty choose_deconstruct) - apply (subst card_Un_disjoint) - prefer 4 apply (force simp add: constr_bij) - prefer 3 apply force - prefer 2 apply (blast intro: finite_Pow_iff [THEN iffD2] - finite_subset [of _ "Pow (insert x F)" for F]) - apply (blast intro: finite_Pow_iff [THEN iffD2, THEN [2] finite_subset]) - done - qed -qed - - -subsection {* The binomial theorem (courtesy of Tobias Nipkow): *} - -text{* Avigad's version, generalized to any commutative ring *} -theorem binomial_ring: "(a+b::'a::{comm_ring_1,power})^n = - (\k=0..n. (of_nat (n choose k)) * a^k * b^(n-k))" (is "?P n") -proof (induct n) - case 0 then show "?P 0" by simp -next - case (Suc n) - have decomp: "{0..n+1} = {0} Un {n+1} Un {1..n}" - by auto - have decomp2: "{0..n} = {0} Un {1..n}" - by auto - have "(a+b)^(n+1) = - (a+b) * (\k=0..n. of_nat (n choose k) * a^k * b^(n-k))" - using Suc.hyps by simp - also have "\ = a*(\k=0..n. of_nat (n choose k) * a^k * b^(n-k)) + - b*(\k=0..n. of_nat (n choose k) * a^k * b^(n-k))" - by (rule distrib) - also have "\ = (\k=0..n. of_nat (n choose k) * a^(k+1) * b^(n-k)) + - (\k=0..n. of_nat (n choose k) * a^k * b^(n-k+1))" - by (auto simp add: setsum_right_distrib mult_ac) - also have "\ = (\k=0..n. of_nat (n choose k) * a^k * b^(n+1-k)) + - (\k=1..n+1. of_nat (n choose (k - 1)) * a^k * b^(n+1-k))" - by (simp add:setsum_shift_bounds_cl_Suc_ivl Suc_diff_le field_simps - del:setsum_cl_ivl_Suc) - also have "\ = a^(n+1) + b^(n+1) + - (\k=1..n. of_nat (n choose (k - 1)) * a^k * b^(n+1-k)) + - (\k=1..n. of_nat (n choose k) * a^k * b^(n+1-k))" - by (simp add: decomp2) - also have - "\ = a^(n+1) + b^(n+1) + - (\k=1..n. of_nat(n+1 choose k) * a^k * b^(n+1-k))" - by (auto simp add: field_simps setsum_addf [symmetric] choose_reduce_nat) - also have "\ = (\k=0..n+1. of_nat (n+1 choose k) * a^k * b^(n+1-k))" - using decomp by (simp add: field_simps) - finally show "?P (Suc n)" by simp -qed - -text{* Original version for the naturals *} -corollary binomial: "(a+b::nat)^n = (\k=0..n. (of_nat (n choose k)) * a^k * b^(n-k))" - using binomial_ring [of "int a" "int b" n] - by (simp only: of_nat_add [symmetric] of_nat_mult [symmetric] of_nat_power [symmetric] - of_nat_setsum [symmetric] - of_nat_eq_iff of_nat_id) - -subsection{* Pochhammer's symbol : generalized rising factorial *} - -text {* See @{url "http://en.wikipedia.org/wiki/Pochhammer_symbol"} *} - -definition "pochhammer (a::'a::comm_semiring_1) n = - (if n = 0 then 1 else setprod (\n. a + of_nat n) {0 .. n - 1})" - -lemma pochhammer_0 [simp]: "pochhammer a 0 = 1" - by (simp add: pochhammer_def) - -lemma pochhammer_1 [simp]: "pochhammer a 1 = a" - by (simp add: pochhammer_def) - -lemma pochhammer_Suc0 [simp]: "pochhammer a (Suc 0) = a" - by (simp add: pochhammer_def) - -lemma pochhammer_Suc_setprod: "pochhammer a (Suc n) = setprod (\n. a + of_nat n) {0 .. n}" - by (simp add: pochhammer_def) - -lemma setprod_nat_ivl_Suc: "setprod f {0 .. Suc n} = setprod f {0..n} * f (Suc n)" -proof - - have "{0..Suc n} = {0..n} \ {Suc n}" by auto - then show ?thesis by (simp add: field_simps) -qed - -lemma setprod_nat_ivl_1_Suc: "setprod f {0 .. Suc n} = f 0 * setprod f {1.. Suc n}" -proof - - have "{0..Suc n} = {0} \ {1 .. Suc n}" by auto - then show ?thesis by simp -qed - - -lemma pochhammer_Suc: "pochhammer a (Suc n) = pochhammer a n * (a + of_nat n)" -proof (cases n) - case 0 - then show ?thesis by simp -next - case (Suc n) - show ?thesis unfolding Suc pochhammer_Suc_setprod setprod_nat_ivl_Suc .. -qed - -lemma pochhammer_rec: "pochhammer a (Suc n) = a * pochhammer (a + 1) n" -proof (cases "n = 0") - case True - then show ?thesis by (simp add: pochhammer_Suc_setprod) -next - case False - have *: "finite {1 .. n}" "0 \ {1 .. n}" by auto - have eq: "insert 0 {1 .. n} = {0..n}" by auto - have **: "(\n\{1\nat..n}. a + of_nat n) = (\n\{0\nat..n - 1}. a + 1 + of_nat n)" - apply (rule setprod_reindex_cong [where f = Suc]) - using False - apply (auto simp add: fun_eq_iff field_simps) - done - show ?thesis - apply (simp add: pochhammer_def) - unfolding setprod_insert [OF *, unfolded eq] - using ** apply (simp add: field_simps) - done -qed - -lemma pochhammer_fact: "of_nat (fact n) = pochhammer 1 n" - unfolding fact_altdef_nat - apply (cases n) - apply (simp_all add: of_nat_setprod pochhammer_Suc_setprod) - apply (rule setprod_reindex_cong[where f=Suc]) - apply (auto simp add: fun_eq_iff) - done - -lemma pochhammer_of_nat_eq_0_lemma: - assumes "k > n" - shows "pochhammer (- (of_nat n :: 'a:: idom)) k = 0" -proof (cases "n = 0") - case True - then show ?thesis - using assms by (cases k) (simp_all add: pochhammer_rec) -next - case False - from assms obtain h where "k = Suc h" by (cases k) auto - then show ?thesis - by (simp add: pochhammer_Suc_setprod) - (metis Suc_leI Suc_le_mono assms atLeastAtMost_iff less_eq_nat.simps(1)) -qed - -lemma pochhammer_of_nat_eq_0_lemma': - assumes kn: "k \ n" - shows "pochhammer (- (of_nat n :: 'a:: {idom,ring_char_0})) k \ 0" -proof (cases k) - case 0 - then show ?thesis by simp -next - case (Suc h) - then show ?thesis - apply (simp add: pochhammer_Suc_setprod) - using Suc kn apply (auto simp add: algebra_simps) - done -qed - -lemma pochhammer_of_nat_eq_0_iff: - shows "pochhammer (- (of_nat n :: 'a:: {idom,ring_char_0})) k = 0 \ k > n" - (is "?l = ?r") - using pochhammer_of_nat_eq_0_lemma[of n k, where ?'a='a] - pochhammer_of_nat_eq_0_lemma'[of k n, where ?'a = 'a] - by (auto simp add: not_le[symmetric]) - - -lemma pochhammer_eq_0_iff: "pochhammer a n = (0::'a::field_char_0) \ (\k < n. a = - of_nat k)" - apply (auto simp add: pochhammer_of_nat_eq_0_iff) - apply (cases n) - apply (auto simp add: pochhammer_def algebra_simps group_add_class.eq_neg_iff_add_eq_0) - apply (metis leD not_less_eq) - done - - -lemma pochhammer_eq_0_mono: - "pochhammer a n = (0::'a::field_char_0) \ m \ n \ pochhammer a m = 0" - unfolding pochhammer_eq_0_iff by auto - -lemma pochhammer_neq_0_mono: - "pochhammer a m \ (0::'a::field_char_0) \ m \ n \ pochhammer a n \ 0" - unfolding pochhammer_eq_0_iff by auto - -lemma pochhammer_minus: - assumes kn: "k \ n" - shows "pochhammer (- b) k = ((- 1) ^ k :: 'a::comm_ring_1) * pochhammer (b - of_nat k + 1) k" -proof (cases k) - case 0 - then show ?thesis by simp -next - case (Suc h) - have eq: "((- 1) ^ Suc h :: 'a) = setprod (%i. - 1) {0 .. h}" - using setprod_constant[where A="{0 .. h}" and y="- 1 :: 'a"] - by auto - show ?thesis - unfolding Suc pochhammer_Suc_setprod eq setprod_timesf[symmetric] - apply (rule strong_setprod_reindex_cong[where f = "%i. h - i"]) - using Suc - apply (auto simp add: inj_on_def image_def of_nat_diff) - apply (metis atLeast0AtMost atMost_iff diff_diff_cancel diff_le_self) - done -qed - -lemma pochhammer_minus': - assumes kn: "k \ n" - shows "pochhammer (b - of_nat k + 1) k = ((- 1) ^ k :: 'a::comm_ring_1) * pochhammer (- b) k" - unfolding pochhammer_minus[OF kn, where b=b] - unfolding mult_assoc[symmetric] - unfolding power_add[symmetric] - by simp - -lemma pochhammer_same: "pochhammer (- of_nat n) n = - ((- 1) ^ n :: 'a::comm_ring_1) * of_nat (fact n)" - unfolding pochhammer_minus[OF le_refl[of n]] - by (simp add: of_nat_diff pochhammer_fact) - - -subsection{* Generalized binomial coefficients *} - -definition gbinomial :: "'a::field_char_0 \ nat \ 'a" (infixl "gchoose" 65) - where "a gchoose n = - (if n = 0 then 1 else (setprod (\i. a - of_nat i) {0 .. n - 1}) / of_nat (fact n))" - -lemma gbinomial_0 [simp]: "a gchoose 0 = 1" "0 gchoose (Suc n) = 0" - apply (simp_all add: gbinomial_def) - apply (subgoal_tac "(\i\nat\{0\nat..n}. - of_nat i) = (0::'b)") - apply (simp del:setprod_zero_iff) - apply simp - done - -lemma gbinomial_pochhammer: "a gchoose n = (- 1) ^ n * pochhammer (- a) n / of_nat (fact n)" -proof (cases "n = 0") - case True - then show ?thesis by simp -next - case False - from this setprod_constant[of "{0 .. n - 1}" "- (1:: 'a)"] - have eq: "(- (1\'a)) ^ n = setprod (\i. - 1) {0 .. n - 1}" - by auto - from False show ?thesis - by (simp add: pochhammer_def gbinomial_def field_simps - eq setprod_timesf[symmetric]) -qed - -lemma binomial_fact_lemma: "k \ n \ fact k * fact (n - k) * (n choose k) = fact n" -proof (induct n arbitrary: k rule: nat_less_induct) - fix n k assume H: "\mx\m. fact x * fact (m - x) * (m choose x) = - fact m" and kn: "k \ n" - let ?ths = "fact k * fact (n - k) * (n choose k) = fact n" - { assume "n=0" then have ?ths using kn by simp } - moreover - { assume "k=0" then have ?ths using kn by simp } - moreover - { assume nk: "n=k" then have ?ths by simp } - moreover - { fix m h assume n: "n = Suc m" and h: "k = Suc h" and hm: "h < m" - from n have mn: "m < n" by arith - from hm have hm': "h \ m" by arith - from hm h n kn have km: "k \ m" by arith - have "m - h = Suc (m - Suc h)" using h km hm by arith - with km h have th0: "fact (m - h) = (m - h) * fact (m - k)" - by simp - from n h th0 - have "fact k * fact (n - k) * (n choose k) = - k * (fact h * fact (m - h) * (m choose h)) + - (m - h) * (fact k * fact (m - k) * (m choose k))" - by (simp add: field_simps) - also have "\ = (k + (m - h)) * fact m" - using H[rule_format, OF mn hm'] H[rule_format, OF mn km] - by (simp add: field_simps) - finally have ?ths using h n km by simp } - moreover have "n=0 \ k = 0 \ k = n \ (\m h. n = Suc m \ k = Suc h \ h < m)" - using kn by presburger - ultimately show ?ths by blast -qed - -lemma binomial_fact: - assumes kn: "k \ n" - shows "(of_nat (n choose k) :: 'a::field_char_0) = - of_nat (fact n) / (of_nat (fact k) * of_nat (fact (n - k)))" - using binomial_fact_lemma[OF kn] - by (simp add: field_simps of_nat_mult [symmetric]) - -lemma binomial_gbinomial: "of_nat (n choose k) = of_nat n gchoose k" -proof - - { assume kn: "k > n" - then have ?thesis - by (subst binomial_eq_0[OF kn]) - (simp add: gbinomial_pochhammer field_simps pochhammer_of_nat_eq_0_iff) } - moreover - { assume "k=0" then have ?thesis by simp } - moreover - { assume kn: "k \ n" and k0: "k\ 0" - from k0 obtain h where h: "k = Suc h" by (cases k) auto - from h - have eq:"(- 1 :: 'a) ^ k = setprod (\i. - 1) {0..h}" - by (subst setprod_constant) auto - have eq': "(\i\{0..h}. of_nat n + - (of_nat i :: 'a)) = (\i\{n - h..n}. of_nat i)" - apply (rule strong_setprod_reindex_cong[where f="op - n"]) - using h kn - apply (simp_all add: inj_on_def image_iff Bex_def set_eq_iff) - apply clarsimp - apply presburger - apply presburger - apply (simp add: fun_eq_iff field_simps of_nat_add[symmetric] del: of_nat_add) - done - have th0: "finite {1..n - Suc h}" "finite {n - h .. n}" - "{1..n - Suc h} \ {n - h .. n} = {}" and - eq3: "{1..n - Suc h} \ {n - h .. n} = {1..n}" - using h kn by auto - from eq[symmetric] - have ?thesis using kn - apply (simp add: binomial_fact[OF kn, where ?'a = 'a] - gbinomial_pochhammer field_simps pochhammer_Suc_setprod) - apply (simp add: pochhammer_Suc_setprod fact_altdef_nat h - of_nat_setprod setprod_timesf[symmetric] eq' del: One_nat_def power_Suc) - unfolding setprod_Un_disjoint[OF th0, unfolded eq3, of "of_nat:: nat \ 'a"] eq[unfolded h] - unfolding mult_assoc[symmetric] - unfolding setprod_timesf[symmetric] - apply simp - apply (rule strong_setprod_reindex_cong[where f= "op - n"]) - apply (auto simp add: inj_on_def image_iff Bex_def) - apply presburger - apply (subgoal_tac "(of_nat (n - x) :: 'a) = of_nat n - of_nat x") - apply simp - apply (rule of_nat_diff) - apply simp - done - } - moreover - have "k > n \ k = 0 \ (k \ n \ k \ 0)" by arith - ultimately show ?thesis by blast -qed - -lemma gbinomial_1[simp]: "a gchoose 1 = a" - by (simp add: gbinomial_def) - -lemma gbinomial_Suc0[simp]: "a gchoose (Suc 0) = a" - by (simp add: gbinomial_def) - -lemma gbinomial_mult_1: - "a * (a gchoose n) = - of_nat n * (a gchoose n) + of_nat (Suc n) * (a gchoose (Suc n))" (is "?l = ?r") -proof - - have "?r = ((- 1) ^n * pochhammer (- a) n / of_nat (fact n)) * (of_nat n - (- a + of_nat n))" - unfolding gbinomial_pochhammer - pochhammer_Suc fact_Suc of_nat_mult right_diff_distrib power_Suc - by (simp add: field_simps del: of_nat_Suc) - also have "\ = ?l" unfolding gbinomial_pochhammer - by (simp add: field_simps) - finally show ?thesis .. -qed - -lemma gbinomial_mult_1': - "(a gchoose n) * a = of_nat n * (a gchoose n) + of_nat (Suc n) * (a gchoose (Suc n))" - by (simp add: mult_commute gbinomial_mult_1) - -lemma gbinomial_Suc: - "a gchoose (Suc k) = (setprod (\i. a - of_nat i) {0 .. k}) / of_nat (fact (Suc k))" - by (simp add: gbinomial_def) - -lemma gbinomial_mult_fact: - "(of_nat (fact (Suc k)) :: 'a) * ((a::'a::field_char_0) gchoose (Suc k)) = - (setprod (\i. a - of_nat i) {0 .. k})" - by (simp_all add: gbinomial_Suc field_simps del: fact_Suc) - -lemma gbinomial_mult_fact': - "((a::'a::field_char_0) gchoose (Suc k)) * (of_nat (fact (Suc k)) :: 'a) = - (setprod (\i. a - of_nat i) {0 .. k})" - using gbinomial_mult_fact[of k a] - by (subst mult_commute) - - -lemma gbinomial_Suc_Suc: - "((a::'a::field_char_0) + 1) gchoose (Suc k) = a gchoose k + (a gchoose (Suc k))" -proof (cases k) - case 0 - then show ?thesis by simp -next - case (Suc h) - have eq0: "(\i\{1..k}. (a + 1) - of_nat i) = (\i\{0..h}. a - of_nat i)" - apply (rule strong_setprod_reindex_cong[where f = Suc]) - using Suc - apply auto - done - - have "of_nat (fact (Suc k)) * (a gchoose k + (a gchoose (Suc k))) = - ((a gchoose Suc h) * of_nat (fact (Suc h)) * of_nat (Suc k)) + (\i\{0\nat..Suc h}. a - of_nat i)" - apply (simp add: Suc field_simps del: fact_Suc) - unfolding gbinomial_mult_fact' - apply (subst fact_Suc) - unfolding of_nat_mult - apply (subst mult_commute) - unfolding mult_assoc - unfolding gbinomial_mult_fact - apply (simp add: field_simps) - done - also have "\ = (\i\{0..h}. a - of_nat i) * (a + 1)" - unfolding gbinomial_mult_fact' setprod_nat_ivl_Suc - by (simp add: field_simps Suc) - also have "\ = (\i\{0..k}. (a + 1) - of_nat i)" - using eq0 - by (simp add: Suc setprod_nat_ivl_1_Suc) - also have "\ = of_nat (fact (Suc k)) * ((a + 1) gchoose (Suc k))" - unfolding gbinomial_mult_fact .. - finally show ?thesis by (simp del: fact_Suc) -qed - - -lemma binomial_symmetric: - assumes kn: "k \ n" - shows "n choose k = n choose (n - k)" -proof- - from kn have kn': "n - k \ n" by arith - from binomial_fact_lemma[OF kn] binomial_fact_lemma[OF kn'] - have "fact k * fact (n - k) * (n choose k) = - fact (n - k) * fact (n - (n - k)) * (n choose (n - k))" by simp - then show ?thesis using kn by simp -qed - -(* Contributed by Manuel Eberl *) -(* Alternative definition of the binomial coefficient as \i n" - shows "of_nat (n choose k) = (\ii = (\i n` unfolding fact_eq_rev_setprod_nat of_nat_setprod - by (auto simp add: setprod_dividef intro!: setprod_cong of_nat_diff[symmetric]) - finally show ?thesis . -next - case False - then show ?thesis by simp -qed - -lemma binomial_ge_n_over_k_pow_k: - fixes k n :: nat - and x :: "'a :: linordered_field_inverse_zero" - assumes "0 < k" - and "k \ n" - shows "(of_nat n / of_nat k :: 'a) ^ k \ of_nat (n choose k)" -proof - - have "(of_nat n / of_nat k :: 'a) ^ k = (\i \ of_nat (n choose k)" - unfolding binomial_altdef_of_nat[OF `k\n`] - proof (safe intro!: setprod_mono) - fix i :: nat - assume "i < k" - from assms have "n * i \ i * k" by simp - then have "n * k - n * i \ n * k - i * k" by arith - then have "n * (k - i) \ (n - i) * k" - by (simp add: diff_mult_distrib2 nat_mult_commute) - then have "of_nat n * of_nat (k - i) \ of_nat (n - i) * (of_nat k :: 'a)" - unfolding of_nat_mult[symmetric] of_nat_le_iff . - with assms show "of_nat n / of_nat k \ of_nat (n - i) / (of_nat (k - i) :: 'a)" - using `i < k` by (simp add: field_simps) - qed (simp add: zero_le_divide_iff) - finally show ?thesis . -qed - -lemma binomial_le_pow: - assumes "r \ n" - shows "n choose r \ n ^ r" -proof - - have "n choose r \ fact n div fact (n - r)" - using `r \ n` by (subst binomial_fact_lemma[symmetric]) auto - with fact_div_fact_le_pow [OF assms] show ?thesis by auto -qed - -lemma binomial_altdef_nat: "(k::nat) \ n \ - n choose k = fact n div (fact k * fact (n - k))" - by (subst binomial_fact_lemma [symmetric]) auto - -end diff -r 39bcdf19dd14 -r 608c157d743d src/HOL/Library/Formal_Power_Series.thy --- a/src/HOL/Library/Formal_Power_Series.thy Mon Jan 27 17:13:33 2014 +0000 +++ b/src/HOL/Library/Formal_Power_Series.thy Wed Jan 29 12:51:37 2014 +0000 @@ -5,7 +5,7 @@ header{* A formalization of formal power series *} theory Formal_Power_Series -imports Binomial +imports "~~/src/HOL/Number_Theory/Binomial" begin diff -r 39bcdf19dd14 -r 608c157d743d src/HOL/Library/Library.thy --- a/src/HOL/Library/Library.thy Mon Jan 27 17:13:33 2014 +0000 +++ b/src/HOL/Library/Library.thy Wed Jan 29 12:51:37 2014 +0000 @@ -3,7 +3,6 @@ imports AList BigO - Binomial Bit BNF_Decl Boolean_Algebra diff -r 39bcdf19dd14 -r 608c157d743d src/HOL/ROOT --- a/src/HOL/ROOT Mon Jan 27 17:13:33 2014 +0000 +++ b/src/HOL/ROOT Wed Jan 29 12:51:37 2014 +0000 @@ -258,8 +258,8 @@ theories [document = false] (* Preliminaries from set and number theory *) "~~/src/HOL/Library/FuncSet" - "~~/src/HOL/Old_Number_Theory/Primes" - "~~/src/HOL/Library/Binomial" + "~~/src/HOL/Number_Theory/Primes" + "~~/src/HOL/Number_Theory/Binomial" "~~/src/HOL/Library/Permutation" theories (*** New development, based on explicit structures ***)