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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Let x3 of type ι be given.
Let x4 of type ι be given.
Let x5 of type ι be given.
Let x6 of type ι be given.
Let x7 of type ι be given.
Assume H0: SNo x0.
Assume H1: SNo x1.
Assume H2: SNo x2.
Assume H3: SNo x3.
Assume H4: SNo x4.
Assume H5: SNo x5.
Assume H6: SNo x6.
Assume H7: SNo x7.
Assume H8: SNoLt (add_SNo x0 x5) (add_SNo x6 x4).
Assume H9: SNoLt x1 (add_SNo x7 x3).
Assume H10: SNoLt (add_SNo x6 x7) (add_SNo x2 x5).
Claim L11: ...
...
Claim L12: ...
...
Claim L13: ...
...
Claim L14: ...
...
Claim L15: ...
...
Apply add_SNo_Lt1_cancel with add_SNo x0 x1, x6, add_SNo x2 (add_SNo x3 x4) leaving 4 subgoals.
The subproof is completed by applying L12.
The subproof is completed by applying H6.
The subproof is completed by applying L13.
Apply SNoLt_tra with add_SNo (add_SNo x0 x1) x6, add_SNo x0 (add_SNo x2 (add_SNo x3 x5)), add_SNo (add_SNo x2 (add_SNo x3 x4)) x6 leaving 5 subgoals.
Apply SNo_add_SNo with add_SNo x0 x1, x6 leaving 2 subgoals.
The subproof is completed by applying L12.
The subproof is completed by applying H6.
Apply SNo_add_SNo with x0, add_SNo x2 (add_SNo x3 x5) leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying L14.
Apply SNo_add_SNo with add_SNo x2 (add_SNo x3 x4), x6 leaving 2 subgoals.
The subproof is completed by applying L13.
The subproof is completed by applying H6.
Apply add_SNo_assoc with x0, x1, x6, λ x8 x9 . SNoLt x8 (add_SNo x0 (add_SNo x2 (add_SNo x3 x5))) leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H6.
Apply add_SNo_com with x1, x6, λ x8 x9 . SNoLt (add_SNo x0 x9) (add_SNo x0 (add_SNo x2 (add_SNo x3 x5))) leaving 3 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H6.
Apply add_SNo_Lt2 with x0, add_SNo x6 x1, add_SNo x2 (add_SNo x3 x5) leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying L15.
The subproof is completed by applying L14.
The subproof is completed by applying L11.
Claim L16: ...
...
Claim L17: add_SNo (add_SNo x2 (add_SNo x3 x4)) x6 = add_SNo (add_SNo x2 x3) (add_SNo x6 x4)
set y8 to be ...
set y9 to be ...
Claim L17: ∀ x10 : ι → ο . x10 y9x10 y8
Let x10 of type ιο be given.
Assume H17: x10 (add_SNo (add_SNo x4 x5) (add_SNo y8 x6)).
set y11 to be ...
set y12 to be ...
Claim L18: ...
...
set y13 to be ...
Apply L18 with λ x14 . y13 x14 y12y13 y12 x14 leaving 2 subgoals.
Assume H19: y13 y12 y12.
The subproof is completed by applying H19.
set y14 to be ...
Apply add_SNo_assoc with add_SNo x7 y8, y9, y11, λ x15 x16 . y14 x16 x15 leaving 4 subgoals.
Apply SNo_add_SNo with x7, y8 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
The subproof is completed by applying H7.
set y15 to be ...
set y16 to be ...
Claim L19: ∀ x17 : ι → ο . x17 y16x17 y15
Let x17 of type ιο be given.
Assume H19: x17 (add_SNo ... ...).
...
set y17 to be λ x17 . y16
Apply L19 with λ x18 . y17 x18 y16y17 y16 x18 leaving 2 subgoals.
Assume H20: y17 y16 y16.
The subproof is completed by applying H20.
The subproof is completed by applying L19.
Let x10 of type ιιο be given.
Apply L17 with λ x11 . x10 x11 y9x10 y9 x11.
Assume H18: x10 y9 y9.
The subproof is completed by applying H18.
Apply L16 with λ x8 x9 . SNoLt x9 (add_SNo (add_SNo x2 (add_SNo x3 x4)) x6).
Apply L17 with λ x8 x9 . SNoLt (add_SNo (add_SNo x2 x3) (add_SNo x0 x5)) x9.
Apply add_SNo_Lt2 with add_SNo x2 x3, add_SNo x0 x5, add_SNo x6 x4 leaving 4 subgoals.
Apply SNo_add_SNo with x2, x3 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
Apply SNo_add_SNo with x0, x5 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H5.
Apply SNo_add_SNo with x6, x4 leaving 2 subgoals.
The subproof is completed by applying H6.
The subproof is completed by applying H4.
The subproof is completed by applying H8.