proposition: Theorem oppℝ_addℝ_realizer : forall e, oppℝ_addℝ↓e ⊩ ∀x, Cauchy x → (x + (- x) ≡ embed_ℚ_ℝ 0)%Re.
proof: Proof. intro e. start. apply (prop_guard Ht1). intros [_ ?]. subst. apply Ht0. eexists; split; [| eok]. Kmove. exists ε₁0, ε₂0. Ksolve; try now apply Qclt_le_weak. subst. apply (prop_subst_stack Hπ). clear_realizers. intro Hle. rewrite <- Qcplus_0_r at 1. apply Qcplus_le_compat. now replace (q₁0 + q₂ - 0) with (q₁0 - - q₂) by ring. ok. Qed.
imports: QArith Psatz ClassicalRealizability.Kbase ClassicalRealizability.Rationals ClassicalRealizability.Real_definitions ClassicalRealizability.Real_relations