-- Lean 4 proof step from Mathlib.Algebra.Order.Rearrangement (goal)
Goal:
ι : Type u_1
α : Type u_2
β : Type u_3
inst : Semiring α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
inst_3 : ExistsAddOfLE α
inst_4 : AddCommMonoid β
inst_5 : LinearOrder β
inst_6 : IsOrderedCancelAddMonoid β
inst_7 : Module α β
inst_8 : PosSMulMono α β
s : Finset ι
σ : Perm ι
f : ι → α
g : ι → β
hfg : MonovaryOn f g (↑s : Set ι)
hσ : {x : ι | (σ : (a : ι) → ι) x ≠ x} ⊆ (↑s : Set ι)
empty :
  ∀ {σ : Perm ι} (hfg : MonovaryOn f g (↑∅ : Set ι)) (hσ : {x : ι | (σ : (a : ι) → ι) x ≠ x} ⊆ (↑∅ : Set ι)),
    LE.le.{u_3} (α := β) (∑ i ∈ ∅, f i • g ((σ : (a : ι) → ι) i)) (∑ i ∈ ∅, f i • g i)
insert insert₁ :
  ∀ (a : ι) (s : Finset ι) (has : a ∉ s)
    (hamax :
      ∀ x ∈ s,
        LE.le (α := Lex (β × α)) ((toLex : (a : β × α) → Lex (β × α)) (g x, f x) : Lex (β × α))
          ((toLex : (a : β × α) → Lex (β × α)) (g a, f a) : Lex (β × α)))
    (hind :
      ∀ {σ : Perm ι} (hfg : MonovaryOn f g (↑s : Set ι)) (hσ : {x : ι | (σ : (a : ι) → ι) x ≠ x} ⊆ (↑s : Set ι)),
        LE.le.{u_3} (α := β) (∑ i ∈ s, f i • g ((σ : (a : ι) → ι) i)) (∑ i ∈ s, f i • g i))
    {σ : Perm ι} (hfg : MonovaryOn f g (↑(Insert.insert a s) : Set ι))
    (hσ : {x : ι | (σ : (a : ι) → ι) x ≠ x} ⊆ (↑(Insert.insert a s) : Set ι)),
    LE.le.{u_3} (α := β) (∑ i ∈ Insert.insert a s, f i • g ((σ : (a : ι) → ι) i)) (∑ i ∈ Insert.insert a s, f i • g i)
insert₂ :
  ∀ (a : ι) (s : Finset ι) (has : a ∉ s)
    (hamax :
      ∀ x ∈ s,
        LE.le (α := Lex (β × α)) ((toLex : (a : β × α) → Lex (β × α)) (g x, f x) : Lex (β × α))
          ((toLex : (a : β × α) → Lex (β × α)) (g a, f a) : Lex (β × α)))
    (hind :
      ∀ {σ : Perm ι} (hfg : MonovaryOn f g (↑s : Set ι)) (hσ : {x : ι | (σ : (a : ι) → ι) x ≠ x} ⊆ (↑s : Set ι)),
        LE.le.{u_3} (α := β) (∑ i ∈ s, f i • g ((σ : (a : ι) → ι) i)) (∑ i ∈ s, f i • g i))
    {σ : Perm ι} (hfg : MonovaryOn f g (↑(Insert.insert a s) : Set ι))
    (hσ : {x : ι | (σ : (a : ι) → ι) x ≠ x} ⊆ (↑(Insert.insert a s) : Set ι)),
    let τ := Equiv.trans σ (Equiv.swap a ((σ : (a : ι) → ι) a));
    ∀ (hτ : τ = Equiv.trans σ (Equiv.swap a ((σ : (a : ι) → ι) a))),
      LE.le.{u_3} (α := β) (∑ i ∈ Insert.insert a s, f i • g ((σ : (a : ι) → ι) i)) (∑ i ∈ Insert.insert a s, f i • g i)
⊢ LE.le.{u_3} (α := β) (∑ i ∈ s, f i • g ((σ : (a : ι) → ι) i)) (∑ i ∈ s, f i • g i)
have insert₃ : ∀ (a : ι) (s : Finset ι) (has : a ∉ s)
  (hamax :
    ∀ x ∈ s,
      LE.le (α := Lex (β × α)) ((toLex : (a : β × α) → Lex (β × α)) (g x, f x) : Lex (β × α))
        ((toLex : (a : β × α) → Lex (β × α)) (g a, f a) : Lex (β × α)))
  (hind :
    ∀ {σ : Perm ι} (hfg : MonovaryOn f g (↑s : Set ι)) (hσ : {x : ι | (σ : (a : ι) → ι) x ≠ x} ⊆ (↑s : Set ι)),
      LE.le.{u_3} (α := β) (∑ i ∈ s, f i • g ((σ : (a : ι) → ι) i)) (∑ i ∈ s, f i • g i))
  {σ : Perm ι} (hfg : MonovaryOn f g (↑(Insert.insert a s) : Set ι))
  (hσ : {x : ι | (σ : (a : ι) → ι) x ≠ x} ⊆ (↑(Insert.insert a s) : Set ι)),
  let τ := Equiv.trans σ (Equiv.swap a ((σ : (a : ι) → ι) a));
  ∀ (hτ : τ = Equiv.trans σ (Equiv.swap a ((σ : (a : ι) → ι) a)))
    (hτs : {x : ι | (τ : (a : ι) → ι) x ≠ x} ⊆ (↑s : Set ι)),
    LE.le.{u_3} (α := β) (∑ i ∈ Insert.insert a s, f i • g ((σ : (a : ι) → ι) i)) (∑ i ∈ Insert.insert a s, f i • g i)
Tactic: classical
  induction s using induction_on_max_value fun i ↦ toLex (g i, f i) generalizing σ with
  | empty => simp only [le_rfl, Finset.sum_empty]
  | insert a s has hamax hind => ?_
  set τ : Perm ι := σ.trans (swap a (σ a)) with hτ
  have hτs : {x | τ x ≠ x} ⊆ s := by
    intro x hx
    simp only [τ, Ne, Set.mem_setOf_eq, Equiv.swap_comp_apply] at hx
    split_ifs at hx with h₁ h₂
    · obtain rfl | hax := eq_or_ne x a
      · contradiction
      · exact mem_of_mem_insert_of_ne (hσ fun h ↦ hax <| h.symm.trans h₁) hax
    · exact (hx <| σ.injective h₂.symm).elim
    · exact mem_of_mem_insert_of_ne (hσ hx) (ne_of_apply_ne _ h₂)
  specialize hind (hfg.subset <| subset_insert _ _) hτs
  simp_rw [sum_insert has]
  grw [← hind]
  obtain hσa | hσa := eq_or_ne a (σ a)
  · rw [hτ, ← hσa, swap_self, trans_refl]
  have h1s : σ.symm a ∈ s := by
    rw [Ne, ← inv_eq_iff_eq] at hσa
    refine mem_of_mem_insert_of_ne (hσ fun h ↦ hσa ?_) hσa
    rwa [apply_symm_apply, eq_comm] at h
  simp only [← s.sum_erase_add _ h1s, add_comm]
  rw [← add_assoc, ← add_assoc]
  simp only [hτ, swap_apply_left, Function.comp_apply, Equiv.coe_trans, apply_symm_apply]
  refine add_le_add (smul_add_smul_le_smul_add_smul' ?_ ?_) (sum_congr rfl fun x hx ↦ ?_).le
  · specialize hamax (σ.symm a) h1s
    rw [Prod.Lex.toLex_le_toLex] at hamax
    rcases hamax with hamax | hamax
    · exact hfg (mem_insert_of_mem h1s) (mem_insert_self _ _) hamax
    · exact hamax.2
  · specialize hamax (σ a) (mem_of_mem_insert_of_ne (hσ <| σ.injective.ne hσa.symm) hσa.symm)
    rw [Prod.Lex.toLex_le_toLex] at hamax
    rcases hamax with hamax | hamax
    · exact hamax.le
    · exact hamax.1.le
  · rw [mem_erase, Ne, eq_symm_apply] at hx
    rw [swap_apply_of_ne_of_ne hx.1 (σ.injective.ne _)]
    rintro rfl
    exact has hx.2
Cleared hypotheses: s, σ, hfg, hσ