Let x0 of type ι → ((ι → ο) → ο) → (ι → ι → ι) → (ι → ο) → (ι → ο) → ι be given.
Let x1 of type ι be given.
Let x2 of type (ι → ο) → ο be given.
Let x3 of type ι → ι → ι be given.
Let x4 of type ι → ο be given.
Let x5 of type ι → ο be given.
Assume H0:
∀ x6 : (ι → ο) → ο . (∀ x7 : ι → ο . (∀ x8 . x7 x8 ⟶ prim1 x8 x1) ⟶ iff (x2 x7) (x6 x7)) ⟶ ∀ x7 : ι → ι → ι . (∀ x8 . prim1 x8 x1 ⟶ ∀ x9 . prim1 x9 x1 ⟶ x3 x8 x9 = x7 x8 x9) ⟶ ∀ x8 : ι → ο . (∀ x9 . prim1 x9 x1 ⟶ iff (x4 x9) (x8 x9)) ⟶ ∀ x9 : ι → ο . (∀ x10 . prim1 x10 x1 ⟶ iff (x5 x10) (x9 x10)) ⟶ x0 x1 x6 x7 x8 x9 = x0 x1 x2 x3 x4 x5.
Apply unknownprop_17bfe4cd292f60ecc42e0dfc2a362b02ca89e63b2d4cfd7b5a3963c133151c4c with
x1,
x2,
x3,
x4,
x5,
λ x6 x7 . x0 x6 (decode_c (f482f.. (783bc.. x1 x2 x3 x4 x5) (4ae4a.. 4a7ef..))) (e3162.. (f482f.. (783bc.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. (783bc.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (decode_p (f482f.. (783bc.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))) = x0 x1 x2 x3 x4 x5.
Apply H0 with
decode_c (f482f.. (783bc.. x1 x2 x3 x4 x5) (4ae4a.. 4a7ef..)),
e3162.. (f482f.. (783bc.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. 4a7ef..))),
decode_p (f482f.. (783bc.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))),
decode_p (f482f.. (783bc.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) leaving 4 subgoals.
Let x6 of type ι → ο be given.
Assume H1:
∀ x7 . x6 x7 ⟶ prim1 x7 x1.
Apply unknownprop_b8b08deb4ce9fe08026cdd946e072c126ec647e59db3da144b4ba7b8a3ae34db with
x1,
x2,
x3,
x4,
x5,
x6,
λ x7 x8 : ο . iff (x2 x6) x7 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying iff_refl with x2 x6.
The subproof is completed by applying unknownprop_361f56d0f0f7721253f46e481bb5360cdc5ca15df56553eab122c1c7545a0139 with x1, x2, x3, x4, x5.
Let x6 of type ι be given.
Apply unknownprop_ed9d8ed93a3bd6bbea7e5eef5e37a69bd0d4a2b9d800738495ab1d6c9090ce53 with
x1,
x2,
x3,
x4,
x5,
x6,
λ x7 x8 : ο . iff (x4 x6) x7 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying iff_refl with x4 x6.
Let x6 of type ι be given.
Apply unknownprop_c34a240ffb36057c4f0b75b4a6d2ec4faae5845aa6182897b19bb5dea423b792 with
x1,
x2,
x3,
x4,
x5,
x6,
λ x7 x8 : ο . iff (x5 x6) x7 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying iff_refl with x5 x6.