Let x0 of type ι → ((ι → ο) → ο) → (ι → ι → ο) → (ι → ο) → ο be given.
Let x1 of type ι be given.
Let x2 of type (ι → ο) → ο be given.
Let x3 of type ι → ι → ο be given.
Let x4 of type ι → ο be given.
Assume H0:
∀ x5 : (ι → ο) → ο . (∀ x6 : ι → ο . (∀ x7 . x6 x7 ⟶ prim1 x7 x1) ⟶ iff (x2 x6) (x5 x6)) ⟶ ∀ x6 : ι → ι → ο . (∀ x7 . prim1 x7 x1 ⟶ ∀ x8 . prim1 x8 x1 ⟶ iff (x3 x7 x8) (x6 x7 x8)) ⟶ ∀ x7 : ι → ο . (∀ x8 . prim1 x8 x1 ⟶ iff (x4 x8) (x7 x8)) ⟶ x0 x1 x5 x6 x7 = x0 x1 x2 x3 x4.
Apply unknownprop_53a1434582c46791d97bf0b29daf1e96260f65f75da341b82fe60b82f00728d7 with
x1,
x2,
x3,
x4,
λ x5 x6 . x0 x5 (decode_c (f482f.. (30bff.. x1 x2 x3 x4) (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. (30bff.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. (30bff.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) = x0 x1 x2 x3 x4.
Apply H0 with
decode_c (f482f.. (30bff.. x1 x2 x3 x4) (4ae4a.. 4a7ef..)),
2b2e3.. (f482f.. (30bff.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))),
decode_p (f482f.. (30bff.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) leaving 3 subgoals.
Let x5 of type ι → ο be given.
Assume H1:
∀ x6 . x5 x6 ⟶ prim1 x6 x1.
Apply unknownprop_7708a5a648779c332bd616f6e4513a6548347153ed171f25a7c039b600789f31 with
x1,
x2,
x3,
x4,
x5,
λ x6 x7 : ο . iff (x2 x5) x6 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying iff_refl with x2 x5.
Let x5 of type ι be given.
Let x6 of type ι be given.
Apply unknownprop_580e8f89400098eafc477e78627cca2cb91e5401e413a05dcb445fa113c00c84 with
x1,
x2,
x3,
x4,
x5,
x6,
λ x7 x8 : ο . iff (x3 x5 x6) x7 leaving 3 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
The subproof is completed by applying iff_refl with x3 x5 x6.
Let x5 of type ι be given.
Apply unknownprop_26c09a5fb4cb36f7b0ddea94b180a8c22b92587770fba43cdead9a6362679f2e with
x1,
x2,
x3,
x4,
x5,
λ x6 x7 : ο . iff (x4 x5) x6 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying iff_refl with x4 x5.