Let x0 of type ι → ι → ι → ο be given.

Assume H0: ∀ x1 x2 x3 . 80242.. x1 ⟶ 80242.. x2 ⟶ 80242.. x3 ⟶ (∀ x4 . prim1 x4 (56ded.. (e4431.. x1)) ⟶ x0 x4 x2 x3) ⟶ (∀ x4 . prim1 x4 (56ded.. (e4431.. x2)) ⟶ x0 x1 x4 x3) ⟶ (∀ x4 . prim1 x4 (56ded.. (e4431.. x3)) ⟶ x0 x1 x2 x4) ⟶ (∀ x4 . prim1 x4 (56ded.. (e4431.. x1)) ⟶ ∀ x5 . prim1 x5 (56ded.. (e4431.. x2)) ⟶ x0 x4 x5 x3) ⟶ (∀ x4 . prim1 x4 (56ded.. (e4431.. x1)) ⟶ ∀ x5 . prim1 x5 (56ded.. (e4431.. x3)) ⟶ x0 x4 x2 x5) ⟶ (∀ x4 . prim1 x4 (56ded.. (e4431.. x2)) ⟶ ∀ x5 . prim1 x5 (56ded.. (e4431.. x3)) ⟶ x0 x1 x4 x5) ⟶ (∀ x4 . prim1 x4 (56ded.. (e4431.. x1)) ⟶ ∀ x5 . prim1 x5 (56ded.. (e4431.. x2)) ⟶ ∀ x6 . prim1 x6 (56ded.. (e4431.. x3)) ⟶ x0 x4 x5 x6) ⟶ x0 x1 x2 x3.

Claim L1: ∀ x1 . ordinal x1 ⟶ ∀ x2 . ordinal x2 ⟶ ∀ x3 . ordinal x3 ⟶ ∀ x4 . prim1 x4 (56ded.. x1) ⟶ ∀ x5 . prim1 x5 (56ded.. x2) ⟶ ∀ x6 . prim1 x6 (56ded.. x3) ⟶ x0 x4 x5 x6

Apply ordinal_ind with λ x1 . ∀ x2 . ordinal x2 ⟶ ∀ x3 . ordinal x3 ⟶ ∀ x4 . prim1 x4 (56ded.. x1) ⟶ ∀ x5 . prim1 x5 (56ded.. x2) ⟶ ∀ x6 . prim1 x6 (56ded.. x3) ⟶ x0 x4 x5 x6.

Let x1 of type ι be given.

Assume H1: ordinal x1.

Assume H2: ∀ x2 . prim1 x2 x1 ⟶ ∀ x3 . ordinal x3 ⟶ ∀ x4 . ordinal x4 ⟶ ∀ x5 . prim1 x5 (56ded.. x2) ⟶ ∀ x6 . prim1 x6 (56ded.. x3) ⟶ ∀ x7 . prim1 x7 (56ded.. x4) ⟶ x0 x5 x6 x7.

Apply ordinal_ind with λ x2 . ∀ x3 . ordinal x3 ⟶ ∀ x4 . prim1 x4 (56ded.. x1) ⟶ ∀ x5 . prim1 x5 (56ded.. x2) ⟶ ∀ x6 . prim1 x6 (56ded.. x3) ⟶ x0 x4 x5 x6.

Let x2 of type ι be given.

Assume H3: ordinal x2.

Assume H4: ∀ x3 . prim1 ... ... ⟶ ∀ x4 . ordinal x4 ⟶ ∀ x5 . prim1 x5 (56ded.. x1) ⟶ ∀ x6 . prim1 x6 (56ded.. x3) ⟶ ∀ x7 . prim1 x7 (56ded.. x4) ⟶ x0 x5 x6 x7.

...

Apply unknownprop_569cb471be0809144fbd3507f3df6cc41b74b600f6043368ffb19c3a241db662 with x0.

The subproof is completed by applying L1.

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Proofgold Proof