Let x0 of type ι be given.

Let x1 of type ι → ι be given.

Let x2 of type ι → ι → ο be given.

Let x3 of type ι be given.

Let x4 of type ι be given.

Assume H0: 4b311.. (bd517.. x0 x1 x2 x3 x4).

Apply H0 with λ x5 . x5 = bd517.. x0 x1 x2 x3 x4 ⟶ prim1 x3 x0 leaving 2 subgoals.

Let x5 of type ι be given.

Let x6 of type ι → ι be given.

Assume H1: ∀ x7 . prim1 x7 x5 ⟶ prim1 (x6 x7) x5.

Let x7 of type ι → ι → ο be given.

Let x8 of type ι be given.

Assume H2: prim1 x8 x5.

Let x9 of type ι be given.

Assume H3: prim1 x9 x5.

Assume H4: bd517.. x5 x6 x7 x8 x9 = bd517.. x0 x1 x2 x3 x4.

Apply unknownprop_0dcb6a37c215d05e3e57b1b37c9382b4a74eac48c1ea9ce59d2791534b7bb74f with x5, x0, x6, x1, x7, x2, x8, x3, x9, x4, prim1 x3 x0 leaving 2 subgoals.

The subproof is completed by applying H4.

Assume H5: and (and (and (x5 = x0) (∀ x10 . prim1 x10 x5 ⟶ x6 x10 = x1 x10)) (∀ x10 . prim1 x10 x5 ⟶ ∀ x11 . prim1 x11 x5 ⟶ x7 x10 x11 = x2 x10 x11)) (x8 = x3).

Apply H5 with x9 = x4 ⟶ prim1 x3 x0.

Assume H6: and (and (x5 = x0) (∀ x10 . prim1 x10 x5 ⟶ x6 x10 = x1 x10)) (∀ x10 . prim1 x10 x5 ⟶ ∀ x11 . prim1 x11 x5 ⟶ x7 x10 x11 = x2 x10 x11).

Apply H6 with x8 = x3 ⟶ x9 = x4 ⟶ prim1 x3 x0.

Assume H7: and (x5 = x0) (∀ x10 . prim1 x10 x5 ⟶ x6 x10 = x1 x10).

Apply H7 with (∀ x10 . prim1 x10 x5 ⟶ ∀ x11 . prim1 x11 x5 ⟶ x7 x10 x11 = x2 x10 x11) ⟶ x8 = x3 ⟶ x9 = x4 ⟶ prim1 x3 x0.

Assume H8: x5 = x0.

Assume H9: ∀ x10 . prim1 x10 x5 ⟶ x6 x10 = x1 x10.

Assume H10: ∀ x10 . prim1 x10 x5 ⟶ ∀ x11 . prim1 x11 x5 ⟶ x7 x10 x11 = x2 x10 x11.

Assume H11: x8 = x3.

Assume H12: x9 = x4.

Apply H8 with λ x10 x11 . prim1 x3 x10.

Apply H11 with λ x10 x11 . prim1 x10 x5.

The subproof is completed by applying H2.

Let x5 of type ι → ι → ο be given.

Assume H1: x5 (bd517.. x0 x1 x2 x3 x4) (bd517.. x0 x1 x2 x3 x4).

The subproof is completed by applying H1.

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Proofgold Proof