Theorem: AddMonoidAlgebra.sup_support_list_prod_le
File: Mathlib/Algebra/MonoidAlgebra/Degree.lean

Proof steps:
state_before: R : Type u_1
R' : Type u_2
A : Type u_3
T : Type u_4
B : Type u_5
ι : Type u_6
inst✝¹¹ : SemilatticeSup B
inst✝¹⁰ : OrderBot B
inst✝⁹ : SemilatticeInf T
inst✝⁸ : OrderTop T
inst✝⁷ : Semiring R
inst✝⁶ : AddMonoid A
inst✝⁵ : AddMonoid B
inst✝⁴ : CovariantClass B B (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝³ : CovariantClass B B (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝² : AddMonoid T
inst✝¹ : CovariantClass T T (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass T T (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
degb : A → B
degt : A → T
degb0 : degb 0 ≤ 0
degbm : ∀ (a b : A), degb (a + b) ≤ degb a + degb b
⊢ [].prod.support.sup degb ≤ (List.map (fun f => f.support.sup degb) []).sum
state_after: R : Type u_1
R' : Type u_2
A : Type u_3
T : Type u_4
B : Type u_5
ι : Type u_6
inst✝¹¹ : SemilatticeSup B
inst✝¹⁰ : OrderBot B
inst✝⁹ : SemilatticeInf T
inst✝⁸ : OrderTop T
inst✝⁷ : Semiring R
inst✝⁶ : AddMonoid A
inst✝⁵ : AddMonoid B
inst✝⁴ : CovariantClass B B (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝³ : CovariantClass B B (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝² : AddMonoid T
inst✝¹ : CovariantClass T T (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass T T (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
degb : A → B
degt : A → T
degb0 : degb 0 ≤ 0
degbm : ∀ (a b : A), degb (a + b) ≤ degb a + degb b
⊢ ∀ b ∈ Finsupp.support 1, degb b ≤ 0
tactic: rw [List.map_nil, Finset.sup_le_iff, List.prod_nil, List.sum_nil]
state_before: R : Type u_1
R' : Type u_2
A : Type u_3
T : Type u_4
B : Type u_5
ι : Type u_6
inst✝¹¹ : SemilatticeSup B
inst✝¹⁰ : OrderBot B
inst✝⁹ : SemilatticeInf T
inst✝⁸ : OrderTop T
inst✝⁷ : Semiring R
inst✝⁶ : AddMonoid A
inst✝⁵ : AddMonoid B
inst✝⁴ : CovariantClass B B (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝³ : CovariantClass B B (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝² : AddMonoid T
inst✝¹ : CovariantClass T T (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass T T (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
degb : A → B
degt : A → T
degb0 : degb 0 ≤ 0
degbm : ∀ (a b : A), degb (a + b) ≤ degb a + degb b
⊢ ∀ b ∈ Finsupp.support 1, degb b ≤ 0
state_after: no goals
tactic: exact fun a ha => by rwa [Finset.mem_singleton.mp (Finsupp.support_single_subset ha)]
state_before: R : Type u_1
R' : Type u_2
A : Type u_3
T : Type u_4
B : Type u_5
ι : Type u_6
inst✝¹¹ : SemilatticeSup B
inst✝¹⁰ : OrderBot B
inst✝⁹ : SemilatticeInf T
inst✝⁸ : OrderTop T
inst✝⁷ : Semiring R
inst✝⁶ : AddMonoid A
inst✝⁵ : AddMonoid B
inst✝⁴ : CovariantClass B B (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝³ : CovariantClass B B (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝² : AddMonoid T
inst✝¹ : CovariantClass T T (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass T T (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
degb : A → B
degt : A → T
degb0 : degb 0 ≤ 0
degbm : ∀ (a b : A), degb (a + b) ≤ degb a + degb b
f : R[A]
fs : List R[A]
⊢ (f :: fs).prod.support.sup degb ≤ (List.map (fun f => f.support.sup degb) (f :: fs)).sum
state_after: R : Type u_1
R' : Type u_2
A : Type u_3
T : Type u_4
B : Type u_5
ι : Type u_6
inst✝¹¹ : SemilatticeSup B
inst✝¹⁰ : OrderBot B
inst✝⁹ : SemilatticeInf T
inst✝⁸ : OrderTop T
inst✝⁷ : Semiring R
inst✝⁶ : AddMonoid A
inst✝⁵ : AddMonoid B
inst✝⁴ : CovariantClass B B (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝³ : CovariantClass B B (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝² : AddMonoid T
inst✝¹ : CovariantClass T T (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass T T (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
degb : A → B
degt : A → T
degb0 : degb 0 ≤ 0
degbm : ∀ (a b : A), degb (a + b) ≤ degb a + degb b
f : R[A]
fs : List R[A]
⊢ (f * fs.prod).support.sup degb ≤ f.support.sup degb + (List.map (fun f => f.support.sup degb) fs).sum
tactic: rw [List.prod_cons, List.map_cons, List.sum_cons]
state_before: R : Type u_1
R' : Type u_2
A : Type u_3
T : Type u_4
B : Type u_5
ι : Type u_6
inst✝¹¹ : SemilatticeSup B
inst✝¹⁰ : OrderBot B
inst✝⁹ : SemilatticeInf T
inst✝⁸ : OrderTop T
inst✝⁷ : Semiring R
inst✝⁶ : AddMonoid A
inst✝⁵ : AddMonoid B
inst✝⁴ : CovariantClass B B (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝³ : CovariantClass B B (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝² : AddMonoid T
inst✝¹ : CovariantClass T T (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass T T (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
degb : A → B
degt : A → T
degb0 : degb 0 ≤ 0
degbm : ∀ (a b : A), degb (a + b) ≤ degb a + degb b
f : R[A]
fs : List R[A]
⊢ (f * fs.prod).support.sup degb ≤ f.support.sup degb + (List.map (fun f => f.support.sup degb) fs).sum
state_after: no goals
tactic: exact (sup_support_mul_le (@fun a b => degbm a b) _ _).trans
        (add_le_add_left (sup_support_list_prod_le degb0 degbm fs) _)