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83 lines
2.4 KiB
Plaintext
83 lines
2.4 KiB
Plaintext
theory HSV_chapter4 imports Complex_Main begin
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(* Triangle numbers *)
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fun triangle :: "nat \<Rightarrow> nat" where
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"triangle n = (if n = 0 then 0 else n + triangle (n-1))"
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value "triangle 1"
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value "triangle 2"
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value "triangle 3"
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theorem triangle_closed_form: "triangle n = (n+1) * n div 2"
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apply (induct n)
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apply simp+
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done
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(* Tetrahedral numbers *)
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fun tet :: "nat \<Rightarrow> nat" where
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"tet n = (if n = 0 then 0 else triangle n + tet (n-1))"
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value "tet 1" (* 1 *)
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value "tet 2" (* 4 *)
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value "tet 3" (* 10 *)
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value "tet 4" (* 20 *)
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value "tet 5" (* 35 *)
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value "tet 6" (* 56 *)
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find_theorems "_ div ?x + _ div ?x"
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thm div_add
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(* Proving that closed form is equivalent to recursive definition *)
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theorem "tet n = ((n + 2) * (n + 1) * n) div 6"
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proof (induct n)
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case 0
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show ?case by simp
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next
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case (Suc k)
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(* induction hypothesis *)
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assume IH: "tet k = (k + 2) * (k + 1) * k div 6"
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(* establish a useful fact and label it "*" *)
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have "2 dvd (k + 2) * (k + 1)" by simp
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hence *: "6 dvd (k + 2) * (k + 1) * 3" by presburger
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(* establish another useful fact and label it "**" *)
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have "2 dvd (k + 2) * (k + 1) * k" by simp
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moreover have "3 dvd (k + 2) * (k + 1) * k"
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proof -
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{
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assume "k mod 3 = 0"
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hence "3 dvd k" by presburger
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hence "3 dvd (k + 2) * (k + 1) * k" by fastforce
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} moreover {
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assume "k mod 3 = 1"
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hence "3 dvd (k + 2)" by presburger
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hence "3 dvd (k + 2) * (k + 1) * k" by fastforce
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} moreover {
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assume "k mod 3 = 2"
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hence "3 dvd (k + 1)" by presburger
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hence "3 dvd (k + 2) * (k + 1) * k" by fastforce
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} ultimately
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show "3 dvd (k + 2) * (k + 1) * k" by linarith
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qed
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ultimately have **: "6 dvd (k + 2) * (k + 1) * k" by presburger
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(* the actual proof *)
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have "tet (Suc k) = triangle (Suc k) + tet k"
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by simp
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also have "... = (k + 2) * (k + 1) div 2 + tet k"
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using triangle_closed_form by simp
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also have "... = (k + 2) * (k + 1) div 2 + (k + 2) * (k + 1) * k div 6"
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using IH by simp
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also have "... = ((k + 2) * (k + 1) * 3 + (k + 2) * (k + 1) * k) div 6"
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using div_add[OF * **] by simp
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also have "... = (k + 2) * (k + 1) * (k + 3) div 6"
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by (simp add: distrib_left)
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also have "... = (Suc k + 2) * (Suc k + 1) * Suc k div 6"
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by (metis One_nat_def Suc_1 add.commute add_Suc_shift mult.assoc
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mult.commute numeral_3_eq_3 plus_1_eq_Suc)
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finally show ?case by assumption
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qed
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end |