cases this
n i : Nat
h : i + 1 < n
k : Fin n
hlt : k < { val := i + 1, isLt := h }
hinv✝ : (transposAdj i h)[↑k] > (transposAdj i h)[↑{ val := i + 1, isLt := h }]
hinv :
  (if i = ↑k then { val := i + 1, isLt := h }
    else
      { val := ↑k,
        isLt :=
          (_ : (fun x i => i < n) (transpos { val := i, isLt := (_ : i < n) } { val := i + 1, isLt := h }) ↑k) }) >
    { val := i, isLt := (_ : i < n) }
hik : i = ↑k
this : { val := i, isLt := (_ : i < n) } = k
⊢ { fst := k, snd := { val := i + 1, isLt := h }, lt := hlt, inverted := hinv✝ } =
    transposInversion { val := i, isLt := (_ : i < n) } { val := i + 1, isLt := h } (_ : i < Nat.succ i)
case refl
n i : Nat
h : i + 1 < n
hlt : { val := i, isLt := (_ : i < n) } < { val := i + 1, isLt := h }
hinv✝ : (transposAdj i h)[↑{ val := i, isLt := (_ : i < n) }] > (transposAdj i h)[↑{ val := i + 1, isLt := h }]
hinv :
  (if i = ↑{ val := i, isLt := (_ : i < n) } then { val := i + 1, isLt := h }
    else
      { val := ↑{ val := i, isLt := (_ : i < n) },
        isLt :=
          (_ :
            (fun x i => i < n) (transpos { val := i, isLt := (_ : i < n) } { val := i + 1, isLt := h })
              ↑{ val := i, isLt := (_ : i < n) }) }) >
    { val := i, isLt := (_ : i < n) }
hik : i = ↑{ val := i, isLt := (_ : i < n) }
⊢ { fst := { val := i, isLt := (_ : i < n) }, snd := { val := i + 1, isLt := h }, lt := hlt, inverted := hinv✝ } =
    transposInversion { val := i, isLt := (_ : i < n) } { val := i + 1, isLt := h } (_ : i < Nat.succ i)