src/HOL/Library/NatPair.thy
 author wenzelm Thu May 06 14:14:18 2004 +0200 (2004-05-06) changeset 14706 71590b7733b7 parent 14414 3fd75e96145d child 15131 c69542757a4d permissions -rw-r--r--
tuned document;
```     1 (*  Title:      HOL/Library/NatPair.thy
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```     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 = Main:
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```    10
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```    11 text{*
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```    12   An injective function from @{text "\<nat>\<twosuperior>"} to @{text
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```    13   \<nat>}.  Definition and proofs are from \cite[page
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```    14   85]{Oberschelp:1993}.
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```    15 *}
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```    16
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```    17 constdefs
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```    18   nat2_to_nat:: "(nat * nat) \<Rightarrow> nat"
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```    19   "nat2_to_nat pair \<equiv> let (n,m) = pair in (n+m) * Suc (n+m) div 2 + n"
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```    20
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```    21 lemma dvd2_a_x_suc_a: "2 dvd a * (Suc a)"
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```    22 proof (cases "2 dvd a")
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```    23   case True
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```    24   thus ?thesis by (rule dvd_mult2)
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```    25 next
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```    26   case False
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```    27   hence "Suc (a mod 2) = 2" by (simp add: dvd_eq_mod_eq_0)
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```    28   hence "Suc a mod 2 = 0" by (simp add: mod_Suc)
```
```    29   hence "2 dvd Suc a" by (simp only:dvd_eq_mod_eq_0)
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```    30   thus ?thesis by (rule dvd_mult)
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```    31 qed
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```    32
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```    33 lemma
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```    34   assumes eq: "nat2_to_nat (u,v) = nat2_to_nat (x,y)"
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```    35   shows nat2_to_nat_help: "u+v \<le> x+y"
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```    36 proof (rule classical)
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```    37   assume "\<not> ?thesis"
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```    38   hence contrapos: "x+y < u+v"
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```    39     by simp
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```    40   have "nat2_to_nat (x,y) < (x+y) * Suc (x+y) div 2 + Suc (x + y)"
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```    41     by (unfold nat2_to_nat_def) (simp add: Let_def)
```
```    42   also have "\<dots> = (x+y)*Suc(x+y) div 2 + 2 * Suc(x+y) div 2"
```
```    43     by (simp only: div_mult_self1_is_m)
```
```    44   also have "\<dots> = (x+y)*Suc(x+y) div 2 + 2 * Suc(x+y) div 2
```
```    45     + ((x+y)*Suc(x+y) mod 2 + 2 * Suc(x+y) mod 2) div 2"
```
```    46   proof -
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```    47     have "2 dvd (x+y)*Suc(x+y)"
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```    48       by (rule dvd2_a_x_suc_a)
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```    49     hence "(x+y)*Suc(x+y) mod 2 = 0"
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```    50       by (simp only: dvd_eq_mod_eq_0)
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```    51     also
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```    52     have "2 * Suc(x+y) mod 2 = 0"
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```    53       by (rule mod_mult_self1_is_0)
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```    54     ultimately have
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```    55       "((x+y)*Suc(x+y) mod 2 + 2 * Suc(x+y) mod 2) div 2 = 0"
```
```    56       by simp
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```    57     thus ?thesis
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```    58       by simp
```
```    59   qed
```
```    60   also have "\<dots> = ((x+y)*Suc(x+y) + 2*Suc(x+y)) div 2"
```
```    61     by (rule div_add1_eq [symmetric])
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```    62   also have "\<dots> = ((x+y+2)*Suc(x+y)) div 2"
```
```    63     by (simp only: add_mult_distrib [symmetric])
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```    64   also from contrapos have "\<dots> \<le> ((Suc(u+v))*(u+v)) div 2"
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```    65     by (simp only: mult_le_mono div_le_mono)
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```    66   also have "\<dots> \<le> nat2_to_nat (u,v)"
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```    67     by (unfold nat2_to_nat_def) (simp add: Let_def)
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```    68   finally show ?thesis
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```    69     by (simp only: eq)
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```    70 qed
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```    71
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```    72 theorem nat2_to_nat_inj: "inj nat2_to_nat"
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```    73 proof -
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```    74   {
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```    75     fix u v x y assume "nat2_to_nat (u,v) = nat2_to_nat (x,y)"
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```    76     hence "u+v \<le> x+y" by (rule nat2_to_nat_help)
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```    77     also from prems [symmetric] have "x+y \<le> u+v"
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```    78       by (rule nat2_to_nat_help)
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```    79     finally have eq: "u+v = x+y" .
```
```    80     with prems have ux: "u=x"
```
```    81       by (simp add: nat2_to_nat_def Let_def)
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```    82     with eq have vy: "v=y"
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```    83       by simp
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```    84     with ux have "(u,v) = (x,y)"
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```    85       by simp
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```    86   }
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```    87   hence "\<And>x y. nat2_to_nat x = nat2_to_nat y \<Longrightarrow> x=y"
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```    88     by fast
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```    89   thus ?thesis
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```    90     by (unfold inj_on_def) simp
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```    91 qed
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```    92
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```    93 end
```