basic properties of cpr ...

[helm.git] / matita / matita / contribs / lambdadelta / ground_2 / lib / arith.ma

diff --git a/matita/matita/contribs/lambdadelta/ground_2/lib/arith.ma b/matita/matita/contribs/lambdadelta/ground_2/lib/arith.ma
index 1ba525475622dfe3fb6777c4cbe59847a4584a1d..83322f4d8de57107b0eeb097bb7cd96d3cd83d86 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground_2/lib/arith.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/lib/arith.ma
@@ -23,6 +23,10 @@ interpretation "nat successor" 'Successor m = (S m).
 
 interpretation "nat predecessor" 'Predecessor m = (pred m).
 
+interpretation "nat min" 'and x y = (min x y).
+
+interpretation "nat max" 'or x y = (max x y).
+
 (* Iota equations ***********************************************************)
 
 lemma pred_O: pred 0 = 0.
@@ -31,8 +35,25 @@ normalize // qed.
 lemma pred_S: ∀m. pred (S m) = m.
 // qed.
 
+lemma plus_S1: ∀x,y. ⫯(x+y) = (⫯x) + y.
+// qed.
+
+lemma max_O1: ∀n. n = (0 ∨ n).
+// qed.
+
+lemma max_O2: ∀n. n = (n ∨ 0).
+// qed.
+
+lemma max_SS: ∀n1,n2. ⫯(n1∨n2) = (⫯n1 ∨ ⫯n2).
+#n1 #n2 elim (decidable_le n1 n2) #H normalize
+[ >(le_to_leb_true … H) | >(not_le_to_leb_false … H) ] -H //
+qed.
+
 (* Equations ****************************************************************)
 
+lemma plus_SO: ∀n. n + 1 = ⫯n.
+// qed.
+
 lemma minus_plus_m_m_commutative: ∀n,m:nat. n = m + n - m.
 // qed-.
 
@@ -112,6 +133,12 @@ lemma lt_S_S: ∀x,y. x < y → ⫯x < ⫯y.
 lemma lt_S: ∀n,m. n < m → n < ⫯m.
 /2 width=1 by le_S/ qed.
 
+lemma max_S1_le_S: ∀n1,n2,n. (n1 ∨ n2) ≤ n → (⫯n1 ∨ n2) ≤ ⫯n.
+/4 width=2 by to_max, le_maxr, le_S_S, le_S/ qed-.
+
+lemma max_S2_le_S: ∀n1,n2,n. (n1 ∨ n2) ≤ n → (n1 ∨ ⫯n2) ≤ ⫯n.
+/2 width=1 by max_S1_le_S/ qed-.
+
 lemma arith_j: ∀x,y,z. x-y-1 ≤ x-(y-z)-1.
 /3 width=1 by monotonic_le_minus_l, monotonic_le_minus_r/ qed.
 
@@ -131,9 +158,19 @@ qed.
 
 (* Inversion & forward lemmas ***********************************************)
 
+lemma max_inv_O3: ∀x,y. (x ∨ y) = 0 → 0 = x ∧ 0 = y.
+/4 width=2 by le_maxr, le_maxl, le_n_O_to_eq, conj/
+qed-.
+
+lemma plus_inv_O3: ∀x,y. x + y = 0 → x = 0 ∧ y = 0.
+/2 width=1 by plus_le_0/ qed-.
+
 lemma discr_plus_xy_y: ∀x,y. x + y = y → x = 0.
 // qed-.
 
+lemma discr_plus_x_xy: ∀x,y. x = x + y → y = 0.
+/2 width=2 by le_plus_minus_comm/ qed-.
+
 lemma lt_plus_SO_to_le: ∀x,y. x < y + 1 → x ≤ y.
 /2 width=1 by monotonic_pred/ qed-.
 
@@ -148,6 +185,10 @@ qed-.
 lemma lt_le_false: ∀x,y. x < y → y ≤ x → ⊥.
 /3 width=4 by lt_refl_false, lt_to_le_to_lt/ qed-.
 
+lemma succ_inv_refl_sn: ∀x. ⫯x = x → ⊥.
+#x #H @(lt_le_false x (⫯x)) //
+qed-.
+
 lemma lt_inv_O1: ∀n. 0 < n → ∃m. ⫯m = n.
 * /2 width=2 by ex_intro/
 #H cases (lt_le_false … H) -H //