We will prove add_nat 3 3 = 6.
We prove the intermediate claim H0nat: nat_p 0.
An exact proof term for the current goal is nat_0.
We prove the intermediate claim H1nat: nat_p 1.
An exact proof term for the current goal is nat_1.
We prove the intermediate claim H2nat: nat_p 2.
An exact proof term for the current goal is nat_2.
We prove the intermediate claim H3eq: 3 = ordsucc 2.
Use reflexivity.
rewrite the current goal using H3eq (from left to right) at position 1.
rewrite the current goal using (add_nat_SR 3 2 H2nat) (from left to right) at position 1.
rewrite the current goal using (add_nat_SR 3 1 H1nat) (from left to right) at position 1.
rewrite the current goal using (add_nat_SR 3 0 H0nat) (from left to right) at position 1.
rewrite the current goal using (add_nat_0R 3) (from left to right) at position 1.
Use reflexivity.