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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ιιι be given.
Let x2 of type ιι be given.
Let x3 of type ιο be given.
Assume H0: 0040b.. (1670d.. x0 x1 x2 x3).
Apply H0 with λ x4 . x4 = 1670d.. x0 x1 x2 x3∀ x5 . prim1 x5 x0prim1 (x2 x5) x0 leaving 2 subgoals.
Let x4 of type ι be given.
Let x5 of type ιιι be given.
Assume H1: ∀ x6 . prim1 x6 x4∀ x7 . prim1 x7 x4prim1 (x5 x6 x7) x4.
Let x6 of type ιι be given.
Assume H2: ∀ x7 . prim1 x7 x4prim1 (x6 x7) x4.
Let x7 of type ιο be given.
Assume H3: 1670d.. x4 x5 x6 x7 = 1670d.. x0 x1 x2 x3.
Apply unknownprop_abde0c95efb9c5a36a3b1b60fce4f97396bb4d91574381fb7896801234f8a172 with x4, x0, x5, x1, x6, x2, x7, x3, ∀ x8 . prim1 x8 x0prim1 (x2 x8) x0 leaving 2 subgoals.
The subproof is completed by applying H3.
Assume H4: and (and (x4 = x0) (∀ x8 . prim1 x8 x4∀ x9 . prim1 x9 x4x5 x8 x9 = x1 x8 x9)) (∀ x8 . prim1 x8 x4x6 x8 = x2 x8).
Apply H4 with (∀ x8 . prim1 x8 x4x7 x8 = x3 x8)∀ x8 . prim1 x8 x0prim1 (x2 x8) x0.
Assume H5: and (x4 = x0) (∀ x8 . prim1 x8 x4∀ x9 . prim1 x9 x4x5 x8 x9 = x1 x8 x9).
Apply H5 with (∀ x8 . prim1 x8 x4x6 x8 = x2 x8)(∀ x8 . prim1 x8 x4x7 x8 = x3 x8)∀ x8 . prim1 x8 x0prim1 (x2 x8) x0.
Assume H6: x4 = x0.
Assume H7: ∀ x8 . prim1 x8 x4∀ x9 . prim1 x9 x4x5 x8 x9 = x1 x8 x9.
Assume H8: ∀ x8 . prim1 x8 x4x6 x8 = x2 x8.
Assume H9: ∀ x8 . prim1 x8 x4x7 x8 = x3 x8.
Apply H6 with λ x8 x9 . ∀ x10 . prim1 x10 x8prim1 (x2 x10) x8.
Let x8 of type ι be given.
Assume H10: prim1 x8 x4.
Apply H8 with x8, λ x9 x10 . prim1 x9 x4 leaving 2 subgoals.
The subproof is completed by applying H10.
Apply H2 with x8.
The subproof is completed by applying H10.
Let x4 of type ιιο be given.
Assume H1: x4 (1670d.. x0 x1 x2 x3) (1670d.. x0 x1 x2 x3).
The subproof is completed by applying H1.