proposition: Lemma one_ap_zero : ([1]:R) [#] [0].
proof: Proof. Proof ring_non_triv R. Definition is_zero_rht S (op : CSetoid_bin_op S) Zero : Prop := forall x, op x Zero [=] Zero. Definition is_zero_lft S (op : CSetoid_bin_op S) Zero : Prop := forall x, op Zero x [=] Zero. Arguments is_zero_rht [S]. Arguments is_zero_lft [S]. Lemma cring_mult_zero : forall x : R, x[*][0] [=] [0]. Proof. intro x. apply cg_cancel_lft with (x[*][0]). astepr (x[*][0]). Step_final (x[*] ([0][+][0])). Qed.
imports: CoRN.tactics.CornTac CoRN.algebra.CSums CoRN.algebra.CAbMonoids Coq.Sorting.Permutation CoRN.util.SetoidPermutation Coq.Setoids.Setoid Coq.Classes.Morphisms Coq(Lia)