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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 → ι be given.
Assume H1: ∀ x3 x4 . x1 x3 ⟶ x1 x4 ⟶ x1 (x2 x3 x4).
Assume H2: ∀ x3 x4 x5 . x1 x3 ⟶ x1 x4 ⟶ x1 x5 ⟶ x2 (x2 x3 x4) x5 = x2 x3 (x2 x4 x5).
Let x3 of type ι be given.
Let x4 of type ι be given.
Let x5 of type ι be given.
Assume H3: CD_carr x0 x1 x3.
Assume H4: CD_carr x0 x1 x4.
Assume H5: CD_carr x0 x1 x5.
Claim L6: ...
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Claim L7: ...
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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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Apply CD_proj0_2 with x0, x1, x2 (CD_proj0 x0 x1 x3) (CD_proj0 x0 x1 x4), x2 (CD_proj1 x0 x1 x3) (CD_proj1 x0 x1 x4), λ x6 x7 . pair_tag x0 (x2 x7 (CD_proj0 x0 x1 x5)) (x2 (CD_proj1 x0 x1 (pair_tag x0 (x2 (CD_proj0 x0 x1 x3) (CD_proj0 x0 x1 x4)) (x2 (CD_proj1 x0 x1 x3) (CD_proj1 x0 x1 x4)))) (CD_proj1 x0 x1 x5)) = pair_tag x0 (x2 (CD_proj0 x0 x1 x3) (CD_proj0 x0 x1 (pair_tag x0 (x2 (CD_proj0 x0 x1 x4) (CD_proj0 x0 x1 x5)) (x2 (CD_proj1 x0 x1 x4) (CD_proj1 x0 x1 x5))))) (x2 (CD_proj1 x0 x1 x3) (CD_proj1 x0 x1 (pair_tag x0 (x2 (CD_proj0 x0 x1 x4) (CD_proj0 x0 x1 x5)) (x2 (CD_proj1 x0 x1 x4) (CD_proj1 x0 x1 x5))))) leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying L12.
The subproof is completed by applying L13.
Apply CD_proj1_2 with x0, x1, x2 (CD_proj0 x0 x1 x3) (CD_proj0 x0 x1 x4), x2 (CD_proj1 x0 x1 x3) (CD_proj1 x0 x1 x4), λ x6 x7 . pair_tag x0 (x2 (x2 (CD_proj0 x0 x1 x3) (CD_proj0 x0 x1 x4)) (CD_proj0 x0 x1 x5)) (x2 x7 (CD_proj1 x0 x1 x5)) = pair_tag x0 (x2 (CD_proj0 x0 x1 x3) (CD_proj0 x0 x1 (pair_tag x0 (x2 (CD_proj0 x0 x1 x4) (CD_proj0 x0 x1 x5)) (x2 (CD_proj1 x0 x1 x4) (CD_proj1 x0 x1 x5))))) (x2 (CD_proj1 x0 x1 x3) (CD_proj1 x0 x1 (pair_tag x0 ... ...))) leaving 4 subgoals.
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