Require Export GeoCoq.Tarski_dev.Ch06_out_lines.
(* In this file we prove that Col in Tarski neutral dimensionless is a Coinc_theory *)
Section Tarski_is_a_Coinc_theory_for_col.
Context `{TnEQD:Tarski_neutral_dimensionless_with_decidable_point_equality}.
Definition diff : arity Tpoint 2 := fun A B : Tpoint => A <> B.

(* Specifications: *)
Lemma diff_perm_1 : forall A B, app_1_n diff A B -> app_n_1 diff B A.
Lemma diff_perm_2 : forall A B (X : cartesianPower Tpoint 0), app_2_n diff A B X -> app_2_n diff B A X.
Definition col : arity Tpoint 3 := Col.
Lemma col_perm_1 : forall A (X : cartesianPower Tpoint 2), app_1_n col A X -> app_n_1 col X A.
Lemma col_perm_2 : forall A B (X : cartesianPower Tpoint 1), app_2_n col A B X -> app_2_n col B A X.
Lemma col_bd : forall A (X : cartesianPower Tpoint 1), app_2_n col A A X.
Lemma col_3 : forall (COL : cartesianPower Tpoint 3) (DIFF : cartesianPower Tpoint 2),
pred_conj col COL DIFF -> app diff DIFF -> app col COL.