/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
    min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
  -- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
  set x := (s, t) with hx
  clear_value x
  simp only [Prod.ext_iff] at hx
  obtain ⟨rfl, rfl⟩ := hx
  refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
    min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
  clear! x
  rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
  simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
    Nat.cast_le] at *
  -- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
  obtain hts | hst := lt_or_le t.card s.card
  · simpa only [← mul_inv_rev, add_comm, card_inv] using
      ih _ _ ht.inv hs.inv
        (devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
          simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
  -- If `s` is a singleton, then the result is trivial.
  obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
  · simp [add_comm]
  -- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
  -- intersects its right translate by `g`.
  obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
    ⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
      mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
  -- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
  -- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
  -- as wanted.
  obtain hsg | hsg := eq_or_ne (op g • s) s
  · have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
      refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
        ⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
      · rw [← hsg, coe_smul_finset, smul_comm]
        exact Set.smul_mem_smul_set hc
      · simp only
        rwa [← op_smul_eq_mul, op_inv, ← Set.mem_smul_set_iff_inv_smul_mem, smul_comm,
          ← coe_smul_finset, hsg]
    refine Or.inl ((minOrder_le_natCard (zpowers_ne_bot.2 hg) <|
      s.finite_toSet.smul_set.subset hS).trans <| WithTop.coe_le_coe.2 <|
        ((Nat.card_mono s.finite_toSet.smul_set hS).trans_eq <| ?_).trans <|
          card_le_card_mul_right _ ht)
    rw [← coe_smul_finset]
    simp [-coe_smul_finset]
  -- Else, we can transform `s`, `t` to `s'`, `t'` and `s''`, `t''`, such that one of `(s', t')` and
  -- `(s'', t'')` is strictly smaller than `(s, t)` according to `DevosMulRel`.
  replace hsg : (s ∩ op g • s).card < s.card := card_lt_card ⟨inter_subset_left _ _, fun h ↦
    hsg <| eq_of_superset_of_card_ge (h.trans <| inter_subset_right _ _) (card_smul_finset _ _).le⟩
  replace aux1 := card_mono $ mulETransformLeft.fst_mul_snd_subset g (s, t)
  replace aux2 := card_mono $ mulETransformRight.fst_mul_snd_subset g (s, t)
  -- If the left translate of `t` by `g⁻¹` is disjoint from `t`, then we're easily done.
  obtain hgt | hgt := disjoint_or_nonempty_inter t (g⁻¹ • t)
  · rw [← card_smul_finset g⁻¹ t]
    refine' Or.inr ((add_le_add_right hst _).trans _)
    rw [← card_union_eq hgt]
    exact (card_le_card_mul_left _ hgs).trans (le_add_of_le_left aux1)
  -- Else, we're done by induction on either `(s', t')` or `(s'', t'')` depending on whether
  -- `|s| + |t| ≤ |s'| + |t'|` or `|s| + |t| < |s''| + |t''|`. One of those two inequalities must
  -- hold since `2 * (|s| + |t|) = |s'| + |t'| + |s''| + |t''|`.
  obtain hstg | hstg := le_or_lt_of_add_le_add (MulETransform.card g (s, t)).ge
  · exact (ih _ _ hgs (hgt.mono inter_subset_union) <| devosMulRel_of_le_of_le aux1 hstg hsg).imp
      (WithTop.coe_le_coe.2 aux1).trans' fun h ↦ hstg.trans <| h.trans <| add_le_add_right aux1 _
  · 

Proof state:
case intro.mk.intro.inr.inr.intro.intro.intro.intro.intro.intro.inr.inr.inr
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
  ∀ (a b : Finset α),
    Finset.Nonempty a →
      Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(card (a * b)) ∨ card a + card b ≤ card (a * b) + 1
hst : card s ≤ card t
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
g : α
hg : g ≠ 1
hgs : Finset.Nonempty (s ∩ op g • s)
hsg : card (s ∩ op g • s) < card s
aux1 : card ((mulETransformLeft g (s, t)).1 * (mulETransformLeft g (s, t)).2) ≤ card ((s, t).1 * (s, t).2)
aux2 : card ((mulETransformRight g (s, t)).1 * (mulETransformRight g (s, t)).2) ≤ card ((s, t).1 * (s, t).2)
hgt : Finset.Nonempty (t ∩ g⁻¹ • t)
hstg : card (s, t).1 + card (s, t).2 < card (mulETransformRight g (s, t)).1 + card (mulETransformRight g (s, t)).2
⊢ minOrder α ≤ ↑(card (s * t)) ∨ card s + card t ≤ card (s * t) + 1

Next tactic:
exact (ih _ _ (hgs.mono inter_subset_union) hgt <| devosMulRel_of_le aux2 hstg).imp
      (WithTop.coe_le_coe.2 aux2).trans' fun h ↦
        hstg.le.trans <| h.trans <| add_le_add_right aux2 _