src/ZF/Bin.thy

--- a/src/ZF/Bin.thy	Thu Aug 21 14:41:05 2014 +0200
+++ b/src/ZF/Bin.thy	Thu Aug 21 14:41:08 2014 +0200
@@ -594,4 +594,92 @@
      "(integ_of(w) = x $* y) \<longleftrightarrow> (x $* y = integ_of(w))"
 by auto
 
+(** To simplify inequalities involving integer negation and literals,
+    such as -x = #3
+**)
+
+lemmas [simp] =
+  zminus_equation [where y = "integ_of(w)"]
+  equation_zminus [where x = "integ_of(w)"]
+  for w
+
+lemmas [iff] =
+  zminus_zless [where y = "integ_of(w)"]
+  zless_zminus [where x = "integ_of(w)"]
+  for w
+
+lemmas [iff] =
+  zminus_zle [where y = "integ_of(w)"]
+  zle_zminus [where x = "integ_of(w)"]
+  for w
+
+lemmas [simp] =
+  Let_def [where s = "integ_of(w)"] for w
+
+
+(*** Simprocs for numeric literals ***)
+
+(** Combining of literal coefficients in sums of products **)
+
+lemma zless_iff_zdiff_zless_0: "(x $< y) \<longleftrightarrow> (x$-y $< #0)"
+  by (simp add: zcompare_rls)
+
+lemma eq_iff_zdiff_eq_0: "[| x \<in> int; y \<in> int |] ==> (x = y) \<longleftrightarrow> (x$-y = #0)"
+  by (simp add: zcompare_rls)
+
+lemma zle_iff_zdiff_zle_0: "(x $<= y) \<longleftrightarrow> (x$-y $<= #0)"
+  by (simp add: zcompare_rls)
+
+
+(** For combine_numerals **)
+
+lemma left_zadd_zmult_distrib: "i$*u $+ (j$*u $+ k) = (i$+j)$*u $+ k"
+  by (simp add: zadd_zmult_distrib zadd_ac)
+
+
+(** For cancel_numerals **)
+
+lemmas rel_iff_rel_0_rls =
+  zless_iff_zdiff_zless_0 [where y = "u $+ v"]
+  eq_iff_zdiff_eq_0 [where y = "u $+ v"]
+  zle_iff_zdiff_zle_0 [where y = "u $+ v"]
+  zless_iff_zdiff_zless_0 [where y = n]
+  eq_iff_zdiff_eq_0 [where y = n]
+  zle_iff_zdiff_zle_0 [where y = n]
+  for u v (* FIXME n (!?) *)
+
+lemma eq_add_iff1: "(i$*u $+ m = j$*u $+ n) \<longleftrightarrow> ((i$-j)$*u $+ m = intify(n))"
+  apply (simp add: zdiff_def zadd_zmult_distrib)
+  apply (simp add: zcompare_rls)
+  apply (simp add: zadd_ac)
+  done
+
+lemma eq_add_iff2: "(i$*u $+ m = j$*u $+ n) \<longleftrightarrow> (intify(m) = (j$-i)$*u $+ n)"
+  apply (simp add: zdiff_def zadd_zmult_distrib)
+  apply (simp add: zcompare_rls)
+  apply (simp add: zadd_ac)
+  done
+
+lemma less_add_iff1: "(i$*u $+ m $< j$*u $+ n) \<longleftrightarrow> ((i$-j)$*u $+ m $< n)"
+  apply (simp add: zdiff_def zadd_zmult_distrib zadd_ac rel_iff_rel_0_rls)
+  done
+
+lemma less_add_iff2: "(i$*u $+ m $< j$*u $+ n) \<longleftrightarrow> (m $< (j$-i)$*u $+ n)"
+  apply (simp add: zdiff_def zadd_zmult_distrib zadd_ac rel_iff_rel_0_rls)
+  done
+
+lemma le_add_iff1: "(i$*u $+ m $<= j$*u $+ n) \<longleftrightarrow> ((i$-j)$*u $+ m $<= n)"
+  apply (simp add: zdiff_def zadd_zmult_distrib)
+  apply (simp add: zcompare_rls)
+  apply (simp add: zadd_ac)
+  done
+
+lemma le_add_iff2: "(i$*u $+ m $<= j$*u $+ n) \<longleftrightarrow> (m $<= (j$-i)$*u $+ n)"
+  apply (simp add: zdiff_def zadd_zmult_distrib)
+  apply (simp add: zcompare_rls)
+  apply (simp add: zadd_ac)
+  done
+
+ML_file "int_arith.ML"
+
 end