author  avigad 
Fri, 17 Jul 2009 13:12:18 0400  
changeset 32047  c141f139ce26 
parent 32039  400a519bc888 
child 32558  e6e1fc2e73cb 
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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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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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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136 
apply (subgoal_tac "1 = int 1") 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

137 
apply (erule ssubst) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

138 
apply (subst zle_int) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

139 
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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140 
done 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

141 

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

142 
lemma dvd_fact_nat [rule_format]: "1 <= m \<longrightarrow> m <= n \<longrightarrow> m dvd fact (n::nat)" 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

143 
apply (induct n) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

144 
apply force 
32047  145 
apply (auto simp only: fact_Suc) 
32036
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

146 
apply (subgoal_tac "m = Suc n") 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

147 
apply (erule ssubst) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

148 
apply (rule dvd_triv_left) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

149 
apply auto 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

150 
done 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

151 

8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
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diff
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152 
lemma dvd_fact_int [rule_format]: "1 <= m \<longrightarrow> m <= n \<longrightarrow> m dvd fact (n::int)" 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

153 
apply (case_tac "1 <= n") 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

154 
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

155 
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

156 
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
changeset

157 
apply auto 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

158 
done 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

159 

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

160 
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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diff
changeset

161 
{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

162 
by auto 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

163 

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

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.
avigad
parents:
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diff
changeset

165 
by auto 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

166 

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

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.
avigad
parents:
30242
diff
changeset

168 
by auto 
15094
a7d1a3fdc30d
conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents:
12196
diff
changeset

169 

32036
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

170 
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

171 
apply (induct n) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

172 
apply force 
32047  173 
apply (subst fact_Suc) 
32036
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

174 
apply (subst interval_Suc) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

175 
apply auto 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

176 
done 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
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diff
changeset

177 

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

178 
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

179 
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

180 
apply force 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

181 
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
changeset

182 
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

183 
apply auto 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

184 
done 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

185 

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

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.
avigad
parents:
30242
diff
changeset

187 
apply (drule le_imp_less_or_eq) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

188 
apply (auto dest!: less_imp_Suc_add) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

189 
apply (induct_tac k, auto) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

190 
done 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

191 

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

192 
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:
30242
diff
changeset

193 
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

194 

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

195 
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

196 
apply (case_tac "m >= 0") 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

197 
apply auto 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

198 
apply (frule fact_gt_zero_int) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

199 
apply arith 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

200 
done 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

201 

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

202 
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:
30242
diff
changeset

203 
fact (m + k) >= fact m" 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

204 
apply (case_tac "m < 0") 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

205 
apply auto 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

206 
apply (induct k rule: int_ge_induct) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

207 
apply auto 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

208 
apply (subst add_assoc [symmetric]) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

209 
apply (subst fact_plus_one_int) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

210 
apply auto 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

211 
apply (erule order_trans) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

212 
apply (subst mult_le_cancel_right1) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

213 
apply (subgoal_tac "fact (m + i) >= 0") 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

214 
apply arith 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

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

216 
done 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

217 

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

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.
avigad
parents:
30242
diff
changeset

219 
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

220 
apply auto 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

221 
done 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

222 

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

223 
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

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.
avigad
parents:
30242
diff
changeset

225 
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

226 
apply (induct_tac k, auto) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

227 
done 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

228 

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

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.
avigad
parents:
30242
diff
changeset

230 
fact m < fact ((m + 1) + k)" 
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 (simp add: fact_plus_one_int) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

233 
apply (subst mult_less_cancel_right1) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

234 
apply (insert fact_gt_zero_int [of m], arith) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

235 
apply (subst (2) fact_reduce_int) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

236 
apply (auto simp add: add_ac) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

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

238 
apply (subst mult_le_cancel_right1) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

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

240 
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

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

242 
apply (rule fact_ge_zero_int) 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

243 
done 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

244 

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

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.
avigad
parents:
30242
diff
changeset

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.
avigad
parents:
30242
diff
changeset

247 
apply auto 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

248 
done 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

249 

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

250 
lemma fact_num_eq_if_nat: "fact (m::nat) = 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

251 
(if m=0 then 1 else m * fact (m  1))" 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

252 
by (cases m) auto 
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_add_num_eq_if_nat: 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

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.
avigad
parents:
30242
diff
changeset

256 
by (cases "m + n") auto 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

257 

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

258 
lemma fact_add_num_eq_if2_nat: 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

259 
"fact ((m::nat) + n) = 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

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.
avigad
parents:
30242
diff
changeset

261 
by (cases m) auto 
8a9228872fbd
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents:
30242
diff
changeset

262 

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

263 

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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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class ordered_semiring_1 = ordered_semiring + semiring_1 
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class ordered_semiring_1_strict = ordered_semiring_strict + semiring_1 
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lemma of_nat_fact_gt_zero [simp]: "(0::'a::{ordered_semidom}) < of_nat(fact n)" by auto 
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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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15131  283 
end 