arithmetics for λδ

[helm.git] / matita / matita / contribs / lambdadelta / ground / arith / nat_le.ma

diff --git a/matita/matita/contribs/lambdadelta/ground/arith/nat_le.ma b/matita/matita/contribs/lambdadelta/ground/arith/nat_le.ma
index 82d285e8ec7bf2bdd310ef18f718ca01192a9781..5f2ea334fa90b1490002674d934231feb5242851 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground/arith/nat_le.ma
+++ b/matita/matita/contribs/lambdadelta/ground/arith/nat_le.ma
@@ -15,7 +15,7 @@
 include "ground/insert_eq/insert_eq_0.ma".
 include "ground/arith/nat_succ.ma".
 
-(* NON-NEGATIVE INTEGERS ****************************************************)
+(* ORDER FOR NON-NEGATIVE INTEGERS ******************************************)
 
 (*** le *)
 (*** le_ind *)
@@ -36,7 +36,7 @@ lemma nle_succ_dx_refl (m): m ≤ ↑m.
 
 (*** le_O_n *)
 lemma nle_zero_sx (m): 𝟎 ≤ m.
-#m @(nat_ind … m) -m /2 width=1 by nle_succ_dx/
+#m @(nat_ind_succ … m) -m /2 width=1 by nle_succ_dx/
 qed.
 
 (*** le_S_S *)
@@ -45,10 +45,12 @@ lemma nle_succ_bi (m) (n): m ≤ n → ↑m ≤ ↑n.
 qed.
 
 (*** le_or_ge *)
-lemma nle_ge_e (m) (n): ∨∨ m ≤ n | n ≤ m.
-#m @(nat_ind … m) -m [ /2 width=1 by or_introl/ ]
-#m #IH #n @(nat_ind … n) -n [ /2 width=1 by or_intror/ ]
-#n #_ elim (IH n) -IH /3 width=2 by nle_succ_bi, or_introl, or_intror/
+lemma nle_ge_dis (m) (n): ∨∨ m ≤ n | n ≤ m.
+#m #n @(nat_ind_succ_2 … m n) -m -n
+[ /2 width=1 by or_introl/
+| /2 width=1 by or_intror/
+| #m #n * /3 width=2 by nle_succ_bi, or_introl, or_intror/
+]
 qed-.
 
 (* Basic inversions *********************************************************)
@@ -75,15 +77,18 @@ lemma nle_inv_zero_dx (m): m ≤ 𝟎 → 𝟎 = m.
 ]
 qed-.
 
+(* Advanced inversions ******************************************************)
+
+lemma nle_inv_succ_zero (m): ↑m ≤ 𝟎 → ⊥.
+/3 width=2 by nle_inv_zero_dx, eq_inv_nzero_succ/ qed-.
+
 lemma nle_inv_succ_sn_refl (m): ↑m ≤ m → ⊥.
-#m @(nat_ind … m) -m [| #m #IH ] #H
+#m @(nat_ind_succ … m) -m [| #m #IH ] #H
 [ /3 width=2 by nle_inv_zero_dx, eq_inv_nzero_succ/
 | /3 width=1 by nle_inv_succ_bi/
 ]
 qed-.
 
-(* Order properties *********************************************************)
-
 (*** le_to_le_to_eq *)
 theorem nle_antisym (m) (n): m ≤ n → n ≤ m → m = n.
 #m #n #H elim H -n //
@@ -93,16 +98,29 @@ lapply (IH H) -IH -H #H destruct
 elim (nle_inv_succ_sn_refl … Hn)
 qed-.
 
+(* Advanced eliminations ****************************************************)
+
+(*** le_elim *)
+lemma nle_ind_alt (Q: relation2 nat nat):
+      (∀n. Q (𝟎) (n)) →
+      (∀m,n. m ≤ n → Q m n → Q (↑m) (↑n)) →
+      ∀m,n. m ≤ n → Q m n.
+#Q #IH1 #IH2 #m #n @(nat_ind_succ_2 … m n) -m -n //
+[ #m #H elim (nle_inv_succ_zero … H)
+| /4 width=1 by nle_inv_succ_bi/
+]
+qed-.
+
+(* Advanced constructions ***************************************************)
+
 (*** transitive_le *)
 theorem nle_trans: Transitive … nle.
 #m #n #H elim H -n /3 width=1 by nle_inv_succ_sn/
 qed-.
 
-(* Advanced constructions ***************************************************)
-
 (*** decidable_le *)
 lemma nle_dec (m) (n): Decidable … (m ≤ n).
-#m #n elim (nle_ge_e m n) [ /2 width=1 by or_introl/ ]
+#m #n elim (nle_ge_dis m n) [ /2 width=1 by or_introl/ ]
 #Hnm elim (eq_nat_dec m n) [ #H destruct /2 width=1 by nle_refl, or_introl/ ]
 /4 width=1 by nle_antisym, or_intror/
 qed-.