update in ground

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

diff --git a/matita/matita/contribs/lambdadelta/ground/arith/nat_plus.ma b/matita/matita/contribs/lambdadelta/ground/arith/nat_plus.ma
index 75349d2da6281f130d0e2a0b3ea8930e379267f0..a9894fc6285e7aaf7ec746ddfebacb208b91885e 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground/arith/nat_plus.ma
+++ b/matita/matita/contribs/lambdadelta/ground/arith/nat_plus.ma
@@ -18,55 +18,69 @@ include "ground/arith/nat_succ_iter.ma".
 
 (*** plus *)
 definition nplus: nat → nat → nat ≝
-           λm,n. nsucc^n m.
+           λm,n. (nsucc^n) m.
 
 interpretation
-  "plus (positive integers"
+  "plus (non-negative integers)"
   'plus m n = (nplus m n).
 
-(* Basic rewrites ***********************************************************)
+(* Basic constructions ******************************************************)
 
 (*** plus_n_O *)
 lemma nplus_zero_dx (m): m = m + 𝟎.
 // qed.
 
-lemma nplus_one_dx (n): ↑n = n + 𝟏.
+(*** plus_SO_dx *)
+lemma nplus_unit_dx (n): ↑n = n + 𝟏.
 // qed.
 
-(* Advanved rewrites (semigroup properties) *********************************)
-
 (*** plus_n_Sm *)
 lemma nplus_succ_dx (m) (n): ↑(m+n) = m + ↑n.
 #m #n @(niter_succ … nsucc)
 qed.
 
+(* Constructions with niter *************************************************)
+
+(*** iter_plus *)
+lemma niter_plus (A) (f) (n1) (n2):
+      f^n2 ∘ f^n1 ≐ f^{A}(n1+n2).
+#A #f #n1 #n2 @(nat_ind_succ … n2) -n2 //
+#n2 #IH <nplus_succ_dx
+@exteq_repl
+/3 width=5 by compose_repl_fwd_sn, compose_repl_fwd_dx/
+qed.
+
+(* Advanced constructions (semigroup properties) ****************************)
+
+(*** plus_S1 *)
 lemma nplus_succ_sn (m) (n): ↑(m+n) = ↑m + n.
 #m #n @(niter_appl … nsucc)
 qed.
 
-(*** plus_O_n.con *)
+(*** plus_O_n *)
 lemma nplus_zero_sn (m): m = 𝟎 + m.
-#m @(nat_ind … m) -m //
+#m @(nat_ind_succ … m) -m //
 qed.
 
 (*** commutative_plus *)
 lemma nplus_comm: commutative … nplus.
-#m @(nat_ind … m) -m //
-qed-.
+#m @(nat_ind_succ … m) -m //
+qed-. (* * gets in the way with auto *)
 
 (*** associative_plus *)
 lemma nplus_assoc: associative … nplus.
-#m #n #o @(nat_ind … o) -o //
+#m #n #o @(nat_ind_succ … o) -o //
 #o #IH <nplus_succ_dx <nplus_succ_dx <nplus_succ_dx <IH -IH //
 qed.
 
-(* Advanced constructions ***************************************************)
+(* Helper constructions *****************************************************)
 
-lemma nplus_one_sn (n): ↑n = 𝟏 + n.
+(*** plus_SO_sn *)
+lemma nplus_unit_sn (n): ↑n = 𝟏 + n.
 #n <nplus_comm // qed.
 
 lemma nplus_succ_shift (m) (n): ↑m + n = m + ↑n.
-// qed-.
+// qed.
 
 (*** assoc_plus1 *)
 lemma nplus_plus_comm_12 (o) (m) (n): m + n + o = n + (m + o).
@@ -75,19 +89,27 @@ lemma nplus_plus_comm_12 (o) (m) (n): m + n + o = n + (m + o).
 (*** plus_plus_comm_23 *)
 lemma nplus_plus_comm_23 (o) (m) (n): o + m + n = o + n + m.
 #o #m #n >nplus_assoc >nplus_assoc <nplus_comm in ⊢ (??(??%)?); //
-qed-.
+qed.
 
 (* Basic inversions *********************************************************)
 
-lemma eq_inv_nzero_plus (m) (n): 𝟎 = m + n → ∧∧ 𝟎 = m & 𝟎 = n.
-#m #n @(nat_ind … n) -n /2 width=1 by conj/
-#n #_ <nplus_succ_dx #H
-elim (eq_inv_nzero_succ … H)
+(*** zero_eq_plus *)
+lemma eq_inv_zero_nplus (m) (n): 𝟎 = m + n → ∧∧ 𝟎 = m & 𝟎 = n.
+#m #n @(nat_ind_succ … n) -n
+[ /2 width=1 by conj/
+| #n #_ <nplus_succ_dx #H
+  elim (eq_inv_zero_nsucc … H)
+]
 qed-.
 
+(*** plus_inv_O3 *)
+lemma eq_inv_nplus_zero (m) (n):
+      m + n = 𝟎 → ∧∧ 𝟎 = m & 𝟎 = n.
+/2 width=1 by eq_inv_zero_nplus/ qed-.
+
 (*** injective_plus_l *)
 lemma eq_inv_nplus_bi_dx (o) (m) (n): m + o = n + o → m = n.
-#o @(nat_ind … o) -o /3 width=1 by eq_inv_nsucc_bi/
+#o @(nat_ind_succ … o) -o /3 width=1 by eq_inv_nsucc_bi/
 qed-.
 
 (*** injective_plus_r *)
@@ -96,10 +118,31 @@ lemma eq_inv_nplus_bi_sn (o) (m) (n): o + m = o + n → m = n.
 /2 width=2 by eq_inv_nplus_bi_dx/
 qed-.
 
+(*** plus_xSy_x_false *)
+lemma succ_nplus_refl_sn (m) (n): m = ↑(m + n) → ⊥.
+#m @(nat_ind_succ … m) -m
+[ /2 width=2 by eq_inv_zero_nsucc/
+| #m #IH #n #H
+  @(IH n) /2 width=1 by eq_inv_nsucc_bi/
+]
+qed-.
+
+(*** discr_plus_xy_y *)
+lemma nplus_refl_dx (m) (n): n = m + n → 𝟎 = m.
+#m #n @(nat_ind_succ … n) -n //
+#n #IH /3 width=1 by eq_inv_nsucc_bi/
+qed-.
+
+(*** discr_plus_x_xy *)
+lemma nplus_refl_sn (m) (n): m = m + n → 𝟎 = n.
+#m #n <nplus_comm
+/2 width=2 by nplus_refl_dx/
+qed-.
+
 (* Advanced eliminations ****************************************************)
 
 (*** nat_ind_plus *)
 lemma nat_ind_plus (Q:predicate …):
       Q (𝟎) → (∀n. Q n → Q (𝟏+n)) → ∀n. Q n.
-#Q #IH1 #IH2 #n @(nat_ind … n) -n /2 width=1 by/
+#Q #IH1 #IH2 #n @(nat_ind_succ … n) -n /2 width=1 by/
 qed-.