author  wenzelm 
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child 50224  aacd6da09825 
permissions  rwrr 
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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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15131  10 
theory Fact 
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imports Main 
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begin 
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class fact = 
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fixes 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 .. 
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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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35644  58 
declare transfer_morphism_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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35644  69 
declare transfer_morphism_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 assms 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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lemma dvd_fact_nat [rule_format]: "1 <= m \<longrightarrow> m <= n \<longrightarrow> m dvd fact (n::nat)" 
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apply (induct n) 
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apply force 
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apply (auto simp only: fact_Suc) 
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apply (subgoal_tac "m = Suc n") 
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143 
apply (erule ssubst) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
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144 
apply (rule dvd_triv_left) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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145 
apply auto 
41550  146 
done 
32036
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

147 

8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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148 
lemma dvd_fact_int [rule_format]: "1 <= m \<longrightarrow> m <= n \<longrightarrow> m dvd fact (n::int)" 
8a9228872fbd
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avigad
parents:
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diff
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149 
apply (case_tac "1 <= n") 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
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diff
changeset

150 
apply (induct n rule: int_ge_induct) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

151 
apply (auto simp add: fact_plus_one_int) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

152 
apply (subgoal_tac "m = i + 1") 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
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153 
apply auto 
41550  154 
done 
32036
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

155 

8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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156 
lemma interval_plus_one_nat: "(i::nat) <= j + 1 \<Longrightarrow> 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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157 
{i..j+1} = {i..j} Un {j+1}" 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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158 
by auto 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

159 

8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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160 
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.
avigad
parents:
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diff
changeset

161 
by auto 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

162 

8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
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163 
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.
avigad
parents:
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diff
changeset

164 
by auto 
15094
a7d1a3fdc30d
conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents:
12196
diff
changeset

165 

32036
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
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166 
lemma fact_altdef_nat: "fact (n::nat) = (PROD i:{1..n}. i)" 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

167 
apply (induct n) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
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168 
apply force 
32047  169 
apply (subst fact_Suc) 
32036
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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170 
apply (subst interval_Suc) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
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171 
apply auto 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
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172 
done 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

173 

8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
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174 
lemma fact_altdef_int: "n >= 0 \<Longrightarrow> fact (n::int) = (PROD i:{1..n}. i)" 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

175 
apply (induct n rule: int_ge_induct) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

176 
apply force 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
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177 
apply (subst fact_plus_one_int, assumption) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
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178 
apply (subst interval_plus_one_int) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

179 
apply auto 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
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180 
done 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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181 

40033
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182 
lemma fact_dvd: "n \<le> m \<Longrightarrow> fact n dvd fact (m::nat)" 
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diff
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183 
by (auto simp add: fact_altdef_nat intro!: setprod_dvd_setprod_subset) 
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added some facts about factorial and dvd, div and mod
bulwahn
parents:
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diff
changeset

184 

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185 
lemma fact_mod: "m \<le> (n::nat) \<Longrightarrow> fact n mod fact m = 0" 
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diff
changeset

186 
by (auto simp add: dvd_imp_mod_0 fact_dvd) 
84200d970bf0
added some facts about factorial and dvd, div and mod
bulwahn
parents:
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diff
changeset

187 

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188 
lemma fact_div_fact: 
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189 
assumes "m \<ge> (n :: nat)" 
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190 
shows "(fact m) div (fact n) = \<Prod>{n + 1..m}" 
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191 
proof  
84200d970bf0
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192 
obtain d where "d = m  n" by auto 
84200d970bf0
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diff
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193 
from assms this have "m = n + d" by auto 
84200d970bf0
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194 
have "fact (n + d) div (fact n) = \<Prod>{n + 1..n + d}" 
84200d970bf0
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bulwahn
parents:
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diff
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195 
proof (induct d) 
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196 
case 0 
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197 
show ?case by simp 
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diff
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198 
next 
84200d970bf0
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199 
case (Suc d') 
84200d970bf0
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diff
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200 
have "fact (n + Suc d') div fact n = Suc (n + d') * fact (n + d') div fact n" 
84200d970bf0
added some facts about factorial and dvd, div and mod
bulwahn
parents:
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201 
by simp 
84200d970bf0
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diff
changeset

202 
also from Suc.hyps have "... = Suc (n + d') * \<Prod>{n + 1..n + d'}" 
84200d970bf0
added some facts about factorial and dvd, div and mod
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parents:
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diff
changeset

203 
unfolding div_mult1_eq[of _ "fact (n + d')"] by (simp add: fact_mod) 
84200d970bf0
added some facts about factorial and dvd, div and mod
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parents:
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diff
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204 
also have "... = \<Prod>{n + 1..n + Suc d'}" 
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added some facts about factorial and dvd, div and mod
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parents:
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diff
changeset

205 
by (simp add: atLeastAtMostSuc_conv setprod_insert) 
84200d970bf0
added some facts about factorial and dvd, div and mod
bulwahn
parents:
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diff
changeset

206 
finally show ?case . 
84200d970bf0
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bulwahn
parents:
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diff
changeset

207 
qed 
84200d970bf0
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parents:
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diff
changeset

208 
from this `m = n + d` show ?thesis by simp 
84200d970bf0
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bulwahn
parents:
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diff
changeset

209 
qed 
84200d970bf0
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parents:
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diff
changeset

210 

32036
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Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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211 
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.
avigad
parents:
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diff
changeset

212 
apply (drule le_imp_less_or_eq) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

213 
apply (auto dest!: less_imp_Suc_add) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

214 
apply (induct_tac k, auto) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

215 
done 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

216 

8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

217 
lemma fact_neg_int [simp]: "m < (0::int) \<Longrightarrow> fact m = 0" 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

218 
unfolding fact_int_def by auto 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

219 

8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

220 
lemma fact_ge_zero_int [simp]: "fact m >= (0::int)" 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

221 
apply (case_tac "m >= 0") 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

222 
apply auto 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

223 
apply (frule fact_gt_zero_int) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

224 
apply arith 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

225 
done 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

226 

8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

227 
lemma fact_mono_int_aux [rule_format]: "k >= (0::int) \<Longrightarrow> 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

228 
fact (m + k) >= fact m" 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

229 
apply (case_tac "m < 0") 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

230 
apply auto 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

231 
apply (induct k rule: int_ge_induct) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

232 
apply auto 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

233 
apply (subst add_assoc [symmetric]) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

234 
apply (subst fact_plus_one_int) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

235 
apply auto 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

236 
apply (erule order_trans) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

237 
apply (subst mult_le_cancel_right1) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

238 
apply (subgoal_tac "fact (m + i) >= 0") 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

239 
apply arith 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

240 
apply auto 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

241 
done 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

242 

8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

243 
lemma fact_mono_int: "(m::int) <= n \<Longrightarrow> fact m <= fact n" 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

244 
apply (insert fact_mono_int_aux [of "n  m" "m"]) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

245 
apply auto 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

246 
done 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

247 

8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

248 
text{*Note that @{term "fact 0 = fact 1"}*} 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

249 
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.
avigad
parents:
30242
diff
changeset

250 
apply (drule_tac m = m in less_imp_Suc_add, auto) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

251 
apply (induct_tac k, auto) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

252 
done 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

253 

8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

254 
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.
avigad
parents:
30242
diff
changeset

255 
fact m < fact ((m + 1) + k)" 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

256 
apply (induct k rule: int_ge_induct) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

257 
apply (simp add: fact_plus_one_int) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

258 
apply (subst (2) fact_reduce_int) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

259 
apply (auto simp add: add_ac) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

260 
apply (erule order_less_le_trans) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

261 
apply (subst mult_le_cancel_right1) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

262 
apply auto 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

263 
apply (subgoal_tac "fact (i + (1 + m)) >= 0") 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

264 
apply force 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

265 
apply (rule fact_ge_zero_int) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

266 
done 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

267 

8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

268 
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.
avigad
parents:
30242
diff
changeset

269 
apply (insert fact_less_mono_int_aux [of "n  (m + 1)" "m"]) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

270 
apply auto 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

271 
done 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

272 

8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
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lemma fact_num_eq_if_nat: "fact (m::nat) = 
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(if m=0 then 1 else m * fact (m  1))" 
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by (cases m) auto 
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lemma fact_add_num_eq_if_nat: 
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"fact ((m::nat) + n) = (if m + n = 0 then 1 else (m + n) * fact (m + n  1))" 
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by (cases "m + n") auto 
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lemma fact_add_num_eq_if2_nat: 
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"fact ((m::nat) + n) = 
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(if m = 0 then fact n else (m + n) * fact ((m  1) + n))" 
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by (cases m) auto 
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45930  286 
lemma fact_le_power: "fact n \<le> n^n" 
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proof (induct n) 

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case (Suc n) 

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

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then show ?case by (simp add: add_le_mono) 

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qed simp 

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subsection {* @{term fact} and @{term of_nat} *} 
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lemma of_nat_fact_not_zero [simp]: "of_nat (fact n) \<noteq> (0::'a::semiring_char_0)" 
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by auto 
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lemma of_nat_fact_gt_zero [simp]: "(0::'a::{linordered_semidom}) < of_nat(fact n)" by auto 
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lemma of_nat_fact_ge_zero [simp]: "(0::'a::linordered_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::linordered_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::linordered_field) \<le> inverse (of_nat (fact n))" 
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by (auto intro: order_less_imp_le) 
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15131  309 
end 