
diff -r ccaadfcf6941 -r 8a9228872fbd src/HOL/Fact.thy
--- a/src/HOL/Fact.thy	Thu Jul 09 17:34:59 2009 +0200
+++ b/src/HOL/Fact.thy	Fri Jul 10 10:45:30 2009 -0400
@@ -2,25 +2,266 @@
     Author      : Jacques D. Fleuriot
     Copyright   : 1998  University of Cambridge
     Conversion to Isar and new proofs by Lawrence C Paulson, 2004
+    The integer version of factorial and other additions by Jeremy Avigad.
 *)
 
 header{*Factorial Function*}
 
 theory Fact
-imports Main
+imports NatTransfer
 begin
 
-consts fact :: "nat => nat"
-primrec
-  fact_0:     "fact 0 = 1"
-  fact_Suc:   "fact (Suc n) = (Suc n) * fact n"
+class fact =
+
+fixes 
+  fact :: "'a \<Rightarrow> 'a"
+
+instantiation nat :: fact
+
+begin 
+
+fun
+  fact_nat :: "nat \<Rightarrow> nat"
+where
+  fact_0_nat: "fact_nat 0 = Suc 0"
+| fact_Suc_nat: "fact_nat (Suc x) = Suc x * fact x"
+
+instance proof qed
+
+end
+
+(* definitions for the integers *)
+
+instantiation int :: fact
+
+begin 
+
+definition
+  fact_int :: "int \<Rightarrow> int"
+where  
+  "fact_int x = (if x >= 0 then int (fact (nat x)) else 0)"
+
+instance proof qed
+
+end
+
+
+subsection {* Set up Transfer *}
+
+lemma transfer_nat_int_factorial:
+  "(x::int) >= 0 \<Longrightarrow> fact (nat x) = nat (fact x)"
+  unfolding fact_int_def
+  by auto
+
+
+lemma transfer_nat_int_factorial_closure:
+  "x >= (0::int) \<Longrightarrow> fact x >= 0"
+  by (auto simp add: fact_int_def)
+
+declare TransferMorphism_nat_int[transfer add return: 
+    transfer_nat_int_factorial transfer_nat_int_factorial_closure]
+
+lemma transfer_int_nat_factorial:
+  "fact (int x) = int (fact x)"
+  unfolding fact_int_def by auto
+
+lemma transfer_int_nat_factorial_closure:
+  "is_nat x \<Longrightarrow> fact x >= 0"
+  by (auto simp add: fact_int_def)
+
+declare TransferMorphism_int_nat[transfer add return: 
+    transfer_int_nat_factorial transfer_int_nat_factorial_closure]
 
 
-lemma fact_gt_zero [simp]: "0 < fact n"
-by (induct n) auto
+subsection {* Factorial *}
+
+lemma fact_0_int [simp]: "fact (0::int) = 1"
+  by (simp add: fact_int_def)
+
+lemma fact_1_nat [simp]: "fact (1::nat) = 1"
+  by simp
+
+lemma fact_Suc_0_nat [simp]: "fact (Suc 0) = Suc 0"
+  by simp
+
+lemma fact_1_int [simp]: "fact (1::int) = 1"
+  by (simp add: fact_int_def)
+
+lemma fact_plus_one_nat: "fact ((n::nat) + 1) = (n + 1) * fact n"
+  by simp
+
+lemma fact_plus_one_int: 
+  assumes "n >= 0"
+  shows "fact ((n::int) + 1) = (n + 1) * fact n"
+
+  using prems unfolding fact_int_def 
+  by (simp add: nat_add_distrib algebra_simps int_mult)
+
+lemma fact_reduce_nat: "(n::nat) > 0 \<Longrightarrow> fact n = n * fact (n - 1)"
+  apply (subgoal_tac "n = Suc (n - 1)")
+  apply (erule ssubst)
+  apply (subst fact_Suc_nat)
+  apply simp_all
+done
+
+lemma fact_reduce_int: "(n::int) > 0 \<Longrightarrow> fact n = n * fact (n - 1)"
+  apply (subgoal_tac "n = (n - 1) + 1")
+  apply (erule ssubst)
+  apply (subst fact_plus_one_int)
+  apply simp_all
+done
+
+lemma fact_nonzero_nat [simp]: "fact (n::nat) \<noteq> 0"
+  apply (induct n)
+  apply (auto simp add: fact_plus_one_nat)
+done
+
+lemma fact_nonzero_int [simp]: "n >= 0 \<Longrightarrow> fact (n::int) ~= 0"
+  by (simp add: fact_int_def)
+
+lemma fact_gt_zero_nat [simp]: "fact (n :: nat) > 0"
+  by (insert fact_nonzero_nat [of n], arith)
+
+lemma fact_gt_zero_int [simp]: "n >= 0 \<Longrightarrow> fact (n :: int) > 0"
+  by (auto simp add: fact_int_def)
+
+lemma fact_ge_one_nat [simp]: "fact (n :: nat) >= 1"
+  by (insert fact_nonzero_nat [of n], arith)
+
+lemma fact_ge_Suc_0_nat [simp]: "fact (n :: nat) >= Suc 0"
+  by (insert fact_nonzero_nat [of n], arith)
+
+lemma fact_ge_one_int [simp]: "n >= 0 \<Longrightarrow> fact (n :: int) >= 1"
+  apply (auto simp add: fact_int_def)
+  apply (subgoal_tac "1 = int 1")
+  apply (erule ssubst)
+  apply (subst zle_int)
+  apply auto
+done
+
+lemma dvd_fact_nat [rule_format]: "1 <= m \<longrightarrow> m <= n \<longrightarrow> m dvd fact (n::nat)"
+  apply (induct n)
+  apply force
+  apply (auto simp only: fact_Suc_nat)
+  apply (subgoal_tac "m = Suc n")
+  apply (erule ssubst)
+  apply (rule dvd_triv_left)
+  apply auto
+done
+
+lemma dvd_fact_int [rule_format]: "1 <= m \<longrightarrow> m <= n \<longrightarrow> m dvd fact (n::int)"
+  apply (case_tac "1 <= n")
+  apply (induct n rule: int_ge_induct)
+  apply (auto simp add: fact_plus_one_int)
+  apply (subgoal_tac "m = i + 1")
+  apply auto
+done
+
+lemma interval_plus_one_nat: "(i::nat) <= j + 1 \<Longrightarrow> 
+  {i..j+1} = {i..j} Un {j+1}"
+  by auto
+
+lemma interval_Suc: "i <= Suc j \<Longrightarrow> {i..Suc j} = {i..j} Un {Suc j}"
+  by auto
+
+lemma interval_plus_one_int: "(i::int) <= j + 1 \<Longrightarrow> {i..j+1} = {i..j} Un {j+1}"
+  by auto
 
-lemma fact_not_eq_zero [simp]: "fact n \<noteq> 0"
-by simp
+lemma fact_altdef_nat: "fact (n::nat) = (PROD i:{1..n}. i)"
+  apply (induct n)
+  apply force
+  apply (subst fact_Suc_nat)
+  apply (subst interval_Suc)
+  apply auto
+done
+
+lemma fact_altdef_int: "n >= 0 \<Longrightarrow> fact (n::int) = (PROD i:{1..n}. i)"
+  apply (induct n rule: int_ge_induct)
+  apply force
+  apply (subst fact_plus_one_int, assumption)
+  apply (subst interval_plus_one_int)
+  apply auto
+done
+
+lemma fact_mono_nat: "(m::nat) \<le> n \<Longrightarrow> 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
+
+lemma fact_neg_int [simp]: "m < (0::int) \<Longrightarrow> fact m = 0"
+  unfolding fact_int_def by auto
+
+lemma fact_ge_zero_int [simp]: "fact m >= (0::int)"
+  apply (case_tac "m >= 0")
+  apply auto
+  apply (frule fact_gt_zero_int)
+  apply arith
+done
+
+lemma fact_mono_int_aux [rule_format]: "k >= (0::int) \<Longrightarrow> 
+    fact (m + k) >= fact m"
+  apply (case_tac "m < 0")
+  apply auto
+  apply (induct k rule: int_ge_induct)
+  apply auto
+  apply (subst add_assoc [symmetric])
+  apply (subst fact_plus_one_int)
+  apply auto
+  apply (erule order_trans)
+  apply (subst mult_le_cancel_right1)
+  apply (subgoal_tac "fact (m + i) >= 0")
+  apply arith
+  apply auto
+done
+
+lemma fact_mono_int: "(m::int) <= n \<Longrightarrow> fact m <= fact n"
+  apply (insert fact_mono_int_aux [of "n - m" "m"])
+  apply auto
+done
+
+text{*Note that @{term "fact 0 = fact 1"}*}
+lemma fact_less_mono_nat: "[| (0::nat) < 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 fact_less_mono_int_aux: "k >= 0 \<Longrightarrow> (0::int) < m \<Longrightarrow>
+    fact m < fact ((m + 1) + k)"
+  apply (induct k rule: int_ge_induct)
+  apply (simp add: fact_plus_one_int)
+  apply (subst mult_less_cancel_right1)
+  apply (insert fact_gt_zero_int [of m], arith)
+  apply (subst (2) fact_reduce_int)
+  apply (auto simp add: add_ac)
+  apply (erule order_less_le_trans)
+  apply (subst mult_le_cancel_right1)
+  apply auto
+  apply (subgoal_tac "fact (i + (1 + m)) >= 0")
+  apply force
+  apply (rule fact_ge_zero_int)
+done
+
+lemma fact_less_mono_int: "(0::int) < m \<Longrightarrow> m < n \<Longrightarrow> fact m < fact n"
+  apply (insert fact_less_mono_int_aux [of "n - (m + 1)" "m"])
+  apply auto
+done
+
+lemma fact_num_eq_if_nat: "fact (m::nat) = 
+  (if m=0 then 1 else m * fact (m - 1))"
+by (cases m) auto
+
+lemma fact_add_num_eq_if_nat:
+  "fact ((m::nat) + 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_nat:
+  "fact ((m::nat) + n) = 
+    (if m = 0 then fact n else (m + n) * fact ((m - 1) + n))"
+by (cases m) auto
+
+
+subsection {* fact and of_nat *}
 
 lemma of_nat_fact_not_zero [simp]: "of_nat (fact n) \<noteq> (0::'a::semiring_char_0)"
 by auto
@@ -33,46 +274,10 @@
 lemma of_nat_fact_ge_zero [simp]: "(0::'a::ordered_semidom) \<le> of_nat(fact n)"
 by simp
 
-lemma fact_ge_one [simp]: "1 \<le> fact n"
-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
-
-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_of_nat_fact_gt_zero [simp]: "(0::'a::ordered_field) < inverse (of_nat (fact n))"
 by (auto simp add: positive_imp_inverse_positive)
 
 lemma inv_of_nat_fact_ge_zero [simp]: "(0::'a::ordered_field) \<le> inverse (of_nat (fact n))"
 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 - Suc 0" in meta_spec, auto)
-done
-
-lemma fact_num0: "fact 0 = 1"
-by auto
-
-lemma fact_num_eq_if: "fact m = (if m=0 then 1 else m * fact (m - 1))"
-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
-
-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
-
 end




