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

tactic: refine' OrderDual.ofDual_le_ofDual.mpr <|
  sup_support_multiset_prod_le (OrderDual.ofDual_le_ofDual.mp _)
    (fun a b => OrderDual.ofDual_le_ofDual.mp (_)) m
state_before: R : Type u_3
A : Type u_2
T : Type u_1
B : Type ?u.80864
ι : Type ?u.80867
inst✝¹¹ : SemilatticeSup B
inst✝¹⁰ : OrderBot B
inst✝⁹ : SemilatticeInf T
inst✝⁸ : OrderTop T
inst✝⁷ : CommSemiring R
inst✝⁶ : AddCommMonoid A
inst✝⁵ : AddCommMonoid 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 : AddCommMonoid 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
degt0 : 0 ≤ degt 0
degtm : ∀ (a b : A), degt a + degt b ≤ degt (a + b)
m : Multiset (AddMonoidAlgebra R A)
⊢ Multiset.sum (Multiset.map (fun f => Finset.inf f.support degt) m) ≤ Finset.inf (Multiset.prod m).support degt

tactic: exact degt0
state_before: case refine'_1
R : Type u_3
A : Type u_2
T : Type u_1
B : Type ?u.80864
ι : Type ?u.80867
inst✝¹¹ : SemilatticeSup B
inst✝¹⁰ : OrderBot B
inst✝⁹ : SemilatticeInf T
inst✝⁸ : OrderTop T
inst✝⁷ : CommSemiring R
inst✝⁶ : AddCommMonoid A
inst✝⁵ : AddCommMonoid 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 : AddCommMonoid 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
degt0 : 0 ≤ degt 0
degtm : ∀ (a b : A), degt a + degt b ≤ degt (a + b)
m : Multiset (AddMonoidAlgebra R A)
⊢ ↑OrderDual.ofDual (degt 0) ≤ ↑OrderDual.ofDual 0

case refine'_2
R : Type u_3
A : Type u_2
T : Type u_1
B : Type ?u.80864
ι : Type ?u.80867
inst✝¹¹ : SemilatticeSup B
inst✝¹⁰ : OrderBot B
inst✝⁹ : SemilatticeInf T
inst✝⁸ : OrderTop T
inst✝⁷ : CommSemiring R
inst✝⁶ : AddCommMonoid A
inst✝⁵ : AddCommMonoid 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 : AddCommMonoid 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
degt0 : 0 ≤ degt 0
degtm : ∀ (a b : A), degt a + degt b ≤ degt (a + b)
m : Multiset (AddMonoidAlgebra R A)
a b : A
⊢ ↑OrderDual.ofDual (degt (a + b)) ≤ ↑OrderDual.ofDual (degt a + degt b)

tactic: exact degtm _ _
state_before: case refine'_2
R : Type u_3
A : Type u_2
T : Type u_1
B : Type ?u.80864
ι : Type ?u.80867
inst✝¹¹ : SemilatticeSup B
inst✝¹⁰ : OrderBot B
inst✝⁹ : SemilatticeInf T
inst✝⁸ : OrderTop T
inst✝⁷ : CommSemiring R
inst✝⁶ : AddCommMonoid A
inst✝⁵ : AddCommMonoid 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 : AddCommMonoid 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
degt0 : 0 ≤ degt 0
degtm : ∀ (a b : A), degt a + degt b ≤ degt (a + b)
m : Multiset (AddMonoidAlgebra R A)
a b : A
⊢ ↑OrderDual.ofDual (degt (a + b)) ≤ ↑OrderDual.ofDual (degt a + degt b)