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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.
Assume H0: SNo x0.
Assume H1: SNoLt 0 x0.
Assume H2: SNo x1.
Assume H3: SNo x2.
Assume H4: SNoLt x1 x2.
Claim L5: add_SNo (mul_SNo 0 x2) (mul_SNo x0 x1) = mul_SNo x0 x1
set y3 to be ...
set y4 to be mul_SNo ... ...
Claim L5: ∀ x5 : ι → ο . x5 y4 ⟶ x5 y3
Let x5 of type ι → ο be given.
Assume H5: x5 (mul_SNo x2 y3).
set y6 to be add_SNo (mul_SNo 0 y4) (mul_SNo x2 y3)
set y7 to be add_SNo 0 (mul_SNo y3 y4)
Claim L6: ∀ x8 : ι → ο . x8 y7 ⟶ x8 y6
Let x8 of type ι → ο be given.
Assume H6: x8 (add_SNo 0 (mul_SNo y4 x5)).
set y9 to be λ x9 . x8
Apply mul_SNo_zeroL with y6, λ x10 x11 . y9 (add_SNo x10 (mul_SNo y4 x5)) (add_SNo x11 (mul_SNo y4 x5)) leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H6.
set y8 to be λ x8 . y7
Apply L6 with λ x9 . y8 x9 y7 ⟶ y8 y7 x9 leaving 2 subgoals.
Assume H7: y8 y7 y7.
The subproof is completed by applying H7.
Apply add_SNo_0L with mul_SNo x5 y6, λ x9 . y8 leaving 2 subgoals.
Apply SNo_mul_SNo with x5, y6 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H3.
The subproof is completed by applying L6.
Let x5 of type ι → ι → ο be given.
Apply L5 with λ x6 . x5 x6 y4 ⟶ x5 y4 x6.
Assume H6: x5 y4 y4.
The subproof is completed by applying H6.
Claim L6: add_SNo (mul_SNo x0 x2) (mul_SNo 0 x1) = mul_SNo x0 x2
set y3 to be add_SNo (mul_SNo x0 x2) (mul_SNo 0 x1)
set y4 to be mul_SNo x1 y3
Claim L6: ∀ x5 : ι → ο . x5 y4 ⟶ x5 y3
Let x5 of type ι → ο be given.
Assume H6: x5 (mul_SNo x2 y4).
set y6 to be add_SNo (mul_SNo x2 y4) (mul_SNo 0 y3)
set y7 to be add_SNo (mul_SNo y3 x5) 0
Claim L7: ∀ x8 : ι → ο . x8 y7 ⟶ x8 y6
Let x8 of type ι → ο be given.
Assume H7: x8 (add_SNo (mul_SNo y4 y6) 0).
set y9 to be λ x9 . x8
Apply mul_SNo_zeroL with x5, λ x10 x11 . y9 (add_SNo (mul_SNo y4 y6) x10) (add_SNo (mul_SNo y4 y6) x11) leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H7.
set y8 to be λ x8 . y7
Apply L7 with λ x9 . y8 x9 y7 ⟶ y8 y7 x9 leaving 2 subgoals.
Assume H8: y8 y7 y7.
The subproof is completed by applying H8.
Apply add_SNo_0R with mul_SNo x5 y7, λ x9 . y8 leaving 2 subgoals.
Apply SNo_mul_SNo with x5, y7 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H4.
The subproof is completed by applying L7.
Let x5 of type ι → ι → ο be given.
Apply L6 with λ x6 . x5 x6 y4 ⟶ x5 y4 x6.
Assume H7: x5 y4 y4.
The subproof is completed by applying H7.
Apply L5 with λ x3 x4 . SNoLt x3 (mul_SNo x0 x2).
Apply L6 with λ x3 x4 . SNoLt (add_SNo (mul_SNo 0 x2) (mul_SNo x0 x1)) x3.
Apply mul_SNo_Lt with x0, x2, 0, x1 leaving 6 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H3.
The subproof is completed by applying SNo_0.
The subproof is completed by applying H2.
The subproof is completed by applying H1.
The subproof is completed by applying H4.
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