src/HOL/Hyperreal/Fact.thy

--- a/src/HOL/Hyperreal/Fact.thy	Sun Oct 21 14:21:54 2007 +0200
+++ b/src/HOL/Hyperreal/Fact.thy	Sun Oct 21 14:53:44 2007 +0200
@@ -17,60 +17,60 @@
 
 
 lemma fact_gt_zero [simp]: "0 < fact n"
-  by (induct n) auto
+by (induct n) auto
 
 lemma fact_not_eq_zero [simp]: "fact n \<noteq> 0"
-  by (simp add: neq0_conv)
+by (simp add: neq0_conv)
 
 lemma real_of_nat_fact_not_zero [simp]: "real (fact n) \<noteq> 0"
-  by auto
+by auto
 
 lemma real_of_nat_fact_gt_zero [simp]: "0 < real(fact n)"
-  by auto
+by auto
 
 lemma real_of_nat_fact_ge_zero [simp]: "0 \<le> real(fact n)"
-  by simp
+by simp
 
 lemma fact_ge_one [simp]: "1 \<le> fact n"
-  by (induct n) auto
+by (induct n) auto
 
 lemma fact_mono: "m \<le> n ==> fact m \<le> fact n"
-  apply (drule le_imp_less_or_eq)
-  apply (auto dest!: less_imp_Suc_add)
-  apply (induct_tac k, auto)
-  done
+apply (drule le_imp_less_or_eq)
+apply (auto dest!: less_imp_Suc_add)
+apply (induct_tac k, auto)
+done
 
 text{*Note that @{term "fact 0 = fact 1"}*}
 lemma fact_less_mono: "[| 0 < m; m < n |] ==> fact m < fact n"
-  apply (drule_tac m = m in less_imp_Suc_add, auto)
-  apply (induct_tac k, auto)
-  done
+apply (drule_tac m = m in less_imp_Suc_add, auto)
+apply (induct_tac k, auto)
+done
 
 lemma inv_real_of_nat_fact_gt_zero [simp]: "0 < inverse (real (fact n))"
-  by (auto simp add: positive_imp_inverse_positive)
+by (auto simp add: positive_imp_inverse_positive)
 
 lemma inv_real_of_nat_fact_ge_zero [simp]: "0 \<le> inverse (real (fact n))"
-  by (auto intro: order_less_imp_le)
+by (auto intro: order_less_imp_le)
 
 lemma fact_diff_Suc [rule_format]:
-    "n < Suc m ==> fact (Suc m - n) = (Suc m - n) * fact (m - n)"
-  apply (induct n arbitrary: m)
-  apply auto
-  apply (drule_tac x = "m - 1" in meta_spec, auto)
-  done
+  "n < Suc m ==> fact (Suc m - n) = (Suc m - n) * fact (m - n)"
+apply (induct n arbitrary: m)
+apply auto
+apply (drule_tac x = "m - 1" in meta_spec, auto)
+done
 
 lemma fact_num0 [simp]: "fact 0 = 1"
-  by auto
+by auto
 
 lemma fact_num_eq_if: "fact m = (if m=0 then 1 else m * fact (m - 1))"
-  by (cases m) auto
+by (cases m) auto
 
 lemma fact_add_num_eq_if:
-    "fact (m + n) = (if m + n = 0 then 1 else (m + n) * fact (m + n - 1))"
-  by (cases "m + n") auto
+  "fact (m + n) = (if m + n = 0 then 1 else (m + n) * fact (m + n - 1))"
+by (cases "m + n") auto
 
 lemma fact_add_num_eq_if2:
-    "fact (m + n) = (if m = 0 then fact n else (m + n) * fact ((m - 1) + n))"
-  by (cases m) auto
+  "fact (m + n) = (if m = 0 then fact n else (m + n) * fact ((m - 1) + n))"
+by (cases m) auto
 
 end