--- a/src/HOL/Library/Extended_Nat.thy Thu Feb 22 20:05:30 2018 +0100
+++ b/src/HOL/Library/Extended_Nat.thy Thu Feb 22 22:58:38 2018 +0000
@@ -383,6 +383,8 @@
by (simp split: enat.splits)
show "a < b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d" for a b c d :: enat
by (cases a b c d rule: enat2_cases[case_product enat2_cases]) auto
+ show "a < b \<Longrightarrow> a + 1 < b + 1"
+ by (metis add_right_mono eSuc_minus_1 eSuc_plus_1 less_le)
qed (simp add: zero_enat_def one_enat_def)
(* BH: These equations are already proven generally for any type in
--- a/src/HOL/Rings.thy Thu Feb 22 20:05:30 2018 +0100
+++ b/src/HOL/Rings.thy Thu Feb 22 22:58:38 2018 +0000
@@ -2245,7 +2245,8 @@
end
-class linordered_nonzero_semiring = ordered_comm_semiring + monoid_mult + linorder + zero_less_one
+class linordered_nonzero_semiring = ordered_comm_semiring + monoid_mult + linorder + zero_less_one +
+ assumes add_mono1: "a < b \<Longrightarrow> a + 1 < b + 1"
begin
subclass zero_neq_one
@@ -2278,7 +2279,15 @@
assumes le_add_diff_inverse2 [simp]: "b \<le> a \<Longrightarrow> a - b + b = a"
begin
-subclass linordered_nonzero_semiring ..
+subclass linordered_nonzero_semiring
+proof
+ show "a + 1 < b + 1" if "a < b" for a b
+ proof (rule ccontr, simp add: not_less)
+ assume "b \<le> a"
+ with that show False
+ by (simp add: )
+ qed
+qed
text \<open>Addition is the inverse of subtraction.\<close>