Proofgold Proof

pf

Let x0 of type ι be given.

Let x1 of type ι → ι → ι be given.

Let x2 of type ι → ι → ι be given.

Let x3 of type ι → ι be given.

Let x4 of type ι → ι be given.

Let x5 of type ι be given.

Assume H0: ∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x1 x6 x7 = x2 x6 x7.

Assume H1: ∀ x6 . prim1 x6 x0 ⟶ x3 x6 = x4 x6.

Claim L2: eb53d.. x0 x1 = eb53d.. x0 x2

Apply unknownprop_dbb26cd650ab4f49eb7a56474ebdc3b4a46f79fd66ae7129c609d055a8e1a150 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..) (eb53d.. x0 x1) (If_i (x8 = 4ae4a.. (4ae4a.. 4a7ef..)) (0fc90.. 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..)) (0fc90.. x0 x4) x5))).

Claim L3: 0fc90.. x0 x3 = 0fc90.. x0 x4

Apply unknownprop_075a71193d04eff3936ee7246a228619e3dbe0ea2b9d96d40e9b467470ee4a92 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..) (eb53d.. x0 x1) (If_i (x8 = 4ae4a.. (4ae4a.. 4a7ef..)) (0fc90.. x0 x3) x5))) = 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (λ x8 . If_i (x8 = 4a7ef..) x0 (If_i (x8 = 4ae4a.. 4a7ef..) (eb53d.. 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..) (eb53d.. x0 x1) (If_i (x7 = 4ae4a.. (4ae4a.. 4a7ef..)) (0fc90.. x0 x3) x5)))) (0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (λ x7 . If_i (x7 = 4a7ef..) x0 (If_i (x7 = 4ae4a.. 4a7ef..) (eb53d.. x0 x1) (If_i (x7 = 4ae4a.. (4ae4a.. 4a7ef..)) (0fc90.. x0 x3) x5)))).

The subproof is completed by applying H4.

■