src/HOL/Hyperreal/Fact.thy
author nipkow
Wed, 18 Aug 2004 11:09:40 +0200
changeset 15140 322485b816ac
parent 15131 c69542757a4d
child 15241 a3949068537e
permissions -rw-r--r--
import -> imports
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
15094
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
     1
(*  Title       : Fact.thy
12196
a3be6b3a9c0b new theories from Jacques Fleuriot
paulson
parents:
diff changeset
     2
    Author      : Jacques D. Fleuriot
a3be6b3a9c0b new theories from Jacques Fleuriot
paulson
parents:
diff changeset
     3
    Copyright   : 1998  University of Cambridge
15094
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
     4
    Conversion to Isar and new proofs by Lawrence C Paulson, 2004
12196
a3be6b3a9c0b new theories from Jacques Fleuriot
paulson
parents:
diff changeset
     5
*)
a3be6b3a9c0b new theories from Jacques Fleuriot
paulson
parents:
diff changeset
     6
15094
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
     7
header{*Factorial Function*}
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
     8
15131
c69542757a4d New theory header syntax.
nipkow
parents: 15094
diff changeset
     9
theory Fact
15140
322485b816ac import -> imports
nipkow
parents: 15131
diff changeset
    10
imports Real
15131
c69542757a4d New theory header syntax.
nipkow
parents: 15094
diff changeset
    11
begin
15094
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    12
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    13
consts fact :: "nat => nat"
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    14
primrec
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    15
   fact_0:     "fact 0 = 1"
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    16
   fact_Suc:   "fact (Suc n) = (Suc n) * fact n"
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    17
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    18
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    19
lemma fact_gt_zero [simp]: "0 < fact n"
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    20
by (induct "n", auto)
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    21
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    22
lemma fact_not_eq_zero [simp]: "fact n \<noteq> 0"
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    23
by simp
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    24
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    25
lemma real_of_nat_fact_not_zero [simp]: "real (fact n) \<noteq> 0"
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    26
by auto
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    27
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    28
lemma real_of_nat_fact_gt_zero [simp]: "0 < real(fact n)"
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    29
by auto
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    30
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    31
lemma real_of_nat_fact_ge_zero [simp]: "0 \<le> real(fact n)"
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    32
by simp
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    33
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    34
lemma fact_ge_one [simp]: "1 \<le> fact n"
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    35
by (induct "n", auto)
12196
a3be6b3a9c0b new theories from Jacques Fleuriot
paulson
parents:
diff changeset
    36
15094
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    37
lemma fact_mono: "m \<le> n ==> fact m \<le> fact n"
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    38
apply (drule le_imp_less_or_eq)
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    39
apply (auto dest!: less_imp_Suc_add)
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    40
apply (induct_tac "k", auto)
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    41
done
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    42
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    43
text{*Note that @{term "fact 0 = fact 1"}*}
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    44
lemma fact_less_mono: "[| 0 < m; m < n |] ==> fact m < fact n"
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    45
apply (drule_tac m = m in less_imp_Suc_add, auto)
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    46
apply (induct_tac "k", auto)
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    47
done
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    48
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    49
lemma inv_real_of_nat_fact_gt_zero [simp]: "0 < inverse (real (fact n))"
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    50
by (auto simp add: positive_imp_inverse_positive)
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    51
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    52
lemma inv_real_of_nat_fact_ge_zero [simp]: "0 \<le> inverse (real (fact n))"
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    53
by (auto intro: order_less_imp_le)
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    54
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    55
lemma fact_diff_Suc [rule_format]:
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    56
     "\<forall>m. n < Suc m --> fact (Suc m - n) = (Suc m - n) * fact (m - n)"
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    57
apply (induct n, auto)
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    58
apply (drule_tac x = "m - 1" in spec, auto)
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    59
done
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    60
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    61
lemma fact_num0 [simp]: "fact 0 = 1"
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    62
by auto
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    63
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    64
lemma fact_num_eq_if: "fact m = (if m=0 then 1 else m * fact (m - 1))"
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    65
by (case_tac "m", auto)
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    66
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    67
lemma fact_add_num_eq_if:
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    68
     "fact (m+n) = (if (m+n = 0) then 1 else (m+n) * (fact (m + n - 1)))"
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    69
by (case_tac "m+n", auto)
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    70
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    71
lemma fact_add_num_eq_if2:
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    72
     "fact (m+n) = (if m=0 then fact n else (m+n) * (fact ((m - 1) + n)))"
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    73
by (case_tac "m", auto)
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
    74
12196
a3be6b3a9c0b new theories from Jacques Fleuriot
paulson
parents:
diff changeset
    75
15131
c69542757a4d New theory header syntax.
nipkow
parents: 15094
diff changeset
    76
end