fixing ennreal using add_mono1; shifting results from linordered_semidom to linordered_nonzero_semiring

--- a/src/HOL/Library/Extended_Nonnegative_Real.thy	Thu Feb 22 22:58:38 2018 +0000
+++ b/src/HOL/Library/Extended_Nonnegative_Real.thy	Fri Feb 23 09:28:14 2018 +0000
@@ -11,7 +11,7 @@
 lemma ereal_ineq_diff_add:
   assumes "b \<noteq> (-\<infinity>::ereal)" "a \<ge> b"
   shows "a = b + (a-b)"
-by (metis add.commute assms(1) assms(2) ereal_eq_minus_iff ereal_minus_le_iff ereal_plus_eq_PInfty)
+by (metis add.commute assms ereal_eq_minus_iff ereal_minus_le_iff ereal_plus_eq_PInfty)
 
 lemma Limsup_const_add:
   fixes c :: "'a::{complete_linorder, linorder_topology, topological_monoid_add, ordered_ab_semigroup_add}"
@@ -345,7 +345,11 @@
      (transfer ; auto intro: add_mono mult_mono mult_ac ereal_left_distrib ereal_mult_left_mono)+
 
 instance ennreal :: linordered_nonzero_semiring
-  proof qed (transfer; simp)
+proof
+  fix a b::ennreal
+  show "a < b \<Longrightarrow> a + 1 < b + 1"
+    by transfer (simp add: add_right_mono ereal_add_cancel_right less_le)
+qed (transfer; simp)
 
 instance ennreal :: strict_ordered_ab_semigroup_add
 proof

--- a/src/HOL/Nat.thy	Thu Feb 22 22:58:38 2018 +0000
+++ b/src/HOL/Nat.thy	Fri Feb 23 09:28:14 2018 +0000
@@ -1774,7 +1774,7 @@
 
 class ring_char_0 = ring_1 + semiring_char_0
 
-context linordered_semidom
+context linordered_nonzero_semiring
 begin
 
 lemma of_nat_0_le_iff [simp]: "0 \<le> of_nat n"
@@ -1783,8 +1783,21 @@
 lemma of_nat_less_0_iff [simp]: "\<not> of_nat m < 0"
   by (simp add: not_less)
 
+lemma of_nat_mono[simp]: "i \<le> j \<Longrightarrow> of_nat i \<le> of_nat j"
+  by (auto simp: le_iff_add intro!: add_increasing2)
+
 lemma of_nat_less_iff [simp]: "of_nat m < of_nat n \<longleftrightarrow> m < n"
-  by (induct m n rule: diff_induct) (simp_all add: add_pos_nonneg)
+proof(induct m n rule: diff_induct)
+  case (1 m) then show ?case
+    by auto
+next
+  case (2 n) then show ?case
+    by (simp add: add_pos_nonneg)
+next
+  case (3 m n)
+  then show ?case
+    by (auto simp: add_commute [of 1] add_mono1 not_less add_right_mono leD)
+qed
 
 lemma of_nat_le_iff [simp]: "of_nat m \<le> of_nat n \<longleftrightarrow> m \<le> n"
   by (simp add: not_less [symmetric] linorder_not_less [symmetric])
@@ -1795,7 +1808,7 @@
 lemma of_nat_less_imp_less: "of_nat m < of_nat n \<Longrightarrow> m < n"
   by simp
 
-text \<open>Every \<open>linordered_semidom\<close> has characteristic zero.\<close>
+text \<open>Every \<open>linordered_nonzero_semiring\<close> has characteristic zero.\<close>
 
 subclass semiring_char_0
   by standard (auto intro!: injI simp add: eq_iff)
@@ -1810,6 +1823,14 @@
 
 end
 
+
+context linordered_semidom
+begin
+subclass linordered_nonzero_semiring ..
+subclass semiring_char_0 ..
+end
+
+
 context linordered_idom
 begin