-- Mathlib/Algebra/Order/Star/Basic.lean
lemma StarModule.smul_lt_smul_of_pos {a b : A} {c : R} (hab : a < b) (hc : 0 < c) :
    c • a < c • b

Proof state:
case le
R : Type u
A : Type u_1
inst✝¹² : Semiring R
inst✝¹¹ : PartialOrder R
inst✝¹⁰ : StarRing R
inst✝⁹ : StarOrderedRing R
inst✝⁸ : NonUnitalRing A
inst✝⁷ : StarRing A
inst✝⁶ : PartialOrder A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Module R A
inst✝³ : StarModule R A
inst✝² : NoZeroSMulDivisors R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
a b : A
c : R
hab : 0 < b - a
hc : 0 < c
⊢ 0 ≤ c • (b - a)

case ne
R : Type u
A : Type u_1
inst✝¹² : Semiring R
inst✝¹¹ : PartialOrder R
inst✝¹⁰ : StarRing R
inst✝⁹ : StarOrderedRing R
inst✝⁸ : NonUnitalRing A
inst✝⁷ : StarRing A
inst✝⁶ : PartialOrder A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Module R A
inst✝³ : StarModule R A
inst✝² : NoZeroSMulDivisors R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
a b : A
c : R
hab : 0 < b - a
hc : 0 < c
⊢ 0 ≠ c • (b - a)

Tactic:
case le =>
  have hab := le_of_lt hab
  rw [StarOrderedRing.nonneg_iff] at hab ⊢
  refine AddSubmonoid.closure_induction ?mem ?zero ?add hab
  case mem =>
    intro x hx
    have hc := le_of_lt hc
    rw [StarOrderedRing.nonneg_iff] at hc
    refine AddSubmonoid.closure_induction ?memc ?zeroc ?addc hc
    case memc =>
      intro c' hc'
      obtain ⟨z, hz⟩ := hc'
      obtain ⟨y, hy⟩ := hx
      apply AddSubmonoid.subset_closure
      refine ⟨z • y, ?_⟩
      simp only [star_smul, smul_mul_smul_comm, hz, hy]
    case zeroc => simpa only [zero_smul] using zero_mem _
    case addc => exact fun c' d _ _ ↦ by simpa only [add_smul] using add_mem
  case zero => simpa only [smul_zero] using zero_mem _
  case add => exact fun x y _ _ ↦ by simpa only [smul_add] using add_mem

Result:
case ne
R : Type u
A : Type u_1
inst✝¹² : Semiring R
inst✝¹¹ : PartialOrder R
inst✝¹⁰ : StarRing R
inst✝⁹ : StarOrderedRing R
inst✝⁸ : NonUnitalRing A
inst✝⁷ : StarRing A
inst✝⁶ : PartialOrder A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Module R A
inst✝³ : StarModule R A
inst✝² : NoZeroSMulDivisors R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
a b : A
c : R
hab : 0 < b - a
hc : 0 < c
⊢ 0 ≠ c • (b - a)

Proof state:
R : Type u
A : Type u_1
inst✝¹² : Semiring R
inst✝¹¹ : PartialOrder R
inst✝¹⁰ : StarRing R
inst✝⁹ : StarOrderedRing R
inst✝⁸ : NonUnitalRing A
inst✝⁷ : StarRing A
inst✝⁶ : PartialOrder A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Module R A
inst✝³ : StarModule R A
inst✝² : NoZeroSMulDivisors R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
a b : A
c : R
hab✝ : 0 < b - a
hc : 0 < c
hab : b - a ∈ AddSubmonoid.closure (range fun s => star s * s)
x : A
hx : x ∈ range fun s => star s * s
⊢ c • x ∈ AddSubmonoid.closure (range fun s => star s * s)

Tactic:
have hc := le_of_lt hc

Result:
R : Type u
A : Type u_1
inst✝¹² : Semiring R
inst✝¹¹ : PartialOrder R
inst✝¹⁰ : StarRing R
inst✝⁹ : StarOrderedRing R
inst✝⁸ : NonUnitalRing A
inst✝⁷ : StarRing A
inst✝⁶ : PartialOrder A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Module R A
inst✝³ : StarModule R A
inst✝² : NoZeroSMulDivisors R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
a b : A
c : R
hab✝ : 0 < b - a
hc✝ : 0 < c
hab : b - a ∈ AddSubmonoid.closure (range fun s => star s * s)
x : A
hx : x ∈ range fun s => star s * s
hc : 0 ≤ c
⊢ c • x ∈ AddSubmonoid.closure (range fun s => star s * s)

Proof state:
case zeroc
R : Type u
A : Type u_1
inst✝¹² : Semiring R
inst✝¹¹ : PartialOrder R
inst✝¹⁰ : StarRing R
inst✝⁹ : StarOrderedRing R
inst✝⁸ : NonUnitalRing A
inst✝⁷ : StarRing A
inst✝⁶ : PartialOrder A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Module R A
inst✝³ : StarModule R A
inst✝² : NoZeroSMulDivisors R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
a b : A
c : R
hab✝ : 0 < b - a
hc✝ : 0 < c
hab : b - a ∈ AddSubmonoid.closure (range fun s => star s * s)
x : A
hx : x ∈ range fun s => star s * s
hc : c ∈ AddSubmonoid.closure (range fun s => star s * s)
⊢ 0 • x ∈ AddSubmonoid.closure (range fun s => star s * s)

case addc
R : Type u
A : Type u_1
inst✝¹² : Semiring R
inst✝¹¹ : PartialOrder R
inst✝¹⁰ : StarRing R
inst✝⁹ : StarOrderedRing R
inst✝⁸ : NonUnitalRing A
inst✝⁷ : StarRing A
inst✝⁶ : PartialOrder A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Module R A
inst✝³ : StarModule R A
inst✝² : NoZeroSMulDivisors R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
a b : A
c : R
hab✝ : 0 < b - a
hc✝ : 0 < c
hab : b - a ∈ AddSubmonoid.closure (range fun s => star s * s)
x : A
hx : x ∈ range fun s => star s * s
hc : c ∈ AddSubmonoid.closure (range fun s => star s * s)
⊢ ∀ (x_1 y : R),
    x_1 ∈ AddSubmonoid.closure (range fun s => star s * s) →
      y ∈ AddSubmonoid.closure (range fun s => star s * s) →
        x_1 • x ∈ AddSubmonoid.closure (range fun s => star s * s) →
          y • x ∈ AddSubmonoid.closure (range fun s => star s * s) →
            (x_1 + y) • x ∈ AddSubmonoid.closure (range fun s => star s * s)

Tactic:
case zeroc => simpa only [zero_smul] using zero_mem _

Result:
case addc
R : Type u
A : Type u_1
inst✝¹² : Semiring R
inst✝¹¹ : PartialOrder R
inst✝¹⁰ : StarRing R
inst✝⁹ : StarOrderedRing R
inst✝⁸ : NonUnitalRing A
inst✝⁷ : StarRing A
inst✝⁶ : PartialOrder A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Module R A
inst✝³ : StarModule R A
inst✝² : NoZeroSMulDivisors R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
a b : A
c : R
hab✝ : 0 < b - a
hc✝ : 0 < c
hab : b - a ∈ AddSubmonoid.closure (range fun s => star s * s)
x : A
hx : x ∈ range fun s => star s * s
hc : c ∈ AddSubmonoid.closure (range fun s => star s * s)
⊢ ∀ (x_1 y : R),
    x_1 ∈ AddSubmonoid.closure (range fun s => star s * s) →
      y ∈ AddSubmonoid.closure (range fun s => star s * s) →
        x_1 • x ∈ AddSubmonoid.closure (range fun s => star s * s) →
          y • x ∈ AddSubmonoid.closure (range fun s => star s * s) →
            (x_1 + y) • x ∈ AddSubmonoid.closure (range fun s => star s * s)

Proof state:
R : Type u
A : Type u_1
inst✝¹² : Semiring R
inst✝¹¹ : PartialOrder R
inst✝¹⁰ : StarRing R
inst✝⁹ : StarOrderedRing R
inst✝⁸ : NonUnitalRing A
inst✝⁷ : StarRing A
inst✝⁶ : PartialOrder A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Module R A
inst✝³ : StarModule R A
inst✝² : NoZeroSMulDivisors R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
a b : A
c : R
hab✝ : 0 < b - a
hc : 0 < c
hab : 0 ≤ b - a
⊢ 0 ≤ c • (b - a)

Tactic:
rw [StarOrderedRing.nonneg_iff] at hab ⊢

Result:
R : Type u
A : Type u_1
inst✝¹² : Semiring R
inst✝¹¹ : PartialOrder R
inst✝¹⁰ : StarRing R
inst✝⁹ : StarOrderedRing R
inst✝⁸ : NonUnitalRing A
inst✝⁷ : StarRing A
inst✝⁶ : PartialOrder A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Module R A
inst✝³ : StarModule R A
inst✝² : NoZeroSMulDivisors R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
a b : A
c : R
hab✝ : 0 < b - a
hc : 0 < c
hab : b - a ∈ AddSubmonoid.closure (range fun s => star s * s)
⊢ c • (b - a) ∈ AddSubmonoid.closure (range fun s => star s * s)