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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.
Assume H0: SNo x0.
Assume H1: SNo x1.
Assume H2: SNo x2.
Assume H3: SNo x3.
Assume H4: SNo x4.
Assume H5: SNo x5.
Claim L6: ...
...
Claim L7: ...
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Claim L8: ...
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Claim L9: ...
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Claim L10: ...
...
Claim L11: ∀ x6 . SNo x6 ⟶ SNo (add_SNo x3 x6)
Let x6 of type ι be given.
Assume H11: SNo x6.
Apply SNo_add_SNo with x3, x6 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H11.
Claim L12: ∀ x6 . SNo x6 ⟶ SNo (add_SNo x3 (add_SNo x6 x5))
Let x6 of type ι be given.
Assume H12: SNo x6.
Apply L11 with add_SNo x6 x5.
Apply SNo_add_SNo with x6, x5 leaving 2 subgoals.
The subproof is completed by applying H12.
The subproof is completed by applying H5.
Claim L13: SNo (add_SNo x3 (add_SNo x4 x5))
Apply L12 with x4.
The subproof is completed by applying H4.
Assume H14: SNoLt (add_SNo x3 (add_SNo x4 x5)) 0.
Assume H15: SNoLe (add_SNo x1 (minus_SNo x0)) x3.
Assume H16: SNoLe (add_SNo x2 (minus_SNo x1)) x4.
Assume H17: SNoLe (add_SNo x0 (minus_SNo x2)) x5.
Apply idl_negcycle_2 with x0, add_SNo x1 x2, add_SNo x3 x2, add_SNo x5 (minus_SNo x1) leaving 7 subgoals.
The subproof is completed by applying H0.
Apply SNo_add_SNo with x1, x2 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Apply SNo_add_SNo with x3, x2 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H2.
Apply SNo_add_SNo with x5, minus_SNo x1 leaving 2 subgoals.
The subproof is completed by applying H5.
The subproof is completed by applying L7.
Apply add_SNo_com with x5, minus_SNo x1, λ x6 x7 . SNoLt (add_SNo (add_SNo x3 x2) x7) 0 leaving 3 subgoals.
The subproof is completed by applying H5.
The subproof is completed by applying L7.
Apply add_SNo_assoc with x3, x2, add_SNo (minus_SNo x1) x5, λ x6 x7 . SNoLt x6 0 leaving 4 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H2.
Apply SNo_add_SNo with minus_SNo x1, x5 leaving 2 subgoals.
The subproof is completed by applying L7.
The subproof is completed by applying H5.
Apply add_SNo_assoc with x2, minus_SNo x1, x5, λ x6 x7 . SNoLt (add_SNo x3 x7) 0 leaving 4 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying L7.
The subproof is completed by applying H5.
Apply SNoLeLt_tra with add_SNo x3 (add_SNo (add_SNo x2 (minus_SNo x1)) x5), add_SNo x3 (add_SNo x4 x5), 0 leaving 5 subgoals.
Apply L12 with add_SNo x2 (minus_SNo x1).
The subproof is completed by applying L9.
The subproof is completed by applying L13.
The subproof is completed by applying SNo_0.
Apply add_SNo_Le2 with x3, add_SNo (add_SNo x2 (minus_SNo x1)) x5, add_SNo x4 x5 leaving 4 subgoals.
The subproof is completed by applying H3.
Apply SNo_add_SNo with add_SNo x2 (minus_SNo x1), x5 leaving 2 subgoals.
The subproof is completed by applying L9.
The subproof is completed by applying H5.
The subproof is completed by applying L10.
Apply add_SNo_Le1 with add_SNo x2 (minus_SNo x1), x5, x4 leaving 4 subgoals.
The subproof is completed by applying L9.
The subproof is completed by applying H5.
The subproof is completed by applying H4.
The subproof is completed by applying H16.
The subproof is completed by applying H14.
Apply add_SNo_com_3b_1_2 with x1, x2, minus_SNo x0, λ x6 x7 . SNoLe x7 (add_SNo x3 x2) leaving 4 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
The subproof is completed by applying L6.
Apply add_SNo_Le1 with add_SNo x1 (minus_SNo x0), x2, x3 leaving 4 subgoals.
Apply SNo_add_SNo with x1, minus_SNo x0 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying L6.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
The subproof is completed by applying H15.
Apply add_SNo_com with x1, x2, λ x6 x7 . SNoLe (add_SNo x0 (minus_SNo x7)) (add_SNo x5 (minus_SNo x1)) leaving 3 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Apply minus_add_SNo_distr with x2, x1, λ x6 x7 . SNoLe (add_SNo x0 x7) (add_SNo x5 (minus_SNo x1)) leaving 3 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H1.
Apply add_SNo_assoc with x0, minus_SNo x2, minus_SNo x1, λ x6 x7 . SNoLe x7 (add_SNo x5 (minus_SNo x1)) leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying L8.
The subproof is completed by applying L7.
Apply add_SNo_Le1 with add_SNo x0 (minus_SNo x2), minus_SNo x1, x5 leaving 4 subgoals.
Apply SNo_add_SNo with x0, minus_SNo x2 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying L8.
The subproof is completed by applying L7.
The subproof is completed by applying H5.
The subproof is completed by applying H17.
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