Let x and y be given.
Assume Hx.
Apply Hx to the current goal.
Let k be given.
Assume H.
Apply H to the current goal.
We prove the intermediate
claim Lek:
SNo (eps_ k).
An
exact proof term for the current goal is
SNo_eps_ k Hk.
Assume H.
Apply H to the current goal.
Let m be given.
Assume H.
Apply H to the current goal.
We prove the intermediate
claim Lm:
SNo m.
An
exact proof term for the current goal is
int_SNo m Hm.
We prove the intermediate
claim Lekm:
SNo (eps_ k * m).
Assume Hy.
Apply Hy to the current goal.
Let l be given.
Assume H.
Apply H to the current goal.
We prove the intermediate
claim Lel:
SNo (eps_ l).
An
exact proof term for the current goal is
SNo_eps_ l Hl.
Assume H.
Apply H to the current goal.
Let n be given.
Assume H.
Apply H to the current goal.
We prove the intermediate
claim Ln:
SNo n.
An
exact proof term for the current goal is
int_SNo n Hn.
We prove the intermediate
claim Leln:
SNo (eps_ l * n).
We use
(k + l) to
witness the existential quantifier.
Apply andI to the current goal.
We use
(2 ^ l * m + 2 ^ k * n) to
witness the existential quantifier.
We prove the intermediate
claim L2l:
2 ^ l ∈ int.
We prove the intermediate
claim L2lm:
2 ^ l * m ∈ int.
An exact proof term for the current goal is L2l.
An exact proof term for the current goal is Hm.
We prove the intermediate
claim L2k:
2 ^ k ∈ int.
We prove the intermediate
claim L2kn:
2 ^ k * n ∈ int.
An exact proof term for the current goal is L2k.
An exact proof term for the current goal is Hn.
Apply andI to the current goal.
An exact proof term for the current goal is L2lm.
An exact proof term for the current goal is L2kn.
Use f_equal.
An exact proof term for the current goal is Hxkm.
An exact proof term for the current goal is Hyln.
∎