src/HOL/Hyperreal/Fact.thy
changeset 15094 a7d1a3fdc30d
parent 12196 a3be6b3a9c0b
child 15131 c69542757a4d
--- a/src/HOL/Hyperreal/Fact.thy	Fri Jul 30 18:37:58 2004 +0200
+++ b/src/HOL/Hyperreal/Fact.thy	Sat Jul 31 20:54:23 2004 +0200
@@ -1,14 +1,74 @@
-(*  Title       : Fact.thy 
+(*  Title       : Fact.thy
     Author      : Jacques D. Fleuriot
     Copyright   : 1998  University of Cambridge
-    Description : Factorial function
+    Conversion to Isar and new proofs by Lawrence C Paulson, 2004
 *)
 
-Fact = NatStar + 
+header{*Factorial Function*}
+
+theory Fact = Real:
+
+consts fact :: "nat => nat"
+primrec
+   fact_0:     "fact 0 = 1"
+   fact_Suc:   "fact (Suc n) = (Suc n) * fact n"
+
+
+lemma fact_gt_zero [simp]: "0 < fact n"
+by (induct "n", auto)
+
+lemma fact_not_eq_zero [simp]: "fact n \<noteq> 0"
+by simp
+
+lemma real_of_nat_fact_not_zero [simp]: "real (fact n) \<noteq> 0"
+by auto
+
+lemma real_of_nat_fact_gt_zero [simp]: "0 < real(fact n)"
+by auto
+
+lemma real_of_nat_fact_ge_zero [simp]: "0 \<le> real(fact n)"
+by simp
+
+lemma fact_ge_one [simp]: "1 \<le> fact n"
+by (induct "n", auto)
 
-consts fact :: nat => nat 
-primrec 
-   fact_0     "fact 0 = 1"
-   fact_Suc   "fact (Suc n) = (Suc n) * fact n" 
+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
+
+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
+
+lemma inv_real_of_nat_fact_gt_zero [simp]: "0 < inverse (real (fact n))"
+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)
+
+lemma fact_diff_Suc [rule_format]:
+     "\<forall>m. n < Suc m --> fact (Suc m - n) = (Suc m - n) * fact (m - n)"
+apply (induct n, auto)
+apply (drule_tac x = "m - 1" in spec, auto)
+done
+
+lemma fact_num0 [simp]: "fact 0 = 1"
+by auto
+
+lemma fact_num_eq_if: "fact m = (if m=0 then 1 else m * fact (m - 1))"
+by (case_tac "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 (case_tac "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 (case_tac "m", auto)
+
 
 end
\ No newline at end of file