src/HOL/Number_Theory/Binomial.thy

declare lex_prod_def [code del]

(*  Title:      Binomial.thy
    Authors:    Lawrence C. Paulson, Jeremy Avigad, Tobias Nipkow


Defines the "choose" function, and establishes basic properties.

The original theory "Binomial" was by Lawrence C. Paulson, based on
the work of Andy Gordon and Florian Kammueller. The approach here,
which derives the definition of binomial coefficients in terms of the
factorial function, is due to Jeremy Avigad. The binomial theorem was
formalized by Tobias Nipkow.

*)


header {* Binomial *}

theory Binomial
imports Cong Fact
begin


subsection {* Main definitions *}

class binomial =

fixes 
  binomial :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "choose" 65)

(* definitions for the natural numbers *)

instantiation nat :: binomial

begin 

fun
  binomial_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat"
where
  "binomial_nat n k =
   (if k = 0 then 1 else
    if n = 0 then 0 else
      (binomial (n - 1) k) + (binomial (n - 1) (k - 1)))"

instance proof qed

end

(* definitions for the integers *)

instantiation int :: binomial

begin 

definition
  binomial_int :: "int => int \<Rightarrow> int"
where
  "binomial_int n k = (if n \<ge> 0 \<and> k \<ge> 0 then int (binomial (nat n) (nat k))
      else 0)"
instance proof qed

end


subsection {* Set up Transfer *}

lemma transfer_nat_int_binomial:
  "(n::int) >= 0 \<Longrightarrow> k >= 0 \<Longrightarrow> binomial (nat n) (nat k) = 
      nat (binomial n k)"
  unfolding binomial_int_def 
  by auto

lemma transfer_nat_int_binomial_closure:
  "n >= (0::int) \<Longrightarrow> k >= 0 \<Longrightarrow> binomial n k >= 0"
  by (auto simp add: binomial_int_def)

declare transfer_morphism_nat_int[transfer add return: 
    transfer_nat_int_binomial transfer_nat_int_binomial_closure]

lemma transfer_int_nat_binomial:
  "binomial (int n) (int k) = int (binomial n k)"
  unfolding fact_int_def binomial_int_def by auto

lemma transfer_int_nat_binomial_closure:
  "is_nat n \<Longrightarrow> is_nat k \<Longrightarrow> binomial n k >= 0"
  by (auto simp add: binomial_int_def)

declare transfer_morphism_int_nat[transfer add return: 
    transfer_int_nat_binomial transfer_int_nat_binomial_closure]


subsection {* Binomial coefficients *}

lemma choose_zero_nat [simp]: "(n::nat) choose 0 = 1"
  by simp

lemma choose_zero_int [simp]: "n \<ge> 0 \<Longrightarrow> (n::int) choose 0 = 1"
  by (simp add: binomial_int_def)

lemma zero_choose_nat [rule_format,simp]: "ALL (k::nat) > n. n choose k = 0"
  by (induct n rule: induct'_nat, auto)

lemma zero_choose_int [rule_format,simp]: "(k::int) > n \<Longrightarrow> n choose k = 0"
  unfolding binomial_int_def apply (case_tac "n < 0")
  apply force
  apply (simp del: binomial_nat.simps)
done

lemma choose_reduce_nat: "(n::nat) > 0 \<Longrightarrow> 0 < k \<Longrightarrow>
    (n choose k) = ((n - 1) choose k) + ((n - 1) choose (k - 1))"
  by simp

lemma choose_reduce_int: "(n::int) > 0 \<Longrightarrow> 0 < k \<Longrightarrow>
    (n choose k) = ((n - 1) choose k) + ((n - 1) choose (k - 1))"
  unfolding binomial_int_def apply (subst choose_reduce_nat)
    apply (auto simp del: binomial_nat.simps 
      simp add: nat_diff_distrib)
done

lemma choose_plus_one_nat: "((n::nat) + 1) choose (k + 1) = 
    (n choose (k + 1)) + (n choose k)"
  by (simp add: choose_reduce_nat)

lemma choose_Suc_nat: "(Suc n) choose (Suc k) = 
    (n choose (Suc k)) + (n choose k)"
  by (simp add: choose_reduce_nat One_nat_def)

lemma choose_plus_one_int: "n \<ge> 0 \<Longrightarrow> k \<ge> 0 \<Longrightarrow> ((n::int) + 1) choose (k + 1) = 
    (n choose (k + 1)) + (n choose k)"
  by (simp add: binomial_int_def choose_plus_one_nat nat_add_distrib del: binomial_nat.simps)

declare binomial_nat.simps [simp del]

lemma choose_self_nat [simp]: "((n::nat) choose n) = 1"
  by (induct n rule: induct'_nat, auto simp add: choose_plus_one_nat)

lemma choose_self_int [simp]: "n \<ge> 0 \<Longrightarrow> ((n::int) choose n) = 1"
  by (auto simp add: binomial_int_def)

lemma choose_one_nat [simp]: "(n::nat) choose 1 = n"
  by (induct n rule: induct'_nat, auto simp add: choose_reduce_nat)

lemma choose_one_int [simp]: "n \<ge> 0 \<Longrightarrow> (n::int) choose 1 = n"
  by (auto simp add: binomial_int_def)

lemma plus_one_choose_self_nat [simp]: "(n::nat) + 1 choose n = n + 1"
  apply (induct n rule: induct'_nat, force)
  apply (case_tac "n = 0")
  apply auto
  apply (subst choose_reduce_nat)
  apply (auto simp add: One_nat_def)  
  (* natdiff_cancel_numerals introduces Suc *)
done

lemma Suc_choose_self_nat [simp]: "(Suc n) choose n = Suc n"
  using plus_one_choose_self_nat by (simp add: One_nat_def)

lemma plus_one_choose_self_int [rule_format, simp]: 
    "(n::int) \<ge> 0 \<longrightarrow> n + 1 choose n = n + 1"
   by (auto simp add: binomial_int_def nat_add_distrib)

(* bounded quantification doesn't work with the unicode characters? *)
lemma choose_pos_nat [rule_format]: "ALL k <= (n::nat). 
    ((n::nat) choose k) > 0"
  apply (induct n rule: induct'_nat) 
  apply force
  apply clarify
  apply (case_tac "k = 0")
  apply force
  apply (subst choose_reduce_nat)
  apply auto
done

lemma choose_pos_int: "n \<ge> 0 \<Longrightarrow> k >= 0 \<Longrightarrow> k \<le> n \<Longrightarrow>
    ((n::int) choose k) > 0"
  by (auto simp add: binomial_int_def choose_pos_nat)

lemma binomial_induct [rule_format]: "(ALL (n::nat). P n n) \<longrightarrow> 
    (ALL n. P (n + 1) 0) \<longrightarrow> (ALL n. (ALL k < n. P n k \<longrightarrow> P n (k + 1) \<longrightarrow>
    P (n + 1) (k + 1))) \<longrightarrow> (ALL k <= n. P n k)"
  apply (induct n rule: induct'_nat)
  apply auto
  apply (case_tac "k = 0")
  apply auto
  apply (case_tac "k = n + 1")
  apply auto
  apply (drule_tac x = n in spec) back back 
  apply (drule_tac x = "k - 1" in spec) back back back
  apply auto
done

lemma choose_altdef_aux_nat: "(k::nat) \<le> n \<Longrightarrow> 
    fact k * fact (n - k) * (n choose k) = fact n"
  apply (rule binomial_induct [of _ k n])
  apply auto
proof -
  fix k :: nat and n
  assume less: "k < n"
  assume ih1: "fact k * fact (n - k) * (n choose k) = fact n"
  hence one: "fact (k + 1) * fact (n - k) * (n choose k) = (k + 1) * fact n"
    by (subst fact_plus_one_nat, auto)
  assume ih2: "fact (k + 1) * fact (n - (k + 1)) * (n choose (k + 1)) = 
      fact n"
  with less have "fact (k + 1) * fact ((n - (k + 1)) + 1) * 
      (n choose (k + 1)) = (n - k) * fact n"
    by (subst (2) fact_plus_one_nat, auto)
  with less have two: "fact (k + 1) * fact (n - k) * (n choose (k + 1)) = 
      (n - k) * fact n" by simp
  have "fact (k + 1) * fact (n - k) * (n + 1 choose (k + 1)) =
      fact (k + 1) * fact (n - k) * (n choose (k + 1)) + 
      fact (k + 1) * fact (n - k) * (n choose k)" 
    by (subst choose_reduce_nat, auto simp add: field_simps)
  also note one
  also note two
  also with less have "(n - k) * fact n + (k + 1) * fact n= fact (n + 1)" 
    apply (subst fact_plus_one_nat)
    apply (subst left_distrib [symmetric])
    apply simp
    done
  finally show "fact (k + 1) * fact (n - k) * (n + 1 choose (k + 1)) = 
    fact (n + 1)" .
qed

lemma choose_altdef_nat: "(k::nat) \<le> n \<Longrightarrow> 
    n choose k = fact n div (fact k * fact (n - k))"
  apply (frule choose_altdef_aux_nat)
  apply (erule subst)
  apply (simp add: mult_ac)
done


lemma choose_altdef_int: 
  assumes "(0::int) <= k" and "k <= n"
  shows "n choose k = fact n div (fact k * fact (n - k))"
  
  apply (subst tsub_eq [symmetric], rule prems)
  apply (rule choose_altdef_nat [transferred])
  using prems apply auto
done

lemma choose_dvd_nat: "(k::nat) \<le> n \<Longrightarrow> fact k * fact (n - k) dvd fact n"
  unfolding dvd_def apply (frule choose_altdef_aux_nat)
  (* why don't blast and auto get this??? *)
  apply (rule exI)
  apply (erule sym)
done

lemma choose_dvd_int: 
  assumes "(0::int) <= k" and "k <= n"
  shows "fact k * fact (n - k) dvd fact n"
 
  apply (subst tsub_eq [symmetric], rule prems)
  apply (rule choose_dvd_nat [transferred])
  using prems apply auto
done

(* generalizes Tobias Nipkow's proof to any commutative semiring *)
theorem binomial: "(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 rule: induct'_nat)
  show "?P 0" by simp
next
  fix n
  assume ih: "?P 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 decomp3: "{1..n+1} = {n+1} 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 ih by (simp add: power_plus_one)
  also have "... =  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 "... = (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 (subst (1 2) power_plus_one, simp add: setsum_right_distrib mult_ac)
  also have "... = (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 
             power_Suc field_simps One_nat_def del:setsum_cl_ivl_Suc)
  also have "... = 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 decomp3)
  also have
      "... = 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 "... = (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 (n + 1)" by simp
qed

lemma set_explicit: "{S. S = T \<and> P S} = (if P T then {T} else {})"
  by auto

lemma card_subsets_nat [rule_format]:
  fixes S :: "'a set"
  assumes "finite S"
  shows "ALL k. card {T. T \<le> S \<and> card T = k} = card S choose k" 
      (is "?P S")
using `finite S`
proof (induct set: finite)
  show "?P {}" by (auto simp add: set_explicit)
  next fix x :: "'a" and F
  assume iassms: "finite F" "x ~: F"
  assume ih: "?P F"
  show "?P (insert x F)" (is "ALL k. ?Q k")
  proof
    fix k
    show "card {T. T \<subseteq> (insert x F) \<and> card T = k} = 
        card (insert x F) choose k" (is "?Q k")
    proof (induct k rule: induct'_nat)
      from iassms have "{T. T \<le> (insert x F) \<and> card T = 0} = {{}}"
        apply auto
        apply (subst (asm) card_0_eq)
        apply (auto elim: finite_subset)
        done
      thus "?Q 0" 
        by auto
      next fix k
      show "?Q (k + 1)"
      proof -
        from iassms have fin: "finite (insert x F)" by auto
        hence "{ T. T \<subseteq> insert x F \<and> card T = k + 1} =
          {T. T \<le> F & card T = k + 1} Un 
          {T. T \<le> insert x F & x : T & card T = k + 1}"
          by (auto intro!: subsetI)
        with iassms fin have "card ({T. T \<le> insert x F \<and> card T = k + 1}) = 
          card ({T. T \<subseteq> F \<and> card T = k + 1}) + 
          card ({T. T \<subseteq> insert x F \<and> x : T \<and> card T = k + 1})"
          apply (subst card_Un_disjoint [symmetric])
          apply auto
          (* note: nice! Didn't have to say anything here *)
          done
        also from ih have "card ({T. T \<subseteq> F \<and> card T = k + 1}) = 
          card F choose (k+1)" by auto
        also have "card ({T. T \<subseteq> insert x F \<and> x : T \<and> card T = k + 1}) =
          card ({T. T <= F & card T = k})"
        proof -
          let ?f = "%T. T Un {x}"
          from iassms have "inj_on ?f {T. T <= F & card T = k}"
            unfolding inj_on_def by (auto intro!: subsetI)
          hence "card ({T. T <= F & card T = k}) = 
            card(?f ` {T. T <= F & card T = k})"
            by (rule card_image [symmetric])
          also from iassms fin have "?f ` {T. T <= F & card T = k} = 
            {T. T \<subseteq> insert x F \<and> x : T \<and> card T = k + 1}"
            unfolding image_def 
            (* I can't figure out why this next line takes so long *)
            apply auto
            apply (frule (1) finite_subset, force)
            apply (rule_tac x = "xa - {x}" in exI)
            apply (subst card_Diff_singleton)
            apply (auto elim: finite_subset)
            done
          finally show ?thesis by (rule sym)
        qed
        also from ih have "card ({T. T <= F & card T = k}) = card F choose k"
          by auto
        finally have "card ({T. T \<le> insert x F \<and> card T = k + 1}) = 
          card F choose (k + 1) + (card F choose k)".
        with iassms choose_plus_one_nat show ?thesis
          by (auto simp del: card.insert)
      qed
    qed
  qed
qed

end