-- Module: Mathlib.Topology.Homotopy.HomotopyGroup
Goal:
N : Type u_1
X : Type u_2
inst : TopologicalSpace X
x : X
inst_1 : DecidableEq N
inst_2 : Nontrivial N
h : ∃ (y : N), y ≠ Classical.arbitrary N := ⋯
a_1 : HomotopyGroup N X x
a : (↑(Ω^ N X x) : Type (max u_2 u_1))
b_1 : HomotopyGroup N X x
b : (↑(Ω^ N X x) : Type (max u_2 u_1))
c_1 : HomotopyGroup N X x
c : (↑(Ω^ N X x) : Type (max u_2 u_1))
d_1 : HomotopyGroup N X x
d : (↑(Ω^ N X x) : Type (max u_2 u_1))
⊢ Eq (α := (↑(Ω^ N X x) : Type (max u_2 u_1)))
    ((Equiv.symm (toEquiv (loopHomeo (Classical.choose (p := fun (x : N) ↦ x ≠ Classical.arbitrary N) h))) :
          (a : Path { as := const }.as { as := const }.as) → (↑(Ω^ N X x) : Type (max u_2 u_1)))
        (Path.trans (y := FundamentalGroupoid.as { as := const })
          ((toEquiv (loopHomeo (Classical.choose (p := fun (x : N) ↦ x ≠ Classical.arbitrary N) h)) :
              (a : (↑(Ω^ N X x) : Type (max u_2 u_1))) → Path { as := const }.as { as := const }.as)
            ((Equiv.symm (toEquiv (loopHomeo (Classical.arbitrary N))) :
                (a : Path { as := const }.as { as := const }.as) → (↑(Ω^ N X x) : Type (max u_2 u_1)))
              (Path.trans (y := FundamentalGroupoid.as { as := const })
                ((toEquiv (loopHomeo (Classical.arbitrary N)) :
                    (a : (↑(Ω^ N X x) : Type (max u_2 u_1))) → Path { as := const }.as { as := const }.as)
                  d)
                ((toEquiv (loopHomeo (Classical.arbitrary N)) :
                    (a : (↑(Ω^ N X x) : Type (max u_2 u_1))) → Path { as := const }.as { as := const }.as)
                  c))))
          ((toEquiv (loopHomeo (Classical.choose (p := fun (x : N) ↦ x ≠ Classical.arbitrary N) h)) :
              (a : (↑(Ω^ N X x) : Type (max u_2 u_1))) → Path { as := const }.as { as := const }.as)
            ((Equiv.symm (toEquiv (loopHomeo (Classical.arbitrary N))) :
                (a : Path { as := const }.as { as := const }.as) → (↑(Ω^ N X x) : Type (max u_2 u_1)))
              (Path.trans (y := FundamentalGroupoid.as { as := const })
                ((toEquiv (loopHomeo (Classical.arbitrary N)) :
                    (a : (↑(Ω^ N X x) : Type (max u_2 u_1))) → Path { as := const }.as { as := const }.as)
                  b)
                ((toEquiv (loopHomeo (Classical.arbitrary N)) :
                    (a : (↑(Ω^ N X x) : Type (max u_2 u_1))) → Path { as := const }.as { as := const }.as)
                  a))))) :
      (↑(Ω^ N X x) : Type (max u_2 u_1)))
    ((Equiv.symm (toEquiv (loopHomeo (Classical.arbitrary N))) :
          (a : Path { as := const }.as { as := const }.as) → (↑(Ω^ N X x) : Type (max u_2 u_1)))
        (Path.trans (y := FundamentalGroupoid.as { as := const })
          ((toEquiv (loopHomeo (Classical.arbitrary N)) :
              (a : (↑(Ω^ N X x) : Type (max u_2 u_1))) → Path { as := const }.as { as := const }.as)
            ((Equiv.symm (toEquiv (loopHomeo (Classical.choose (p := fun (x : N) ↦ x ≠ Classical.arbitrary N) h))) :
                (a : Path { as := const }.as { as := const }.as) → (↑(Ω^ N X x) : Type (max u_2 u_1)))
              (Path.trans (y := FundamentalGroupoid.as { as := const })
                ((toEquiv (loopHomeo (Classical.choose (p := fun (x : N) ↦ x ≠ Classical.arbitrary N) h)) :
                    (a : (↑(Ω^ N X x) : Type (max u_2 u_1))) → Path { as := const }.as { as := const }.as)
                  d)
                ((toEquiv (loopHomeo (Classical.choose (p := fun (x : N) ↦ x ≠ Classical.arbitrary N) h)) :
                    (a : (↑(Ω^ N X x) : Type (max u_2 u_1))) → Path { as := const }.as { as := const }.as)
                  b))))
          ((toEquiv (loopHomeo (Classical.arbitrary N)) :
              (a : (↑(Ω^ N X x) : Type (max u_2 u_1))) → Path { as := const }.as { as := const }.as)
            ((Equiv.symm (toEquiv (loopHomeo (Classical.choose (p := fun (x : N) ↦ x ≠ Classical.arbitrary N) h))) :
                (a : Path { as := const }.as { as := const }.as) → (↑(Ω^ N X x) : Type (max u_2 u_1)))
              (Path.trans (y := FundamentalGroupoid.as { as := const })
                ((toEquiv (loopHomeo (Classical.choose (p := fun (x : N) ↦ x ≠ Classical.arbitrary N) h)) :
                    (a : (↑(Ω^ N X x) : Type (max u_2 u_1))) → Path { as := const }.as { as := const }.as)
                  c)
                ((toEquiv (loopHomeo (Classical.choose (p := fun (x : N) ↦ x ≠ Classical.arbitrary N) h)) :
                    (a : (↑(Ω^ N X x) : Type (max u_2 u_1))) → Path { as := const }.as { as := const }.as)
                  a))))) :
      (↑(Ω^ N X x) : Type (max u_2 u_1)))

Tactic:
simp only [fromLoop_trans_toLoop, transAt_distrib <| Classical.choose_spec h, coe_toEquiv,
        loopHomeo_apply, coe_symm_toEquiv, loopHomeo_symm_apply]

Premises used: GenLoop.transAt_distrib, Homeomorph.coe_symm_toEquiv, GenLoop.loopHomeo_apply, Classical.choose_spec, GenLoop.loopHomeo_symm_apply, Homeomorph.coe_toEquiv, GenLoop.fromLoop_trans_toLoop