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Proofgold Proof

pf
Let x0 of type ι(ιιι) → ιι be given.
Let x1 of type ι be given.
Let x2 of type ιι be given.
Assume H0: In_rec_G_ii x0 x1 x2.
Apply H0 with λ x3 . λ x4 : ι → ι . ∃ x5 : ι → ι → ι . and (∀ x6 . x6x3In_rec_G_ii x0 x6 (x5 x6)) (x4 = x0 x3 x5).
Let x3 of type ι be given.
Let x4 of type ιιι be given.
Assume H1: ∀ x5 . x5x3∃ x6 : ι → ι → ι . and (∀ x7 . x7x5In_rec_G_ii x0 x7 (x6 x7)) (x4 x5 = x0 x5 x6).
Let x5 of type ο be given.
Assume H2: ∀ x6 : ι → ι → ι . and (∀ x7 . x7x3In_rec_G_ii x0 x7 (x6 x7)) (x0 x3 x4 = x0 x3 x6)x5.
Apply H2 with x4.
Apply andI with ∀ x6 . x6x3In_rec_G_ii x0 x6 (x4 x6), x0 x3 x4 = x0 x3 x4 leaving 2 subgoals.
Let x6 of type ι be given.
Assume H3: x6x3.
Apply exandE_iii with λ x7 : ι → ι → ι . ∀ x8 . x8x6In_rec_G_ii x0 x8 (x7 x8), λ x7 : ι → ι → ι . x4 x6 = x0 x6 x7, In_rec_G_ii x0 x6 (x4 x6) leaving 2 subgoals.
Apply H1 with x6.
The subproof is completed by applying H3.
Let x7 of type ιιι be given.
Assume H4: ∀ x8 . x8x6In_rec_G_ii x0 x8 (x7 x8).
Assume H5: x4 x6 = x0 x6 x7.
Apply H5 with λ x8 x9 : ι → ι . In_rec_G_ii x0 x6 x9.
Apply In_rec_G_ii_c with x0, x6, x7.
The subproof is completed by applying H4.
Let x6 of type (ιι) → (ιι) → ο be given.
Assume H3: x6 (x0 x3 x4) (x0 x3 x4).
The subproof is completed by applying H3.