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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H0: x1 ∈ int.
Let x2 of type ι be given.
Assume H1: x2 ∈ int.
Let x3 of type ι be given.
Assume H2: x3 ∈ int.
Let x4 of type ι be given.
Assume H3: x4 ∈ int.
Let x5 of type ι be given.
Assume H4: x5 ∈ int.
Let x6 of type ι be given.
Assume H5: x6 ∈ int.
Let x7 of type ι be given.
Assume H6: x7 ∈ int.
Let x8 of type ι be given.
Assume H7: x8 ∈ int.
Assume H8: divides_int x0 (add_SNo x2 (minus_SNo x1)).
Assume H9: divides_int x0 (add_SNo x4 (minus_SNo x3)).
Assume H10: divides_int x0 (add_SNo x6 (minus_SNo x5)).
Assume H11: divides_int x0 (add_SNo x8 (minus_SNo x7)).
Assume H12: divides_int x0 (add_SNo x1 (add_SNo x3 (add_SNo x5 x7))).
Claim L13: SNo x6
Apply int_SNo with x6.
The subproof is completed by applying H5.
Claim L14: SNo x8
Apply int_SNo with x8.
The subproof is completed by applying H7.
Apply unknownprop_82d8b16cbabe15f33566315da037f391b292861be9631cc7d9815c42bac38696 with x0, x2, x1, add_SNo x4 (add_SNo x6 x8) leaving 5 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H0.
Apply unknownprop_a9dff3eed75edafaffb9658311eccbd582a4d2f793113dc0569b6d2e05bbb080 with x4, x6, x8 leaving 3 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H5.
The subproof is completed by applying H7.
The subproof is completed by applying H8.
Apply unknownprop_61887ed89638f3e8ae2bf6a2c384a905c1377bda9906e7801b339098548e1a07 with x0, x1, add_SNo x3 (add_SNo x5 x7), add_SNo x4 (add_SNo x6 x8) leaving 5 subgoals.
The subproof is completed by applying H0.
Apply unknownprop_a9dff3eed75edafaffb9658311eccbd582a4d2f793113dc0569b6d2e05bbb080 with x3, x5, x7 leaving 3 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H4.
The subproof is completed by applying H6.
Apply unknownprop_a9dff3eed75edafaffb9658311eccbd582a4d2f793113dc0569b6d2e05bbb080 with x4, x6, x8 leaving 3 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H5.
The subproof is completed by applying H7.
Apply minus_add_SNo_distr_3 with x4, x6, x8, λ x9 x10 . divides_int x0 (add_SNo (add_SNo x3 (add_SNo x5 x7)) x10) leaving 4 subgoals.
Apply int_SNo with x4.
The subproof is completed by applying H3.
The subproof is completed by applying L13.
The subproof is completed by applying L14.
Apply add_SNo_com_4_inner_mid with x3, add_SNo x5 x7, minus_SNo x4, add_SNo (minus_SNo x6) (minus_SNo x8), λ x9 x10 . divides_int x0 x10 leaving 5 subgoals.
Apply int_SNo with x3.
The subproof is completed by applying H2.
Apply SNo_add_SNo with x5, x7 leaving 2 subgoals.
Apply int_SNo with x5.
The subproof is completed by applying H4.
Apply int_SNo with x7.
The subproof is completed by applying H6.
Apply SNo_minus_SNo with x4.
Apply int_SNo with x4.
The subproof is completed by applying H3.
Apply SNo_add_SNo with minus_SNo x6, minus_SNo x8 leaving 2 subgoals.
Apply SNo_minus_SNo with x6.
The subproof is completed by applying L13.
Apply SNo_minus_SNo with x8.
The subproof is completed by applying L14.
Apply add_SNo_com_4_inner_mid with x5, x7, minus_SNo x6, minus_SNo x8, λ x9 x10 . divides_int x0 (add_SNo (add_SNo x3 (minus_SNo x4)) x10) leaving 5 subgoals.
Apply int_SNo with x5.
The subproof is completed by applying H4.
Apply int_SNo with x7.
The subproof is completed by applying H6.
Apply SNo_minus_SNo with x6.
The subproof is completed by applying L13.
Apply SNo_minus_SNo with x8.
The subproof is completed by applying L14.
Apply divides_int_add_SNo_3 with x0, add_SNo x3 (minus_SNo x4), add_SNo x5 (minus_SNo x6), add_SNo x7 (minus_SNo x8) leaving 3 subgoals.
Apply divides_int_diff_SNo_rev with x0, x4, x3 leaving 3 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H2.
The subproof is completed by applying H9.
Apply divides_int_diff_SNo_rev with x0, x6, x5 leaving 3 subgoals.
The subproof is completed by applying H5.
The subproof is completed by applying H4.
The subproof is completed by applying H10.
Apply divides_int_diff_SNo_rev with x0, x8, x7 leaving 3 subgoals.
The subproof is completed by applying H7.
The subproof is completed by applying H6.
The subproof is completed by applying H11.
The subproof is completed by applying H12.
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