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.

Assume H0: ∀ x6 : ι → ο . (∀ x7 . x6 x7 ⟶ prim1 x7 x0) ⟶ iff (x1 x6) (x2 x6).

Assume H1: ∀ x6 . prim1 x6 x0 ⟶ iff (x3 x6) (x4 x6).

Claim L2: e0e40.. x0 x1 = e0e40.. x0 x2

Apply unknownprop_35ee954b0de81ace4d484d57278ef6dea3fd2cb486e752fb5784dfc8cd9b7c4a with x0, x1, x2.

The subproof is completed by applying H0.

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

Claim L3: 1216a.. x0 x3 = 1216a.. x0 x4

Apply unknownprop_b3f0546588eaaffbfada457a1b6c239e49888b4c546cfd7ac96a6a43fd9db95e with x0, x3, x4.

The subproof is completed by applying H1.

Apply L3 with λ x6 x7 . 0fc90.. (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) x5))) = 0fc90.. (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..)) x6 x5))).

Let x6 of type ι → ι → ο be given.

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

The subproof is completed by applying H4.

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