src/HOL/Library/NatPair.thy
 author haftmann Mon Jul 07 08:47:17 2008 +0200 (2008-07-07) changeset 27487 c8a6ce181805 parent 27368 9f90ac19e32b child 28070 f329e59cebab permissions -rw-r--r--
absolute imports of HOL/*.thy theories
```     1 (*  Title:      HOL/Library/NatPair.thy
```
```     2     ID:         \$Id\$
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```     3     Author:     Stefan Richter
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```     4     Copyright   2003 Technische Universitaet Muenchen
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```     5 *)
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```     6
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```     7 header {* Pairs of Natural Numbers *}
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```     8
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```     9 theory NatPair
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```    10 imports Plain "~~/src/HOL/Presburger"
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```    11 begin
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```    12
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```    13 text{*
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```    14   An injective function from @{text "\<nat>\<twosuperior>"} to @{text \<nat>}.  Definition
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```    15   and proofs are from \cite[page 85]{Oberschelp:1993}.
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```    16 *}
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```    17
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```    18 definition
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```    19   nat2_to_nat:: "(nat * nat) \<Rightarrow> nat" where
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```    20   "nat2_to_nat pair = (let (n,m) = pair in (n+m) * Suc (n+m) div 2 + n)"
```
```    21
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```    22 lemma dvd2_a_x_suc_a: "2 dvd a * (Suc a)"
```
```    23 proof (cases "2 dvd a")
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```    24   case True
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```    25   then show ?thesis by (rule dvd_mult2)
```
```    26 next
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```    27   case False
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```    28   then have "Suc (a mod 2) = 2" by (simp add: dvd_eq_mod_eq_0)
```
```    29   then have "Suc a mod 2 = 0" by (simp add: mod_Suc)
```
```    30   then have "2 dvd Suc a" by (simp only:dvd_eq_mod_eq_0)
```
```    31   then show ?thesis by (rule dvd_mult)
```
```    32 qed
```
```    33
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```    34 lemma
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```    35   assumes eq: "nat2_to_nat (u,v) = nat2_to_nat (x,y)"
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```    36   shows nat2_to_nat_help: "u+v \<le> x+y"
```
```    37 proof (rule classical)
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```    38   assume "\<not> ?thesis"
```
```    39   then have contrapos: "x+y < u+v"
```
```    40     by simp
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```    41   have "nat2_to_nat (x,y) < (x+y) * Suc (x+y) div 2 + Suc (x + y)"
```
```    42     by (unfold nat2_to_nat_def) (simp add: Let_def)
```
```    43   also have "\<dots> = (x+y)*Suc(x+y) div 2 + 2 * Suc(x+y) div 2"
```
```    44     by (simp only: div_mult_self1_is_m)
```
```    45   also have "\<dots> = (x+y)*Suc(x+y) div 2 + 2 * Suc(x+y) div 2
```
```    46     + ((x+y)*Suc(x+y) mod 2 + 2 * Suc(x+y) mod 2) div 2"
```
```    47   proof -
```
```    48     have "2 dvd (x+y)*Suc(x+y)"
```
```    49       by (rule dvd2_a_x_suc_a)
```
```    50     then have "(x+y)*Suc(x+y) mod 2 = 0"
```
```    51       by (simp only: dvd_eq_mod_eq_0)
```
```    52     also
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```    53     have "2 * Suc(x+y) mod 2 = 0"
```
```    54       by (rule mod_mult_self1_is_0)
```
```    55     ultimately have
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```    56       "((x+y)*Suc(x+y) mod 2 + 2 * Suc(x+y) mod 2) div 2 = 0"
```
```    57       by simp
```
```    58     then show ?thesis
```
```    59       by simp
```
```    60   qed
```
```    61   also have "\<dots> = ((x+y)*Suc(x+y) + 2*Suc(x+y)) div 2"
```
```    62     by (rule div_add1_eq [symmetric])
```
```    63   also have "\<dots> = ((x+y+2)*Suc(x+y)) div 2"
```
```    64     by (simp only: add_mult_distrib [symmetric])
```
```    65   also from contrapos have "\<dots> \<le> ((Suc(u+v))*(u+v)) div 2"
```
```    66     by (simp only: mult_le_mono div_le_mono)
```
```    67   also have "\<dots> \<le> nat2_to_nat (u,v)"
```
```    68     by (unfold nat2_to_nat_def) (simp add: Let_def)
```
```    69   finally show ?thesis
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```    70     by (simp only: eq)
```
```    71 qed
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```    72
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```    73 theorem nat2_to_nat_inj: "inj nat2_to_nat"
```
```    74 proof -
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```    75   {
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```    76     fix u v x y
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```    77     assume eq1: "nat2_to_nat (u,v) = nat2_to_nat (x,y)"
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```    78     then have "u+v \<le> x+y" by (rule nat2_to_nat_help)
```
```    79     also from eq1 [symmetric] have "x+y \<le> u+v"
```
```    80       by (rule nat2_to_nat_help)
```
```    81     finally have eq2: "u+v = x+y" .
```
```    82     with eq1 have ux: "u=x"
```
```    83       by (simp add: nat2_to_nat_def Let_def)
```
```    84     with eq2 have vy: "v=y" by simp
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```    85     with ux have "(u,v) = (x,y)" by simp
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```    86   }
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```    87   then have "\<And>x y. nat2_to_nat x = nat2_to_nat y \<Longrightarrow> x=y" by fast
```
```    88   then show ?thesis unfolding inj_on_def by simp
```
```    89 qed
```
```    90
```
```    91 end
```