Proofgold Proof

pf

Let x0 of type ι be given.

Let x1 of type ι → ι be given.

Let x2 of type ι → ι be given.

Let x3 of type ι → ο be given.

Let x4 of type ι → ο be given.

Assume H0: ∀ x5 . prim1 x5 x0 ⟶ x1 x5 = x2 x5.

Assume H1: ∀ x5 . prim1 x5 x0 ⟶ iff (x3 x5) (x4 x5).

Claim L2: 0fc90.. x0 x1 = 0fc90.. x0 x2

Apply unknownprop_075a71193d04eff3936ee7246a228619e3dbe0ea2b9d96d40e9b467470ee4a92 with x0, x1, x2.

The subproof is completed by applying H0.

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

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

Let x5 of type ι → ι → ο be given.

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

The subproof is completed by applying H4.

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