Let x0 of type ι be given.
Apply unknownprop_e9c4cec7fb327dcb17b88acdaf76daee024e49fa71834a13065f86e12e958609 with
2,
x0,
λ x1 x2 . x2 = add_nat (exp_nat 2 x0) (exp_nat 2 x0) leaving 2 subgoals.
The subproof is completed by applying H0.
Apply add_nat_1_1_2 with
λ x1 x2 . mul_nat x1 (exp_nat 2 x0) = add_nat (exp_nat 2 x0) (exp_nat 2 x0).
Apply mul_add_nat_distrR with
1,
1,
exp_nat 2 x0,
λ x1 x2 . x2 = add_nat (exp_nat 2 x0) (exp_nat 2 x0) leaving 4 subgoals.
The subproof is completed by applying nat_1.
The subproof is completed by applying nat_1.
Apply unknownprop_1b73e01bc234ba0e318bc9baf16604d4f0fc12bea57885439d70438de4eb7174 with
2,
x0 leaving 2 subgoals.
The subproof is completed by applying nat_2.
The subproof is completed by applying H0.
Apply unknownprop_6e31f7e63da1d74f4ea3ef967162bc5821029ffe1e451b13664a6e51a570dea7 with
exp_nat 2 x0,
λ x1 x2 . add_nat x2 x2 = add_nat (exp_nat 2 x0) (exp_nat 2 x0) leaving 2 subgoals.
Apply unknownprop_1b73e01bc234ba0e318bc9baf16604d4f0fc12bea57885439d70438de4eb7174 with
2,
x0 leaving 2 subgoals.
The subproof is completed by applying nat_2.
The subproof is completed by applying H0.
Let x1 of type ι → ι → ο be given.
The subproof is completed by applying H1.