Re: [isabelle] finite order allows maxima




On 03/10/2020 10:37, Stepan Holub wrote:
Dear all,

in need (of a particular instance of) the following obvious fact:

lemma (in order) fin_max: "finite A ⟹ a ∈ A ⟹ ∃ s ∈ A. a ≤ s ∧  (∀ b ∈ A. s ≤ b ⟶ s = b)"

This is a useful theorem and I am happy to add it. However, it seems more modular to prove the simpler

lemma (in order) finite_has_max: "finite A ⟹ A ≠ {} ⟹ ∃ s ∈ A. (∀ b ∈ A. s ≤ b ⟶ s = b)"

first and derive

lemma (in order) finite_has_max2: "finite A ⟹ a ∈ A ⟹ ∃ s ∈ A. a ≤ s ∧ (∀ b ∈ A. s ≤ b ⟶ s = b)"

from it in one line. In fact, I wonder if the second one is basic enough to go into Main, but probably yes.

I would add the lemma(s) to Finite_Set using Stepan's proof because that way they are added at the earliest point.

Comments, please.

Tobias

I have proved it for myself (see below). I wonder whether it can be obtained somehow more directly from Main.

Best regards

Stepan

proof (induct "card {b ∈ A. a < b}" arbitrary: a rule: nat_less_induct)
   case 1
  have IH: "⋀ x. x ∈ A ⟹ card {b ∈ A. x < b} < card {b ∈ A. a < b} ⟹ ∃s∈A. x ≤ s ∧ (∀b∈A. s ≤ b ⟶ s = b)"
     by (simp add: "1.hyps" ‹finite A›)
   then show ?case
   proof (cases "∀b∈A. a ≤ b ⟶ a = b")
     assume "∀b∈A. a ≤ b ⟶ a = b"
     thus ?thesis
       using ‹a ∈ A› by blast
   next
     assume "¬ (∀b∈A. a ≤ b ⟶ a = b)"
     then obtain a' where "a' ∈ A" and "a < a'"
       using local.antisym_conv1 by blast
     have "{b ∈ A. a' < b} ⊂ {b ∈ A. a < b}" (is "?Ma' ⊂  ?Ma")
     proof-
       have "a' ∈ ?Ma" and "a' ∉ ?Ma'" and "⋀ c. c ∈ ?Ma' ⟹ c ∈ ?Ma"
         using ‹a < a'› ‹a' ∈ A› by auto
       thus "?Ma' ⊂  ?Ma"
         by blast
     qed
     hence card: "card {b ∈ A. a' < b} < card {b ∈ A. a < b}"
       by (simp add: ‹finite A› psubset_card_mono)
     then obtain s where "s ∈ A" "a' ≤ s ∧ (∀b∈A. s ≤ b ⟶ s = b)"
       using IH[OF ‹a' ∈ A› card] by blast
     hence "a ≤ s ∧ (∀b∈A. s ≤ b ⟶ s = b)"
       using ‹a < a'› by auto
     thus ?thesis
       using ‹s ∈ A› by blast
   qed
qed



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