simp only [dif_pos hx, Array.nodup_push, Array.mem_push, ivs'_nodup, true_and]
n : Nat
x : Permutation n
k : Nat
hk : k < n
ivs : Array (Inversion x)
ivs_nodup : Array.Nodup ivs
ivs_lt : ∀ (iv : Inversion x), iv ∈ ivs → ↑iv.snd < k
l : Nat
hl : l < k
ivs' : Array (Inversion x)
ivs'_nodup : Array.Nodup ivs'
ivs'_lt : ∀ (iv : Inversion x), iv ∈ ivs' → ↑iv.snd < k ∨ ↑iv.snd = k ∧ ↑iv.fst < l
hx : x[l] > x[k]
⊢ Array.Nodup
      (if h : x[l] > x[k] then
        Array.push ivs'
          { fst := { val := l, isLt := (_ : l < n) }, snd := { val := k, isLt := hk },
            lt := (_ : { val := l, isLt := (_ : l < n) } < { val := k, isLt := hk }),
            inverted := (_ : x[↑{ val := l, isLt := (_ : l < n) }] > x[↑{ val := k, isLt := hk }]) }
      else ivs') ∧
    ∀ (iv : Inversion x),
      (iv ∈
          if h : x[l] > x[k] then
            Array.push ivs'
              { fst := { val := l, isLt := (_ : l < n) }, snd := { val := k, isLt := hk },
                lt := (_ : { val := l, isLt := (_ : l < n) } < { val := k, isLt := hk }),
                inverted := (_ : x[↑{ val := l, isLt := (_ : l < n) }] > x[↑{ val := k, isLt := hk }]) }
          else ivs') →
        ↑iv.snd < k ∨ ↑iv.snd = k ∧ ↑iv.fst < l + 1
n : Nat
x : Permutation n
k : Nat
hk : k < n
ivs : Array (Inversion x)
ivs_nodup : Array.Nodup ivs
ivs_lt : ∀ (iv : Inversion x), iv ∈ ivs → ↑iv.snd < k
l : Nat
hl : l < k
ivs' : Array (Inversion x)
ivs'_nodup : Array.Nodup ivs'
ivs'_lt : ∀ (iv : Inversion x), iv ∈ ivs' → ↑iv.snd < k ∨ ↑iv.snd = k ∧ ↑iv.fst < l
hx : x[l] > x[k]
⊢ ¬{ fst := { val := l, isLt := (_ : l < n) }, snd := { val := k, isLt := hk },
          lt := (_ : { val := l, isLt := (_ : l < n) } < { val := k, isLt := hk }),
          inverted := (_ : x[↑{ val := l, isLt := (_ : l < n) }] > x[↑{ val := k, isLt := hk }]) } ∈
        ivs' ∧
    ∀ (iv : Inversion x),
      iv ∈ ivs' ∨
          iv =
            { fst := { val := l, isLt := (_ : l < n) }, snd := { val := k, isLt := hk },
              lt := (_ : { val := l, isLt := (_ : l < n) } < { val := k, isLt := hk }),
              inverted := (_ : x[↑{ val := l, isLt := (_ : l < n) }] > x[↑{ val := k, isLt := hk }]) } →
        ↑iv.snd < k ∨ ↑iv.snd = k ∧ ↑iv.fst < l + 1