Degree of polynomial smaller than a number

Question

I am working on a lemma that shows that the degree of a sum of monomials is always less or equal to n if the exponent of each monomial is less or equal to n.

lemma degree_poly_smaller:
  fixes a :: "('a::comm_ring_1 poly)" and n::nat
  shows "degree (∑x∷nat | x ≤ n . monom (coeff a x) x) ≤ n"
sorry

What I have to so far is the following (please mind that I am a beginner in Isabelle):

lemma degree_smaller:
  fixes a :: "('a::comm_ring_1 poly)" and n::nat
  shows "degree (∑x∷nat | x ≤ n . monom (coeff a x) x) ≤ n"
proof-

 have th01: "⋀k. k ≤ n ⟹ 
             degree (setsum (λ x . monom (coeff a x) k) {x∷nat. x ≤ n})  ≤ n" 
                    by (metis degree_monom_le monom_setsum order_trans)

 have min_lemma1: "k∈{x∷nat. x ≤ n} ⟹ k ≤ n" by simp

 from this th01 have th02: 
     "⋀k. k∈{x∷nat. x ≤ n} ⟹ 
     degree (setsum (λ x . monom (coeff a x) k) {x∷nat. x ≤ n})  ≤ n" 
                                             by (metis mem_Collect_eq)

 have min_lemma2: "(SOME y . y ≤ n) ≤ n" by (metis (full_types) le0 some_eq_ex)

 from this have th03: 
     "degree (∑x∷nat | x ≤ n . monom (coeff a x) (SOME y . y ≤ n)) ≤ n" 
                                                         by (metis th01)

 from min_lemma1 min_lemma2 have min_lemma3: 
     "(SOME y . y∈{x∷nat. x ≤ n}) ≤ n" by (metis (full_types) mem_Collect_eq some_eq_ex)

 from this th01 th02 th03 have th04: 
     "degree (∑x∷nat | x ≤ n . monom (coeff a x) (SOME y . y∈{x∷nat. x ≤ n}) ) ≤ n" 
                                                                      by presburger

Here is the problem, I don't understand why the following lemma is not sufficient to finish the proof. In particular, I would expect the last part (where the sorry is) to be simple enough for sledgehammer to find a proof:

 from this th01 th02 th03 th04  have th05: 
 "degree 
  (setsum (λ i . monom (coeff a i) (SOME y . y∈{x∷nat. x ≤ n})) {x∷nat. x ≤ n})
   ≤ n"
        by linarith

  (* how can I prove this last step ? *)
  from this have 
    "degree (setsum (λ i . monom (coeff a i) i) {x∷nat. x ≤ n}) ≤ n" sorry 

  from this show ?thesis by auto
qed

.
SOLUTION from Brian Huffman's excellent answer:
.

lemma degree_setsum_smaller:
  "finite A ⟹ ∀x∈A. degree (f x) ≤ n ⟹ degree (∑x∈A. f x) ≤ n"
apply(induct rule: finite_induct)
apply(auto)
by (metis degree_add_le)

lemma finiteSetSmallerThanNumber: 
   "finite {x∷nat. x ≤ n}" 
by (metis finite_Collect_le_nat)

lemma degree_smaller:
  fixes a :: "('a::comm_ring_1 poly)" and n::nat
  shows "degree (∑x∷nat | x ≤ n . monom (coeff a x) x) ≤ n"
apply (rule degree_setsum_smaller)
apply(simp add: finiteSetSmallerThanNumber)
by (metis degree_0 degree_monom_eq le0 mem_Collect_eq monom_eq_0_iff) (* from sledgehammer *)

Answer 1

The last step does not follow from your th05. The problem is that you seem to want to unify (SOME y. y∈{x∷nat. x ≤ n}) with the bound variable i. However, in HOL (SOME y. y∈{x∷nat. x ≤ n}) has a single value that depends only on n and not on i. Furthermore, you don't get to choose the value; a theorem using SOME is not the same as a theorem with a universally quantified variable.

My advice is to avoid using SOME altogether, and instead try to prove a generalization of your theorem first:

lemma degree_setsum_smaller:
  "finite A ⟹ ∀x∈A. degree (f x) ≤ n ⟹ degree (∑x∈A. f x) ≤ n"

You should be able to prove degree_setsum_smaller by induction over A (use induct rule: finite_induct), and then use it to prove degree_poly_smaller. Lemma degree_add_le from the polynomial library should be useful.

Answer 2

It is generally considered bad style to use auto as anything other than the last method in an apply script, because such proofs tend to be quite fragile. I would eliminate the "finiteSetSmallerThanNumber" lemma completely, it is a needlessly specific special case of Collect_le_nat. Also, camel case names for theorems are usually not used in Isabelle.

At any rate, this is my suggestion of how to make the proofs nicer:

lemma degree_setsum_smaller:
  "finite A ⟹ ∀x∈A. degree (f x) ≤ n ⟹ degree (∑x∈A. f x) ≤ n"
by (induct rule: finite_induct, simp_all add: degree_add_le)

lemma degree_smaller:
  fixes a :: "('a::comm_ring_1 poly)" and n::nat
  shows "degree (∑x∷nat | x ≤ n . monom (coeff a x) x) ≤ n"
proof (rule degree_setsum_smaller)
  show "finite {x. x ≤ n}" using finite_Collect_le_nat .
  {
    fix x assume "x ≤ n"
    hence "degree (monom (coeff a x) x) ≤ n"
        by (cases "coeff a x = 0", simp_all add: degree_monom_eq)
  }
  thus "∀x∈{x. x≤ n}. degree (monom (coeff a x) x) ≤ n" by simp
qed