src/HOL/Fact.thy
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(*  Title       : Fact.thy
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    Author      : Jacques D. Fleuriot
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    Copyright   : 1998  University of Cambridge
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    Conversion to Isar and new proofs by Lawrence C Paulson, 2004
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    The integer version of factorial and other additions by Jeremy Avigad.
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*)
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header{*Factorial Function*}
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theory Fact
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imports NatTransfer
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begin
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class fact =
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fixes 
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  fact :: "'a \<Rightarrow> 'a"
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instantiation nat :: fact
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begin 
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fun
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  fact_nat :: "nat \<Rightarrow> nat"
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where
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  fact_0_nat: "fact_nat 0 = Suc 0"
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| fact_Suc: "fact_nat (Suc x) = Suc x * fact x"
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instance proof qed
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end
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(* definitions for the integers *)
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instantiation int :: fact
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begin 
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definition
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  fact_int :: "int \<Rightarrow> int"
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where  
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  "fact_int x = (if x >= 0 then int (fact (nat x)) else 0)"
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instance proof qed
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end
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subsection {* Set up Transfer *}
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lemma transfer_nat_int_factorial:
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  "(x::int) >= 0 \<Longrightarrow> fact (nat x) = nat (fact x)"
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  unfolding fact_int_def
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  by auto
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lemma transfer_nat_int_factorial_closure:
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  "x >= (0::int) \<Longrightarrow> fact x >= 0"
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  by (auto simp add: fact_int_def)
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declare TransferMorphism_nat_int[transfer add return: 
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    transfer_nat_int_factorial transfer_nat_int_factorial_closure]
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lemma transfer_int_nat_factorial:
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  "fact (int x) = int (fact x)"
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  unfolding fact_int_def by auto
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lemma transfer_int_nat_factorial_closure:
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  "is_nat x \<Longrightarrow> fact x >= 0"
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  by (auto simp add: fact_int_def)
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declare TransferMorphism_int_nat[transfer add return: 
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    transfer_int_nat_factorial transfer_int_nat_factorial_closure]
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subsection {* Factorial *}
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lemma fact_0_int [simp]: "fact (0::int) = 1"
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  by (simp add: fact_int_def)
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lemma fact_1_nat [simp]: "fact (1::nat) = 1"
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  by simp
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lemma fact_Suc_0_nat [simp]: "fact (Suc 0) = Suc 0"
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  by simp
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lemma fact_1_int [simp]: "fact (1::int) = 1"
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  by (simp add: fact_int_def)
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lemma fact_plus_one_nat: "fact ((n::nat) + 1) = (n + 1) * fact n"
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  by simp
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lemma fact_plus_one_int: 
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  assumes "n >= 0"
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  shows "fact ((n::int) + 1) = (n + 1) * fact n"
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  using prems unfolding fact_int_def 
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  by (simp add: nat_add_distrib algebra_simps int_mult)
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lemma fact_reduce_nat: "(n::nat) > 0 \<Longrightarrow> fact n = n * fact (n - 1)"
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  apply (subgoal_tac "n = Suc (n - 1)")
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  apply (erule ssubst)
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  apply (subst fact_Suc)
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  apply simp_all
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done
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lemma fact_reduce_int: "(n::int) > 0 \<Longrightarrow> fact n = n * fact (n - 1)"
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  apply (subgoal_tac "n = (n - 1) + 1")
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  apply (erule ssubst)
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  apply (subst fact_plus_one_int)
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  apply simp_all
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done
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lemma fact_nonzero_nat [simp]: "fact (n::nat) \<noteq> 0"
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  apply (induct n)
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  apply (auto simp add: fact_plus_one_nat)
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done
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lemma fact_nonzero_int [simp]: "n >= 0 \<Longrightarrow> fact (n::int) ~= 0"
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  by (simp add: fact_int_def)
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lemma fact_gt_zero_nat [simp]: "fact (n :: nat) > 0"
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  by (insert fact_nonzero_nat [of n], arith)
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lemma fact_gt_zero_int [simp]: "n >= 0 \<Longrightarrow> fact (n :: int) > 0"
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  by (auto simp add: fact_int_def)
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lemma fact_ge_one_nat [simp]: "fact (n :: nat) >= 1"
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  by (insert fact_nonzero_nat [of n], arith)
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lemma fact_ge_Suc_0_nat [simp]: "fact (n :: nat) >= Suc 0"
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  by (insert fact_nonzero_nat [of n], arith)
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lemma fact_ge_one_int [simp]: "n >= 0 \<Longrightarrow> fact (n :: int) >= 1"
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  apply (auto simp add: fact_int_def)
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  apply (subgoal_tac "1 = int 1")
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  apply (erule ssubst)
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  apply (subst zle_int)
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  apply auto
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done
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   141
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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lemma dvd_fact_nat [rule_format]: "1 <= m \<longrightarrow> m <= n \<longrightarrow> m dvd fact (n::nat)"
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   143
  apply (induct n)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   144
  apply force
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c141f139ce26 Changed fact_Suc_nat back to fact_Suc
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   145
  apply (auto simp only: fact_Suc)
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   146
  apply (subgoal_tac "m = Suc n")
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   147
  apply (erule ssubst)
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   148
  apply (rule dvd_triv_left)
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   149
  apply auto
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   150
done
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   151
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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lemma dvd_fact_int [rule_format]: "1 <= m \<longrightarrow> m <= n \<longrightarrow> m dvd fact (n::int)"
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   153
  apply (case_tac "1 <= n")
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   154
  apply (induct n rule: int_ge_induct)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   155
  apply (auto simp add: fact_plus_one_int)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   156
  apply (subgoal_tac "m = i + 1")
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   157
  apply auto
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   158
done
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   159
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   160
lemma interval_plus_one_nat: "(i::nat) <= j + 1 \<Longrightarrow> 
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   161
  {i..j+1} = {i..j} Un {j+1}"
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   162
  by auto
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   163
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   164
lemma interval_Suc: "i <= Suc j \<Longrightarrow> {i..Suc j} = {i..j} Un {Suc j}"
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   165
  by auto
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   166
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   167
lemma interval_plus_one_int: "(i::int) <= j + 1 \<Longrightarrow> {i..j+1} = {i..j} Un {j+1}"
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   168
  by auto
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   169
32036
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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lemma fact_altdef_nat: "fact (n::nat) = (PROD i:{1..n}. i)"
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   171
  apply (induct n)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   172
  apply force
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c141f139ce26 Changed fact_Suc_nat back to fact_Suc
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   173
  apply (subst fact_Suc)
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8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   174
  apply (subst interval_Suc)
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   175
  apply auto
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   176
done
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   177
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   178
lemma fact_altdef_int: "n >= 0 \<Longrightarrow> fact (n::int) = (PROD i:{1..n}. i)"
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   179
  apply (induct n rule: int_ge_induct)
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   180
  apply force
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   181
  apply (subst fact_plus_one_int, assumption)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   182
  apply (subst interval_plus_one_int)
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   183
  apply auto
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   184
done
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   185
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   186
lemma fact_mono_nat: "(m::nat) \<le> n \<Longrightarrow> fact m \<le> fact n"
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   187
apply (drule le_imp_less_or_eq)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   188
apply (auto dest!: less_imp_Suc_add)
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   189
apply (induct_tac k, auto)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   190
done
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   191
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   192
lemma fact_neg_int [simp]: "m < (0::int) \<Longrightarrow> fact m = 0"
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   193
  unfolding fact_int_def by auto
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   194
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   195
lemma fact_ge_zero_int [simp]: "fact m >= (0::int)"
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   196
  apply (case_tac "m >= 0")
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   197
  apply auto
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   198
  apply (frule fact_gt_zero_int)
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   199
  apply arith
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   200
done
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   201
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   202
lemma fact_mono_int_aux [rule_format]: "k >= (0::int) \<Longrightarrow> 
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   203
    fact (m + k) >= fact m"
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   204
  apply (case_tac "m < 0")
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   205
  apply auto
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   206
  apply (induct k rule: int_ge_induct)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   207
  apply auto
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   208
  apply (subst add_assoc [symmetric])
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   209
  apply (subst fact_plus_one_int)
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   210
  apply auto
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   211
  apply (erule order_trans)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   212
  apply (subst mult_le_cancel_right1)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   213
  apply (subgoal_tac "fact (m + i) >= 0")
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   214
  apply arith
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   215
  apply auto
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   216
done
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   217
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   218
lemma fact_mono_int: "(m::int) <= n \<Longrightarrow> fact m <= fact n"
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   219
  apply (insert fact_mono_int_aux [of "n - m" "m"])
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   220
  apply auto
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   221
done
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   222
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   223
text{*Note that @{term "fact 0 = fact 1"}*}
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   224
lemma fact_less_mono_nat: "[| (0::nat) < m; m < n |] ==> fact m < fact n"
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   225
apply (drule_tac m = m in less_imp_Suc_add, auto)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   226
apply (induct_tac k, auto)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   227
done
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   228
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   229
lemma fact_less_mono_int_aux: "k >= 0 \<Longrightarrow> (0::int) < m \<Longrightarrow>
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   230
    fact m < fact ((m + 1) + k)"
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   231
  apply (induct k rule: int_ge_induct)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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diff changeset
   232
  apply (simp add: fact_plus_one_int)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   233
  apply (subst mult_less_cancel_right1)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   234
  apply (insert fact_gt_zero_int [of m], arith)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   235
  apply (subst (2) fact_reduce_int)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   236
  apply (auto simp add: add_ac)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   237
  apply (erule order_less_le_trans)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   238
  apply (subst mult_le_cancel_right1)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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parents: 30242
diff changeset
   239
  apply auto
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   240
  apply (subgoal_tac "fact (i + (1 + m)) >= 0")
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
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diff changeset
   241
  apply force
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   242
  apply (rule fact_ge_zero_int)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   243
done
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   244
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   245
lemma fact_less_mono_int: "(0::int) < m \<Longrightarrow> m < n \<Longrightarrow> fact m < fact n"
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   246
  apply (insert fact_less_mono_int_aux [of "n - (m + 1)" "m"])
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   247
  apply auto
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   248
done
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   249
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   250
lemma fact_num_eq_if_nat: "fact (m::nat) = 
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   251
  (if m=0 then 1 else m * fact (m - 1))"
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   252
by (cases m) auto
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   253
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   254
lemma fact_add_num_eq_if_nat:
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   255
  "fact ((m::nat) + n) = (if m + n = 0 then 1 else (m + n) * fact (m + n - 1))"
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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diff changeset
   256
by (cases "m + n") auto
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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diff changeset
   257
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   258
lemma fact_add_num_eq_if2_nat:
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   259
  "fact ((m::nat) + n) = 
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   260
    (if m = 0 then fact n else (m + n) * fact ((m - 1) + n))"
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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diff changeset
   261
by (cases m) auto
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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parents: 30242
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   262
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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   263
32039
400a519bc888 Use term antiquotation to refer to constant names in subsection title.
berghofe
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   264
subsection {* @{term fact} and @{term of_nat} *}
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a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
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   265
29693
708dcf7dec9f moved upwards in thy graph, real related theorems moved to Transcendental.thy
chaieb
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   266
lemma of_nat_fact_not_zero [simp]: "of_nat (fact n) \<noteq> (0::'a::semiring_char_0)"
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
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   267
by auto
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a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
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   268
29693
708dcf7dec9f moved upwards in thy graph, real related theorems moved to Transcendental.thy
chaieb
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   269
class ordered_semiring_1 = ordered_semiring + semiring_1
708dcf7dec9f moved upwards in thy graph, real related theorems moved to Transcendental.thy
chaieb
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   270
class ordered_semiring_1_strict = ordered_semiring_strict + semiring_1
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a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
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   271
29693
708dcf7dec9f moved upwards in thy graph, real related theorems moved to Transcendental.thy
chaieb
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   272
lemma of_nat_fact_gt_zero [simp]: "(0::'a::{ordered_semidom}) < of_nat(fact n)" by auto
708dcf7dec9f moved upwards in thy graph, real related theorems moved to Transcendental.thy
chaieb
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   273
708dcf7dec9f moved upwards in thy graph, real related theorems moved to Transcendental.thy
chaieb
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   274
lemma of_nat_fact_ge_zero [simp]: "(0::'a::ordered_semidom) \<le> of_nat(fact n)"
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by simp
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lemma inv_of_nat_fact_gt_zero [simp]: "(0::'a::ordered_field) < inverse (of_nat (fact n))"
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by (auto simp add: positive_imp_inverse_positive)
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lemma inv_of_nat_fact_ge_zero [simp]: "(0::'a::ordered_field) \<le> inverse (of_nat (fact n))"
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by (auto intro: order_less_imp_le)
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end