Module: Mathlib.Topology.Homotopy.HomotopyGroup

Goal state before:
pp: N : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace X
x : X
inst✝ : DecidableEq N
i j : N
h : i ≠ j
a b c d : ↑(Ω^ N X x)
y✝ : N → ↑I
⊢ (if ↑(y✝ i) ≤ 1 / 2 then
      if ↑(Function.update y✝ i (Set.projIcc 0 1 transAt._proof_3 (2 * ↑(y✝ i))) j) ≤ 1 / 2 then
        a
          (Function.update (Function.update y✝ i (Set.projIcc 0 1 transAt._proof_3 (2 * ↑(y✝ i)))) j
            (Set.projIcc 0 1 transAt._proof_3
              (2 * ↑(Function.update y✝ i (Set.projIcc 0 1 transAt._proof_3 (2 * ↑(y✝ i))) j))))
      else
        b
          (Function.update (Function.update y✝ i (Set.projIcc 0 1 transAt._proof_3 (2 * ↑(y✝ i)))) j
            (Set.projIcc 0 1 transAt._proof_3
              (2 * ↑(Function.update y✝ i (Set.projIcc 0 1 transAt._proof_3 (2 * ↑(y✝ i))) j) - 1)))
    else
      if ↑(Function.update y✝ i (Set.projIcc 0 1 transAt._proof_3 (2 * ↑(y✝ i) - 1)) j) ≤ 1 / 2 then
        c
          (Function.update (Function.update y✝ i (Set.projIcc 0 1 transAt._proof_3 (2 * ↑(y✝ i) - 1))) j
            (Set.projIcc 0 1 transAt._proof_3
              (2 * ↑(Function.update y✝ i (Set.projIcc 0 1 transAt._proof_3 (2 * ↑(y✝ i) - 1)) j))))
      else
        d
          (Function.update (Function.update y✝ i (Set.projIcc 0 1 transAt._proof_3 (2 * ↑(y✝ i) - 1))) j
            (Set.projIcc 0 1 transAt._proof_3
              (2 * ↑(Function.update y✝ i (Set.projIcc 0 1 transAt._proof_3 (2 * ↑(y✝ i) - 1)) j) - 1)))) =
    if ↑(y✝ j) ≤ 1 / 2 then
      if ↑(Function.update y✝ j (Set.projIcc 0 1 transAt._proof_3 (2 * ↑(y✝ j))) i) ≤ 1 / 2 then
        a
          (Function.update (Function.update y✝ j (Set.projIcc 0 1 transAt._proof_3 (2 * ↑(y✝ j)))) i
            (Set.projIcc 0 1 transAt._proof_3
              (2 * ↑(Function.update y✝ j (Set.projIcc 0 1 transAt._proof_3 (2 * ↑(y✝ j))) i))))
      else
        c
          (Function.update (Function.update y✝ j (Set.projIcc 0 1 transAt._proof_3 (2 * ↑(y✝ j)))) i
            (Set.projIcc 0 1 transAt._proof_3
              (2 * ↑(Function.update y✝ j (Set.projIcc 0 1 transAt._proof_3 (2 * ↑(y✝ j))) i) - 1)))
    else
      if ↑(Function.update y✝ j (Set.projIcc 0 1 transAt._proof_3 (2 * ↑(y✝ j) - 1)) i) ≤ 1 / 2 then
        b
          (Function.update (Function.update y✝ j (Set.projIcc 0 1 transAt._proof_3 (2 * ↑(y✝ j) - 1))) i
            (Set.projIcc 0 1 transAt._proof_3
              (2 * ↑(Function.update y✝ j (Set.projIcc 0 1 transAt._proof_3 (2 * ↑(y✝ j) - 1)) i))))
      else
        d
          (Function.update (Function.update y✝ j (Set.projIcc 0 1 transAt._proof_3 (2 * ↑(y✝ j) - 1))) i
            (Set.projIcc 0 1 transAt._proof_3
              (2 * ↑(Function.update y✝ j (Set.projIcc 0 1 transAt._proof_3 (2 * ↑(y✝ j) - 1)) i) - 1)))

Tactic:
Function.update_apply,

Goal state after:
N : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace X
x : X
inst✝ : DecidableEq N
i j : N
h : i ≠ j
a b c d : ↑(Ω^ N X x)
y✝ : N → ↑I
⊢ (if ↑(y✝ i) ≤ 1 / 2 then
      if ↑(if j = i then Set.projIcc 0 1 transAt._proof_3 (2 * ↑(y✝ i)) else y✝ j) ≤ 1 / 2 then
        a
          (Function.update (Function.update y✝ i (Set.projIcc 0 1 transAt._proof_3 (2 * ↑(y✝ i)))) j
            (Set.projIcc 0 1 transAt._proof_3
              (2 * ↑(if j = i then Set.projIcc 0 1 transAt._proof_3 (2 * ↑(y✝ i)) else y✝ j))))
      else
        b
          (Function.update (Function.update y✝ i (Set.projIcc 0 1 transAt._proof_3 (2 * ↑(y✝ i)))) j
            (Set.projIcc 0 1 transAt._proof_3
              (2 * ↑(if j = i then Set.projIcc 0 1 transAt._proof_3 (2 * ↑(y✝ i)) else y✝ j) - 1)))
    else
      if ↑(if j = i then Set.projIcc 0 1 transAt._proof_3 (2 * ↑(y✝ i) - 1) else y✝ j) ≤ 1 / 2 then
        c
          (Function.update (Function.update y✝ i (Set.projIcc 0 1 transAt._proof_3 (2 * ↑(y✝ i) - 1))) j
            (Set.projIcc 0 1 transAt._proof_3
              (2 * ↑(if j = i then Set.projIcc 0 1 transAt._proof_3 (2 * ↑(y✝ i) - 1) else y✝ j))))
      else
        d
          (Function.update (Function.update y✝ i (Set.projIcc 0 1 transAt._proof_3 (2 * ↑(y✝ i) - 1))) j
            (Set.projIcc 0 1 transAt._proof_3
              (2 * ↑(if j = i then Set.projIcc 0 1 transAt._proof_3 (2 * ↑(y✝ i) - 1) else y✝ j) - 1)))) =
    if ↑(y✝ j) ≤ 1 / 2 then
      if ↑(if i = j then Set.projIcc 0 1 transAt._proof_3 (2 * ↑(y✝ j)) else y✝ i) ≤ 1 / 2 then
        a
          (Function.update (Function.update y✝ j (Set.projIcc 0 1 transAt._proof_3 (2 * ↑(y✝ j)))) i
            (Set.projIcc 0 1 transAt._proof_3
              (2 * ↑(if i = j then Set.projIcc 0 1 transAt._proof_3 (2 * ↑(y✝ j)) else y✝ i))))
      else
        c
          (Function.update (Function.update y✝ j (Set.projIcc 0 1 transAt._proof_3 (2 * ↑(y✝ j)))) i
            (Set.projIcc 0 1 transAt._proof_3
              (2 * ↑(if i = j then Set.projIcc 0 1 transAt._proof_3 (2 * ↑(y✝ j)) else y✝ i) - 1)))
    else
      if ↑(if i = j then Set.projIcc 0 1 transAt._proof_3 (2 * ↑(y✝ j) - 1) else y✝ i) ≤ 1 / 2 then
        b
          (Function.update (Function.update y✝ j (Set.projIcc 0 1 transAt._proof_3 (2 * ↑(y✝ j) - 1))) i
            (Set.projIcc 0 1 transAt._proof_3
              (2 * ↑(if i = j then Set.projIcc 0 1 transAt._proof_3 (2 * ↑(y✝ j) - 1) else y✝ i))))
      else
        d
          (Function.update (Function.update y✝ j (Set.projIcc 0 1 transAt._proof_3 (2 * ↑(y✝ j) - 1))) i
            (Set.projIcc 0 1 transAt._proof_3
              (2 * ↑(if i = j then Set.projIcc 0 1 transAt._proof_3 (2 * ↑(y✝ j) - 1) else y✝ i) - 1)))