We will prove Rlt (minus_SNo one_third) one_third.
We prove the intermediate claim H13R: one_third ∈ R.
An exact proof term for the current goal is one_third_in_R.
We prove the intermediate claim H0lt13: Rlt 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).
An exact proof term for the current goal is (inv_nat_pos 3 H3in).
We prove the intermediate claim Hlt13: 0 < one_third.
An exact proof term for the current goal is (RltE_lt 0 one_third H0lt13).
We prove the intermediate claim H13S: SNo one_third.
An exact proof term for the current goal is (real_SNo one_third H13R).
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
We prove the intermediate claim Hm13S: SNo (minus_SNo one_third).
An exact proof term for the current goal is (real_SNo (minus_SNo one_third) Hm13R).
We prove the intermediate claim Hm13lt0: (minus_SNo one_third) < 0.
We will prove (minus_SNo one_third) < 0.
rewrite the current goal using minus_SNo_0 (from right to left) at position 1.
An exact proof term for the current goal is (minus_SNo_Lt_contra 0 one_third SNo_0 H13S Hlt13).
We prove the intermediate claim Hm13lt13: (minus_SNo one_third) < one_third.
An exact proof term for the current goal is (SNoLt_tra (minus_SNo one_third) 0 one_third Hm13S SNo_0 H13S Hm13lt0 Hlt13).
An exact proof term for the current goal is (RltI (minus_SNo one_third) one_third Hm13R H13R Hm13lt13).
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