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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ι → ο be given.
Assume H0: ∀ x2 . x1 x2 ⟶ ∀ x3 . x3 ∈ x2 ⟶ nIn x0 x3.
Let x2 of type ι → ι be given.
Let x3 of type ι → ι be given.
Let x4 of type ι → ι → ι be given.
Let x5 of type ι → ι → ι be given.
Assume H1: ∀ x6 . x1 x6 ⟶ x1 (x2 x6).
Assume H2: ∀ x6 . x1 x6 ⟶ x1 (x3 x6).
Assume H3: ∀ x6 x7 . x1 x6 ⟶ x1 x7 ⟶ x1 (x4 x6 x7).
Assume H4: ∀ x6 x7 . x1 x6 ⟶ x1 x7 ⟶ x1 (x5 x6 x7).
Assume H5: ∀ x6 x7 . x1 x6 ⟶ x1 x7 ⟶ x2 (x4 x6 x7) = x4 (x2 x6) (x2 x7).
Assume H6: ∀ x6 x7 x8 . x1 x6 ⟶ x1 x7 ⟶ x1 x8 ⟶ x4 (x4 x6 x7) x8 = x4 x6 (x4 x7 x8).
Assume H7: ∀ x6 x7 . x1 x6 ⟶ x1 x7 ⟶ x4 x6 x7 = x4 x7 x6.
Assume H8: ∀ x6 x7 x8 . x1 x6 ⟶ x1 x7 ⟶ x1 x8 ⟶ x5 x6 (x4 x7 x8) = x4 (x5 x6 x7) (x5 x6 x8).
Assume H9: ∀ x6 x7 x8 . x1 x6 ⟶ x1 x7 ⟶ x1 x8 ⟶ x5 (x4 x6 x7) x8 = x4 (x5 x6 x8) (x5 x7 x8).
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Let x6 of type ι be given.
Let x7 of type ι be given.
Let x8 of type ι be given.
Assume H11: CD_carr x0 x1 x6.
Assume H12: CD_carr x0 x1 x7.
Assume H13: CD_carr x0 x1 x8.
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Apply CD_proj0_2 with x0, x1, x4 (CD_proj0 x0 x1 x6) (CD_proj0 x0 x1 x7), x4 (CD_proj1 x0 x1 x6) (CD_proj1 x0 x1 x7), λ x9 x10 . pair_tag x0 (x4 (x5 x10 (CD_proj0 x0 x1 x8)) (x2 (x5 (x3 (CD_proj1 x0 x1 x8)) (CD_proj1 x0 x1 (pair_tag x0 (x4 (CD_proj0 x0 x1 x6) (CD_proj0 x0 x1 x7)) (x4 (CD_proj1 x0 x1 x6) (CD_proj1 x0 x1 x7))))))) (x4 (x5 (CD_proj1 x0 x1 x8) x10) (x5 (CD_proj1 x0 x1 (pair_tag x0 (x4 (CD_proj0 x0 x1 x6) (CD_proj0 x0 x1 x7)) (x4 (CD_proj1 x0 x1 x6) (CD_proj1 x0 x1 x7)))) (x3 (CD_proj0 x0 x1 x8)))) = pair_tag x0 (x4 (CD_proj0 x0 x1 (pair_tag x0 (x4 (x5 (CD_proj0 ... ... ...) ...) ...) ...)) ...) ... leaving 4 subgoals.
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