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Proofgold Proof

pf
Assume H0: ∃ x0 . ∃ x1 : ι → ι . MetaCat_initial_p PtdPred UnaryPredHom struct_id struct_comp x0 x1.
Apply H0 with False.
Let x0 of type ι be given.
Assume H1: (λ x1 . ∃ x2 : ι → ι . MetaCat_initial_p PtdPred UnaryPredHom struct_id struct_comp x1 x2) x0.
Apply H1 with False.
Let x1 of type ι → ι be given.
Assume H2: MetaCat_initial_p PtdPred UnaryPredHom struct_id struct_comp x0 x1.
Apply H2 with False.
Assume H3: PtdPred x0.
Apply unknownprop_c86200a2eefb0ff844f50b29d5cbeaa2ee14856a2db63542bcbf63218f0d0f1e with x0, λ x2 . (∀ x3 . PtdPred x3 ⟶ and (UnaryPredHom x2 x3 (x1 x3)) (∀ x4 . UnaryPredHom x2 x3 x4 ⟶ x4 = x1 x3)) ⟶ False leaving 2 subgoals.
The subproof is completed by applying H3.
Let x2 of type ι be given.
Let x3 of type ι → ο be given.
Let x4 of type ι be given.
Assume H4: x4 ∈ x2.
Assume H5: x3 x4.
Assume H6: ∀ x5 . PtdPred x5 ⟶ and (UnaryPredHom (pack_p x2 x3) x5 (x1 x5)) (∀ x6 . UnaryPredHom (pack_p x2 x3) x5 x6 ⟶ x6 = x1 x5).
Claim L7: ...
...
Apply H6 with pack_p 2 (λ x5 . True), False leaving 2 subgoals.
The subproof is completed by applying L7.
Assume H8: UnaryPredHom (pack_p x2 x3) (pack_p 2 (λ x5 . True)) (x1 (pack_p 2 (λ x5 . True))).
Assume H9: ∀ x5 . UnaryPredHom (pack_p x2 x3) (pack_p 2 (λ x6 . True)) x5 ⟶ x5 = x1 (pack_p 2 (λ x6 . True)).
Claim L10: ...
...
Claim L11: ...
...
Apply neq_0_1.
set y5 to be 0
set y6 to be 1
Claim L12: ∀ x7 : ι → ο . x7 y6 ⟶ x7 y5
Let x7 of type ι → ο be given.
Assume H12: x7 1.
set y8 to be ...
Apply beta with x4, λ x9 . 0, y6, λ x9 x10 . y8 x10 x9 leaving 2 subgoals.
The subproof is completed by applying H4.
set y9 to be ...
set y10 to be ...
Claim L13: ∀ x11 : ι → ο . x11 y10 ⟶ x11 y9
Let x11 of type ι → ο be given.
Assume H13: x11 (ap (y6 (pack_p 2 (λ x12 . True))) y9).
set y12 to be ...
Apply H9 with lam x7 (λ x13 . 0), λ x13 x14 . y12 (ap ... ...) ... leaving 2 subgoals.
...
...
set y11 to be λ x11 . y10
Apply L13 with λ x12 . y11 x12 y10 ⟶ y11 y10 x12 leaving 2 subgoals.
Assume H14: y11 y10 y10.
The subproof is completed by applying H14.
set y12 to be ap (x7 (pack_p 2 (λ x12 . True))) y10
set y13 to be ap (lam y9 (λ x13 . 1)) y11
Claim L14: ∀ x14 : ι → ο . x14 y13 ⟶ x14 y12
Let x14 of type ι → ο be given.
Assume H14: x14 (ap (lam y10 (λ x15 . 1)) y12).
set y15 to be λ x15 . x14
set y16 to be λ x16 x17 . y15 (ap x16 y12) (ap x17 y12)
Apply L10 with lam y10 (λ x17 . 1), λ x17 x18 . y16 x18 x17 leaving 2 subgoals.
The subproof is completed by applying H12.
The subproof is completed by applying H14.
set y14 to be λ x14 . y13
Apply L14 with λ x15 . y14 x15 y13 ⟶ y14 y13 x15 leaving 2 subgoals.
Assume H15: y14 y13 y13.
The subproof is completed by applying H15.
Apply beta with y11, λ x15 . 1, y13, λ x15 . y14 leaving 2 subgoals.
The subproof is completed by applying H6.
The subproof is completed by applying L14.
Let x7 of type ι → ι → ο be given.
Apply L12 with λ x8 . x7 x8 y6 ⟶ x7 y6 x8.
Assume H13: x7 y6 y6.
The subproof is completed by applying H13.
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