We will prove 0 < one_third.
We prove the intermediate claim H3nat: nat_p 3.
An exact proof term for the current goal is (nat_ordsucc 2 nat_2).
We prove the intermediate claim H3omega: 3 ∈ ω.
An exact proof term for the current goal is (nat_p_omega 3 H3nat).
We prove the intermediate claim H3non0: 3 ∉ {0}.
Assume H3in: 3 ∈ {0}.
We prove the intermediate claim H30: 3 = 0.
An exact proof term for the current goal is (SingE 0 3 H3in).
We prove the intermediate claim H3ne0: 3 ≠ 0.
We prove the intermediate claim H3def: 3 = ordsucc 2.
Use reflexivity.
rewrite the current goal using H3def (from left to right).
An exact proof term for the current goal is (neq_ordsucc_0 2).
An exact proof term for the current goal is (H3ne0 H30).
We prove the intermediate claim H3in: 3 ∈ ω ∖ {0}.
An exact proof term for the current goal is (setminusI ω {0} 3 H3omega H3non0).
We prove the intermediate claim Hdef: one_third = inv_nat 3.
Use reflexivity.
rewrite the current goal using Hdef (from left to right).
We prove the intermediate claim H0ltR: Rlt 0 (inv_nat 3).
An exact proof term for the current goal is (inv_nat_pos 3 H3in).
An exact proof term for the current goal is (RltE_lt 0 (inv_nat 3) H0ltR).
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