--- 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"
--- 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:
"\<lbrakk> gcd k n = 1; k dvd m*n \<rbrakk> \<Longrightarrow> k dvd m"
apply (insert gcd_mult_distrib2 [of m k n])
txt{*@{subgoals[display,indent=0,margin=65]}*}
--- 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 \<Rightarrow> nat \<Rightarrow> 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<n \<Longrightarrow> 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) \<and> (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 \<longrightarrow> k dvd n \<longrightarrow> 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 \<longrightarrow> k dvd n \<longrightarrow> 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 \<and> 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] \<Rightarrow> bool" where (*gcd as a relation*)
- "is_gcd p m n == p dvd m \<and> p dvd n \<and>
- (ALL d. d dvd m \<and> d dvd n \<longrightarrow> 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: "\<lbrakk> is_gcd m a b; is_gcd n a b \<rbrakk> \<Longrightarrow> 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 \<Longrightarrow> gcd (k*m) n = gcd m n"
- oops
-
-end
--- /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 \<Rightarrow> nat \<Rightarrow> 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<n \<Longrightarrow> 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) \<and> (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 \<longrightarrow> k dvd n \<longrightarrow> 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 \<longrightarrow> k dvd n \<longrightarrow> 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 \<and> 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] \<Rightarrow> bool" where (*gcd as a relation*)
+ "is_gcd p m n == p dvd m \<and> p dvd n \<and>
+ (ALL d. d dvd m \<and> d dvd n \<longrightarrow> 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: "\<lbrakk> is_gcd m a b; is_gcd n a b \<rbrakk> \<Longrightarrow> 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 \<Longrightarrow> gcd (k*m) n = gcd m n"
+ oops
+
+end
--- 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 \<union> -A = UNIV"
by blast
@@ -142,7 +142,7 @@
lemma "{x. x \<in> 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<p & (ALL m. m dvd p --> m=1 | m=p)}"
-lemma "{p*q | p q. p\<in>prime \<and> q\<in>prime} =
+lemma "{p*q | p q. p\<in>prime \<and> q\<in>prime} =
{z. \<exists>p q. z = p*q \<and> p\<in>prime \<and> q\<in>prime}"
by (rule refl)
--- 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 \<Rightarrow> nat \<Rightarrow> 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) \<Longrightarrow> 0 < k \<Longrightarrow>
- (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 \<Longrightarrow> 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 \<le> n \<Longrightarrow> n choose k > 0"
- by (induct n k rule: diff_induct) simp_all
-
-lemma binomial_eq_0_iff [simp]: "n choose k = 0 \<longleftrightarrow> 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 \<longleftrightarrow> k \<le> 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 \<le> n \<Longrightarrow> 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 \<le> n \<Longrightarrow> (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 \<le> n \<Longrightarrow> 0 < k \<Longrightarrow> (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 \<Longrightarrow> card {B. B \<subseteq> A & card B = 0} = 1"
- by (simp cong add: conj_cong add: finite_subset [THEN card_0_eq])
-
-lemma choose_deconstruct: "finite M \<Longrightarrow> x \<notin> M \<Longrightarrow>
- {s. s \<subseteq> insert x M \<and> card s = Suc k} =
- {s. s \<subseteq> M \<and> card s = Suc k} \<union> {s. \<exists>t. t \<subseteq> M \<and> card t = k \<and> 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 "\<And>A. A \<subseteq> B \<Longrightarrow> finite {x. P x A}"
- shows "finite {x. \<exists>A \<subseteq> 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 \<Longrightarrow> x \<notin> A \<Longrightarrow>
- card {B. \<exists>C. C \<subseteq> A \<and> card C = k \<and> B = insert x C} =
- card {B. B \<subseteq> A & card(B) = k}"
- apply (rule card_bij_eq [where f = "\<lambda>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 \<Longrightarrow> card {B. B \<subseteq> A \<and> 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 =
- (\<Sum>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) * (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-k))"
- using Suc.hyps by simp
- also have "\<dots> = a*(\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-k)) +
- b*(\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-k))"
- by (rule distrib)
- also have "\<dots> = (\<Sum>k=0..n. of_nat (n choose k) * a^(k+1) * b^(n-k)) +
- (\<Sum>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 "\<dots> = (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n+1-k)) +
- (\<Sum>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 "\<dots> = a^(n+1) + b^(n+1) +
- (\<Sum>k=1..n. of_nat (n choose (k - 1)) * a^k * b^(n+1-k)) +
- (\<Sum>k=1..n. of_nat (n choose k) * a^k * b^(n+1-k))"
- by (simp add: decomp2)
- also have
- "\<dots> = a^(n+1) + b^(n+1) +
- (\<Sum>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 "\<dots> = (\<Sum>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 = (\<Sum>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 (\<lambda>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 (\<lambda>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} \<union> {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} \<union> {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 \<notin> {1 .. n}" by auto
- have eq: "insert 0 {1 .. n} = {0..n}" by auto
- have **: "(\<Prod>n\<in>{1\<Colon>nat..n}. a + of_nat n) = (\<Prod>n\<in>{0\<Colon>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 \<le> n"
- shows "pochhammer (- (of_nat n :: 'a:: {idom,ring_char_0})) k \<noteq> 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 \<longleftrightarrow> 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) \<longleftrightarrow> (\<exists>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) \<Longrightarrow> m \<ge> n \<Longrightarrow> pochhammer a m = 0"
- unfolding pochhammer_eq_0_iff by auto
-
-lemma pochhammer_neq_0_mono:
- "pochhammer a m \<noteq> (0::'a::field_char_0) \<Longrightarrow> m \<ge> n \<Longrightarrow> pochhammer a n \<noteq> 0"
- unfolding pochhammer_eq_0_iff by auto
-
-lemma pochhammer_minus:
- assumes kn: "k \<le> 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 \<le> 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 \<Rightarrow> nat \<Rightarrow> 'a" (infixl "gchoose" 65)
- where "a gchoose n =
- (if n = 0 then 1 else (setprod (\<lambda>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 "(\<Prod>i\<Colon>nat\<in>{0\<Colon>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\<Colon>'a)) ^ n = setprod (\<lambda>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 \<le> n \<Longrightarrow> fact k * fact (n - k) * (n choose k) = fact n"
-proof (induct n arbitrary: k rule: nat_less_induct)
- fix n k assume H: "\<forall>m<n. \<forall>x\<le>m. fact x * fact (m - x) * (m choose x) =
- fact m" and kn: "k \<le> 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 \<le> m" by arith
- from hm h n kn have km: "k \<le> 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 "\<dots> = (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 \<or> k = 0 \<or> k = n \<or> (\<exists>m h. n = Suc m \<and> k = Suc h \<and> h < m)"
- using kn by presburger
- ultimately show ?ths by blast
-qed
-
-lemma binomial_fact:
- assumes kn: "k \<le> 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 \<le> n" and k0: "k\<noteq> 0"
- from k0 obtain h where h: "k = Suc h" by (cases k) auto
- from h
- have eq:"(- 1 :: 'a) ^ k = setprod (\<lambda>i. - 1) {0..h}"
- by (subst setprod_constant) auto
- have eq': "(\<Prod>i\<in>{0..h}. of_nat n + - (of_nat i :: 'a)) = (\<Prod>i\<in>{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} \<inter> {n - h .. n} = {}" and
- eq3: "{1..n - Suc h} \<union> {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 \<Rightarrow> '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 \<or> k = 0 \<or> (k \<le> n \<and> k \<noteq> 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 "\<dots> = ?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 (\<lambda>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 (\<lambda>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 (\<lambda>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: "(\<Prod>i\<in>{1..k}. (a + 1) - of_nat i) = (\<Prod>i\<in>{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)) + (\<Prod>i\<in>{0\<Colon>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 "\<dots> = (\<Prod>i\<in>{0..h}. a - of_nat i) * (a + 1)"
- unfolding gbinomial_mult_fact' setprod_nat_ivl_Suc
- by (simp add: field_simps Suc)
- also have "\<dots> = (\<Prod>i\<in>{0..k}. (a + 1) - of_nat i)"
- using eq0
- by (simp add: Suc setprod_nat_ivl_1_Suc)
- also have "\<dots> = 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 \<le> n"
- shows "n choose k = n choose (n - k)"
-proof-
- from kn have kn': "n - k \<le> 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 \<Prod>i<k. (n - i) / (k - i) *)
-lemma binomial_altdef_of_nat:
- fixes n k :: nat
- and x :: "'a :: {field_char_0,field_inverse_zero}"
- assumes "k \<le> n"
- shows "of_nat (n choose k) = (\<Prod>i<k. of_nat (n - i) / of_nat (k - i) :: 'a)"
-proof (cases "0 < k")
- case True
- then have "(of_nat (n choose k) :: 'a) = (\<Prod>i<k. of_nat n - of_nat i) / of_nat (fact k)"
- unfolding binomial_gbinomial gbinomial_def
- by (auto simp: gr0_conv_Suc lessThan_Suc_atMost atLeast0AtMost)
- also have "\<dots> = (\<Prod>i<k. of_nat (n - i) / of_nat (k - i) :: 'a)"
- using `k \<le> 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 \<le> n"
- shows "(of_nat n / of_nat k :: 'a) ^ k \<le> of_nat (n choose k)"
-proof -
- have "(of_nat n / of_nat k :: 'a) ^ k = (\<Prod>i<k. of_nat n / of_nat k :: 'a)"
- by (simp add: setprod_constant)
- also have "\<dots> \<le> of_nat (n choose k)"
- unfolding binomial_altdef_of_nat[OF `k\<le>n`]
- proof (safe intro!: setprod_mono)
- fix i :: nat
- assume "i < k"
- from assms have "n * i \<ge> i * k" by simp
- then have "n * k - n * i \<le> n * k - i * k" by arith
- then have "n * (k - i) \<le> (n - i) * k"
- by (simp add: diff_mult_distrib2 nat_mult_commute)
- then have "of_nat n * of_nat (k - i) \<le> 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 \<le> 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 \<le> n"
- shows "n choose r \<le> n ^ r"
-proof -
- have "n choose r \<le> fact n div fact (n - r)"
- using `r \<le> 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) \<le> n \<Longrightarrow>
- n choose k = fact n div (fact k * fact (n - k))"
- by (subst binomial_fact_lemma [symmetric]) auto
-
-end
--- 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
--- 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
--- 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 ***)