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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.
Assume H1: ∀ x5 . x1 x5 ⟶ x1 (x2 x5).
Assume H2: ∀ x5 . x1 x5 ⟶ x1 (x3 x5).
Assume H3: ∀ x5 x6 . x1 x5 ⟶ x1 x6 ⟶ x1 (x4 x5 x6).
Assume H4: ∀ x5 x6 . x1 x5 ⟶ x1 x6 ⟶ x2 (x4 x5 x6) = x4 (x2 x5) (x2 x6).
Assume H5: ∀ x5 x6 . x1 x5 ⟶ x1 x6 ⟶ x3 (x4 x5 x6) = x4 (x3 x5) (x3 x6).
Let x5 of type ι be given.
Let x6 of type ι be given.
Assume H6: CD_carr x0 x1 x5.
Assume H7: CD_carr x0 x1 x6.
Claim L8: ...
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Claim L9: ...
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Claim L10: ...
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Claim L11: ...
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Claim L12: ...
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Claim L13: ...
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Claim L14: ...
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Claim L15: ...
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Claim L16: ...
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Claim L17: ...
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Apply CD_proj0_2 with x0, x1, x4 (CD_proj0 x0 x1 x5) (CD_proj0 x0 x1 x6), x4 (CD_proj1 x0 x1 x5) (CD_proj1 x0 x1 x6), λ x7 x8 . pair_tag x0 (x3 x8) (x2 (CD_proj1 x0 x1 (pair_tag x0 (x4 (CD_proj0 x0 x1 x5) (CD_proj0 x0 x1 x6)) (x4 (CD_proj1 x0 x1 x5) (CD_proj1 x0 x1 x6))))) = pair_tag x0 (x4 (CD_proj0 x0 x1 (pair_tag x0 (x3 (CD_proj0 x0 x1 x5)) (x2 (CD_proj1 x0 x1 x5)))) (CD_proj0 x0 x1 (pair_tag x0 (x3 (CD_proj0 x0 x1 x6)) (x2 (CD_proj1 x0 x1 x6))))) (x4 (CD_proj1 x0 x1 (pair_tag x0 (x3 (CD_proj0 x0 x1 x5)) (x2 (CD_proj1 x0 x1 x5)))) (CD_proj1 x0 x1 (pair_tag x0 (x3 (CD_proj0 x0 x1 x6)) (x2 (CD_proj1 x0 x1 x6))))) leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying L16.
The subproof is completed by applying L17.
Apply CD_proj1_2 with x0, x1, x4 (CD_proj0 x0 x1 x5) (CD_proj0 x0 x1 x6), x4 (CD_proj1 x0 x1 x5) (CD_proj1 x0 x1 x6), λ x7 x8 . pair_tag x0 (x3 (x4 (CD_proj0 x0 x1 x5) (CD_proj0 x0 x1 x6))) (x2 x8) = pair_tag x0 (x4 (CD_proj0 x0 x1 (pair_tag x0 (x3 (CD_proj0 x0 x1 x5)) (x2 (CD_proj1 x0 x1 x5)))) (CD_proj0 x0 x1 (pair_tag x0 (x3 (CD_proj0 x0 x1 x6)) (x2 ...)))) ... leaving 4 subgoals.
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