Let x0 of type ι be given.

Let x1 of type ι be given.

Let x2 of type (ι → ο) → ο be given.

Let x3 of type (ι → ο) → ο be given.

Let x4 of type ι → ι → ι be given.

Let x5 of type ι → ι → ι be given.

Let x6 of type ι → ι → ο be given.

Let x7 of type ι → ι → ο be given.

Assume H0: dd9fd.. x0 x2 x4 x6 = dd9fd.. x1 x3 x5 x7.

Claim L1: x1 = f482f.. (dd9fd.. x0 x2 x4 x6) 4a7ef..

Apply unknownprop_b630500b59e94e3837ce5a8709e7dd4319757dbf2e976fd4f202d3a12a13d78b with dd9fd.. x0 x2 x4 x6, x1, x3, x5, x7.

The subproof is completed by applying H0.

Claim L2: x0 = x1

Apply L1 with λ x8 x9 . x0 = x9.

The subproof is completed by applying unknownprop_56935e6c9a24659b7d13f33246d8a8cc6556d0f02e1a4f33ac268613268a6aae with x0, x2, x4, x6.

Apply and4I with x0 = x1, ∀ x8 : ι → ο . (∀ x9 . x8 x9 ⟶ prim1 x9 x0) ⟶ x2 x8 = x3 x8, ∀ x8 . prim1 x8 x0 ⟶ ∀ x9 . prim1 x9 x0 ⟶ x4 x8 x9 = x5 x8 x9, ∀ x8 . prim1 x8 x0 ⟶ ∀ x9 . prim1 x9 x0 ⟶ x6 x8 x9 = x7 x8 x9 leaving 4 subgoals.

The subproof is completed by applying L2.

Let x8 of type ι → ο be given.

Assume H3: ∀ x9 . x8 x9 ⟶ prim1 x9 x0.

Apply unknownprop_f13df839665187840429cad3e5a5a45d9326857c44bdeb45a93d3db1f138f136 with x0, x2, x4, x6, x8, λ x9 x10 : ο . x10 = x3 x8 leaving 2 subgoals.

The subproof is completed by applying H3.

Claim L4: ∀ x9 . x8 x9 ⟶ prim1 x9 x1

Apply L2 with λ x9 x10 . ∀ x11 . x8 x11 ⟶ prim1 x11 x9.

The subproof is completed by applying H3.

Apply H0 with λ x9 x10 . decode_c (f482f.. x10 (4ae4a.. 4a7ef..)) x8 = x3 x8.

Let x9 of type ο → ο → ο be given.

Apply unknownprop_f13df839665187840429cad3e5a5a45d9326857c44bdeb45a93d3db1f138f136 with x1, x3, x5, x7, x8, λ x10 x11 : ο . x9 x11 x10.

The subproof is completed by applying L4.

Let x8 of type ι be given.

Assume H3: prim1 x8 x0.

Let x9 of type ι be given.

Assume H4: prim1 x9 x0.

Apply unknownprop_da8ba16eb91eec06703da10c3f76f6dae08fa0cf16544ffbb5d83f83bbdcc69f with x0, x2, x4, x6, x8, x9, λ x10 x11 . x11 = x5 x8 x9 leaving 3 subgoals.

The subproof is completed by applying H3.

The subproof is completed by applying H4.

Claim L5: prim1 x8 x1

Apply L2 with λ x10 x11 . prim1 x8 x10.

The subproof is completed by applying H3.

Claim L6: prim1 x9 x1

Apply L2 with λ x10 x11 . prim1 x9 x10.

The subproof is completed by applying H4.

Apply H0 with λ x10 x11 . e3162.. (f482f.. x11 (4ae4a.. (4ae4a.. 4a7ef..))) x8 x9 = x5 x8 x9.

Let x10 of type ι → ι → ο be given.

Apply unknownprop_da8ba16eb91eec06703da10c3f76f6dae08fa0cf16544ffbb5d83f83bbdcc69f with x1, x3, x5, x7, x8, x9, λ x11 x12 . x10 x12 x11 leaving 2 subgoals.

The subproof is completed by applying L5.

The subproof is completed by applying L6.

Let x8 of type ι be given.

Assume H3: prim1 x8 x0.

Let x9 of type ι be given.

Assume H4: prim1 x9 x0.

Apply unknownprop_d6b440500cdc8dabd67854ee8818cafaae46bbd38c8eda0b48535927f1a4607e with x0, x2, x4, x6, x8, x9, λ x10 x11 : ο . x11 = x7 x8 x9 leaving 3 subgoals.

The subproof is completed by applying H3.

The subproof is completed by applying H4.

Claim L5: prim1 x8 x1

Apply L2 with λ x10 x11 . prim1 x8 x10.

The subproof is completed by applying H3.

Claim L6: prim1 x9 x1

Apply L2 with λ x10 x11 . prim1 x9 x10.

The subproof is completed by applying H4.

Apply H0 with λ x10 x11 . 2b2e3.. (f482f.. x11 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x8 x9 = x7 x8 x9.

Let x10 of type ο → ο → ο be given.

Apply unknownprop_d6b440500cdc8dabd67854ee8818cafaae46bbd38c8eda0b48535927f1a4607e with x1, x3, x5, x7, x8, x9, λ x11 x12 : ο . x10 x12 x11 leaving 2 subgoals.

The subproof is completed by applying L5.

The subproof is completed by applying L6.

■

Proofgold Proof