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Proofgold Proof

pf
Let x0 of type ι be given.
Assume H0: nat_p x0.
Apply nat_ind with λ x1 . ∀ x2 : ι → ι . (∀ x3 . x3 ∈ x1 ⟶ x2 x3 ∈ ordsucc x0) ⟶ Sigma_nat x1 (λ x3 . mul_nat (x2 x3) (exp_nat (ordsucc x0) x3)) ∈ exp_nat (ordsucc x0) x1 leaving 2 subgoals.
Let x1 of type ι → ι be given.
Assume H1: ∀ x2 . x2 ∈ 0 ⟶ x1 x2 ∈ ordsucc x0.
Apply unknownprop_2ab88bc3cb73bc8bbab980b6b6fd9d920b44f54ff9f8140f78b2a17b00566385 with λ x2 . mul_nat (x1 x2) (exp_nat (ordsucc x0) x2), λ x2 x3 . x3 ∈ exp_nat (ordsucc x0) 0.
Apply nat_primrec_0 with 1, λ x2 x3 . mul_nat (ordsucc x0) x3, λ x2 x3 . 0 ∈ x3.
The subproof is completed by applying In_0_1.
Let x1 of type ι be given.
Assume H1: nat_p x1.
Assume H2: ∀ x2 : ι → ι . (∀ x3 . x3 ∈ x1 ⟶ x2 x3 ∈ ordsucc x0) ⟶ Sigma_nat x1 (λ x3 . mul_nat (x2 x3) (exp_nat (ordsucc x0) x3)) ∈ exp_nat (ordsucc x0) x1.
Let x2 of type ι → ι be given.
Assume H3: ∀ x3 . x3 ∈ ordsucc x1 ⟶ x2 x3 ∈ ordsucc x0.
Apply unknownprop_32bc7778cd02b216f957f8c5e9693c4b58b7ca04a4ca47b5f3adb522add7dd35 with x1, λ x3 . mul_nat (x2 x3) (exp_nat (ordsucc x0) x3), λ x3 x4 . x4 ∈ exp_nat (ordsucc x0) (ordsucc x1) leaving 2 subgoals.
The subproof is completed by applying H1.
Apply nat_primrec_S with 1, λ x3 x4 . mul_nat (ordsucc x0) x4, x1, λ x3 x4 . add_nat (Sigma_nat x1 (λ x5 . mul_nat (x2 x5) (exp_nat (ordsucc x0) x5))) (mul_nat (x2 x1) (exp_nat (ordsucc x0) x1)) ∈ x4 leaving 2 subgoals.
The subproof is completed by applying H1.
Claim L4: ...
...
Claim L5: ...
...
Apply mul_nat_SL with x0, exp_nat (ordsucc x0) x1, λ x3 x4 . add_nat (Sigma_nat x1 (λ x5 . mul_nat (x2 x5) (exp_nat (ordsucc x0) x5))) (mul_nat (x2 x1) (exp_nat (ordsucc x0) x1)) ∈ x4 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying L5.
Apply add_nat_com with mul_nat x0 (exp_nat (ordsucc x0) x1), exp_nat (ordsucc x0) x1, λ x3 x4 . add_nat (Sigma_nat x1 (λ x5 . mul_nat (x2 x5) (exp_nat (ordsucc x0) x5))) (mul_nat (x2 x1) (exp_nat (ordsucc x0) x1)) ∈ x4 leaving 3 subgoals.
Apply mul_nat_p with x0, exp_nat (ordsucc x0) x1 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying L5.
The subproof is completed by applying L5.
Claim L6: ...
...
Claim L7: ...
...
Claim L8: ...
...
Claim L9: ...
...
Claim L10: add_nat (exp_nat (ordsucc ...) ...) ... ⊆ ...
...
Apply L10 with add_nat (Sigma_nat x1 (λ x3 . mul_nat (x2 x3) (exp_nat (ordsucc x0) x3))) (mul_nat (x2 x1) (exp_nat (ordsucc x0) x1)).
Apply unknownprop_eeaa5555ccfaf9be2474522165cc658a4c21b3dcbef964c1d1aad1f792298727 with exp_nat (ordsucc x0) x1, Sigma_nat x1 (λ x3 . mul_nat (x2 x3) (exp_nat (ordsucc x0) x3)), mul_nat (x2 x1) (exp_nat (ordsucc x0) x1) leaving 3 subgoals.
The subproof is completed by applying L5.
The subproof is completed by applying L6.
Apply mul_nat_p with x2 x1, exp_nat (ordsucc x0) x1 leaving 2 subgoals.
The subproof is completed by applying L7.
The subproof is completed by applying L5.
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