some more work to factorize out uninteresting parts of the proof...

[helm.git] / helm / software / matita / contribs / dama / dama / models / q_support.ma

diff --git a/helm/software/matita/contribs/dama/dama/models/q_support.ma b/helm/software/matita/contribs/dama/dama/models/q_support.ma
index 82832c4bc64ee90cea02b82e867f4a321cc83328..9d73f7ab9b209fdfa887c3aed98764d3cd737265 100644 (file)
--- a/helm/software/matita/contribs/dama/dama/models/q_support.ma
+++ b/helm/software/matita/contribs/dama/dama/models/q_support.ma
@@ -33,6 +33,7 @@ axiom q_minus_r: ∀x,y. y + Qpos x = y - Qneg x.
 axiom q_plus_assoc: ∀x,y,z.x + (y + z) = x + y + z. 
 axiom q_elim_minus: ∀x,y.x - y = x + Qopp y.
 axiom q_elim_opp: ∀x,y.x - Qopp y = x + y.
+axiom q_minus_distrib:∀x,y,z:Q.x - (y + z) = x - y - z.
 
 (* order over Q *)
 axiom qlt : ℚ → ℚ → CProp.
@@ -68,6 +69,8 @@ axiom q_pos_lt_OQ: ∀x.OQ < Qpos x.
 axiom q_le_plus_trans: ∀x,y:Q. OQ ≤ x → OQ ≤ y → OQ ≤ x + y.
 axiom q_le_S: ∀x,y,z.OQ ≤ x → x + y ≤ z → y ≤ z.
 axiom q_eq_to_le: ∀x,y. x = y → x ≤ y.
+axiom q_le_OQ_Qpos: ∀x.OQ ≤ Qpos x.
+       
 
 inductive q_le_elimination (a,b:ℚ) : CProp ≝
 | q_le_from_eq : a = b → q_le_elimination a b
@@ -90,3 +93,33 @@ axiom q_d_sym:  ∀x,y. ⅆ[x,y] = ⅆ[y,x].
 (* integral part *)
 axiom nat_of_q: ℚ → nat.
 
+(* derived *)
+lemma q_lt_canc_plus_r:
+  ∀x,y,z:Q.x + z < y + z → x < y.
+intros; rewrite < (q_plus_OQ y); rewrite < (q_plus_minus z);
+rewrite > q_elim_minus; rewrite > q_plus_assoc;
+apply q_lt_plus; rewrite > q_elim_opp; assumption;
+qed.
+
+lemma q_lt_inj_plus_r:
+  ∀x,y,z:Q.x < y → x + z < y + z.
+intros; apply (q_lt_canc_plus_r ?? (Qopp z));
+do 2 (rewrite < q_plus_assoc;rewrite < q_elim_minus);
+rewrite > q_plus_minus;
+do 2 rewrite > q_plus_OQ; assumption;
+qed.
+
+lemma q_le_inj_plus_r:
+  ∀x,y,z:Q.x ≤ y → x + z ≤ y + z.
+intros;cases (q_le_cases ?? H);
+[1: rewrite > H1; apply q_eq_to_le; reflexivity;
+|2: apply q_lt_to_le; apply q_lt_inj_plus_r; assumption;]
+qed.
+
+lemma q_le_canc_plus_r:
+  ∀x,y,z:Q.x + z ≤ y + z → x ≤ y.
+intros; lapply (q_le_inj_plus_r ?? (Qopp z) H) as H1;
+do 2 rewrite < q_plus_assoc in H1;
+rewrite < q_elim_minus in H1; rewrite > q_plus_minus in H1;
+do 2 rewrite > q_plus_OQ in H1; assumption;
+qed.