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Proofgold Proof

pf
Let x0 of type ι be given.
Assume H0: SNo x0.
Let x1 of type ι → ι be given.
Let x2 of type ι → ι be given.
Assume H1: ∀ x3 . x3 ∈ SNoS_ (SNoLev x0) ⟶ x1 x3 = x2 x3.
Apply nat_ind with λ x3 . SNo_recipaux x0 x1 x3 = SNo_recipaux x0 x2 x3 leaving 2 subgoals.
Apply SNo_recipaux_0 with x0, x2, λ x3 x4 . SNo_recipaux x0 x1 0 = x4.
The subproof is completed by applying SNo_recipaux_0 with x0, x1.
Let x3 of type ι be given.
Assume H2: nat_p x3.
Assume H3: SNo_recipaux x0 x1 x3 = SNo_recipaux x0 x2 x3.
Apply SNo_recipaux_S with x0, x1, x3, λ x4 x5 . x5 = SNo_recipaux x0 x2 (ordsucc x3) leaving 2 subgoals.
The subproof is completed by applying H2.
Apply SNo_recipaux_S with x0, x2, x3, λ x4 x5 . lam 2 (λ x6 . If_i (x6 = 0) (binunion (binunion (ap (SNo_recipaux x0 x1 x3) 0) (SNo_recipauxset (ap (SNo_recipaux x0 x1 x3) 0) x0 (SNoR x0) x1)) (SNo_recipauxset (ap (SNo_recipaux x0 x1 x3) 1) x0 (SNoL_pos x0) x1)) (binunion (binunion (ap (SNo_recipaux x0 x1 x3) 1) (SNo_recipauxset (ap (SNo_recipaux x0 x1 x3) 0) x0 (SNoL_pos x0) x1)) (SNo_recipauxset (ap (SNo_recipaux x0 x1 x3) 1) x0 (SNoR x0) x1))) = x5 leaving 2 subgoals.
The subproof is completed by applying H2.
Apply H3 with λ x4 x5 . lam 2 (λ x6 . If_i (x6 = 0) (binunion (binunion (ap x5 0) (SNo_recipauxset (ap x5 0) x0 (SNoR x0) x1)) (SNo_recipauxset (ap x5 1) x0 (SNoL_pos x0) x1)) (binunion (binunion (ap x5 1) (SNo_recipauxset (ap x5 0) x0 (SNoL_pos x0) x1)) (SNo_recipauxset (ap x5 1) x0 (SNoR x0) x1))) = lam 2 (λ x6 . If_i (x6 = 0) (binunion (binunion (ap (SNo_recipaux x0 x2 x3) 0) (SNo_recipauxset (ap (SNo_recipaux x0 x2 x3) 0) x0 (SNoR x0) x2)) (SNo_recipauxset (ap (SNo_recipaux x0 x2 x3) 1) x0 (SNoL_pos x0) x2)) (binunion (binunion (ap (SNo_recipaux x0 x2 x3) 1) (SNo_recipauxset (ap (SNo_recipaux x0 x2 x3) 0) x0 (SNoL_pos x0) x2)) (SNo_recipauxset (ap (SNo_recipaux x0 x2 x3) 1) x0 (SNoR x0) x2))).
set y4 to be ...
set y5 to be ...
Claim L4: ...
...
set y6 to be ...
Apply L4 with λ x7 . y6 x7 y5 ⟶ y6 y5 x7 leaving 2 subgoals.
Assume H5: y6 ... ....
...
...
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