proposition: Theorem addℝ_assoc_realizer : forall e, addℝ_assoc↓e ⊩ ∀x y z, Cauchy x → Cauchy y → Cauchy z → (x + (y + z) ≡ (x + y) + z)%Re.
proof: Proof. intro e. startn 17. apply addℝ_Cauchy_realizer. do 2 eexists. repeat split. startn 2. apply addℝ_Cauchy_realizer. find. eok. exists ε₁, ε₂. find. startn 1. apply addℝ_shift_realizer. exists x, y, z, ε₁, q₁. split; assumption. Qed.
imports: QArith Psatz ClassicalRealizability.Kbase ClassicalRealizability.Rationals ClassicalRealizability.Real_definitions ClassicalRealizability.Real_relations