Proofgold Proof

pf

Let x0 of type ι be given.

Let x1 of type (ι → ο) → ο be given.

Let x2 of type (ι → ο) → ο be given.

Let x3 of type ι → ο be given.

Let x4 of type ι → ο be given.

Let x5 of type ι be given.

Let x6 of type ι be given.

Assume H0: ∀ x7 : ι → ο . (∀ x8 . x7 x8 ⟶ prim1 x8 x0) ⟶ iff (x1 x7) (x2 x7).

Assume H1: ∀ x7 . prim1 x7 x0 ⟶ iff (x3 x7) (x4 x7).

Claim L2: ...

...

Apply L2 with λ x7 x8 . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x9 . If_i (x9 = 4a7ef..) x0 (If_i (x9 = 4ae4a.. 4a7ef..) (e0e40.. x0 x1) (If_i (x9 = 4ae4a.. (4ae4a.. 4a7ef..)) (1216a.. x0 x3) (If_i (x9 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) x5 x6)))) = 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x9 . If_i (x9 = 4a7ef..) x0 (If_i (x9 = 4ae4a.. 4a7ef..) x7 (If_i (x9 = 4ae4a.. (4ae4a.. 4a7ef..)) (1216a.. x0 x4) (If_i (x9 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) x5 x6)))).

Claim L3: ...

...

Apply L3 with λ x7 x8 . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x9 . If_i (x9 = 4a7ef..) x0 (If_i (x9 = 4ae4a.. 4a7ef..) (e0e40.. x0 x1) (If_i (x9 = 4ae4a.. (4ae4a.. 4a7ef..)) (1216a.. x0 x3) (If_i (x9 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) x5 x6)))) = 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x9 . If_i (x9 = 4a7ef..) x0 (If_i (x9 = 4ae4a.. 4a7ef..) (e0e40.. x0 x1) (If_i (x9 = 4ae4a.. (4ae4a.. 4a7ef..)) x7 (If_i (x9 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) x5 x6)))).

Let x7 of type ι → ι → ο be given.

Assume H4: x7 (0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x8 . If_i (x8 = 4a7ef..) x0 (If_i (x8 = 4ae4a.. 4a7ef..) (e0e40.. x0 x1) (If_i (x8 = 4ae4a.. (4ae4a.. 4a7ef..)) (1216a.. x0 x3) (If_i (x8 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) x5 x6))))) (0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x8 . If_i (x8 = 4a7ef..) x0 (If_i (x8 = 4ae4a.. 4a7ef..) (e0e40.. x0 x1) (If_i (x8 = 4ae4a.. (4ae4a.. 4a7ef..)) (1216a.. x0 x3) (If_i (x8 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) x5 x6))))).

The subproof is completed by applying H4.

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