(*  Title       : Fact.thy

     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

 begin

 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: "fact_nat (Suc x) = Suc x * fact x"

 instance ..

 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 transfer_morphism_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 transfer_morphism_int_nat[transfer add return: 

     transfer_int_nat_factorial transfer_int_nat_factorial_closure]

 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 assms 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)

   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)

   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_altdef_nat: "fact (n::nat) = (PROD i:{1..n}. i)"

   apply (induct n)

   apply force

   apply (subst fact_Suc)

   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_dvd: "n \<le> m \<Longrightarrow> fact n dvd fact (m::nat)"

   by (auto simp add: fact_altdef_nat intro!: setprod_dvd_setprod_subset)

 lemma fact_mod: "m \<le> (n::nat) \<Longrightarrow> fact n mod fact m = 0"

   by (auto simp add: dvd_imp_mod_0 fact_dvd)

 lemma fact_div_fact:

   assumes "m \<ge> (n :: nat)"

   shows "(fact m) div (fact n) = \<Prod>{n + 1..m}"

 proof -

   obtain d where "d = m - n" by auto

   from assms this have "m = n + d" by auto

   have "fact (n + d) div (fact n) = \<Prod>{n + 1..n + d}"

   proof (induct d)

     case 0

     show ?case by simp

   next

     case (Suc d')

     have "fact (n + Suc d') div fact n = Suc (n + d') * fact (n + d') div fact n"

       by simp

     also from Suc.hyps have "... = Suc (n + d') * \<Prod>{n + 1..n + d'}" 

       unfolding div_mult1_eq[of _ "fact (n + d')"] by (simp add: fact_mod)

     also have "... = \<Prod>{n + 1..n + Suc d'}"

       by (simp add: atLeastAtMostSuc_conv setprod_insert)

     finally show ?case .

   qed

   from this `m = n + d` show ?thesis by simp

 qed

 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 (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

 lemma fact_le_power: "fact n \<le> n^n"

 proof (induct n)

   case (Suc n)

   then have "fact n \<le> Suc n ^ n" by (rule le_trans) (simp add: power_mono)

   then show ?case by (simp add: add_le_mono)

 qed simp

 subsection {* @{term fact} and @{term of_nat} *}

 lemma of_nat_fact_not_zero [simp]: "of_nat (fact n) \<noteq> (0::'a::semiring_char_0)"

 by auto

 lemma of_nat_fact_gt_zero [simp]: "(0::'a::{linordered_semidom}) < of_nat(fact n)" by auto

 lemma of_nat_fact_ge_zero [simp]: "(0::'a::linordered_semidom) \<le> of_nat(fact n)"

 by simp

 lemma inv_of_nat_fact_gt_zero [simp]: "(0::'a::linordered_field) < inverse (of_nat (fact n))"

 by (auto simp add: positive_imp_inverse_positive)

 lemma inv_of_nat_fact_ge_zero [simp]: "(0::'a::linordered_field) \<le> inverse (of_nat (fact n))"

 by (auto intro: order_less_imp_le)

 end