
haftmann@22799: (*  Title:      HOL/Library/Char_nat.thy
haftmann@22799:     Author:     Norbert Voelker, Florian Haftmann
haftmann@22799: *)
haftmann@22799: 
haftmann@22799: header {* Mapping between characters and natural numbers *}
haftmann@22799: 
haftmann@22799: theory Char_nat
haftmann@30663: imports List Main
haftmann@22799: begin
haftmann@22799: 
haftmann@22799: text {* Conversions between nibbles and natural numbers in [0..15]. *}
haftmann@22799: 
haftmann@26152: primrec
haftmann@22799:   nat_of_nibble :: "nibble \<Rightarrow> nat" where
wenzelm@23394:     "nat_of_nibble Nibble0 = 0"
haftmann@22799:   | "nat_of_nibble Nibble1 = 1"
haftmann@22799:   | "nat_of_nibble Nibble2 = 2"
haftmann@22799:   | "nat_of_nibble Nibble3 = 3"
haftmann@22799:   | "nat_of_nibble Nibble4 = 4"
haftmann@22799:   | "nat_of_nibble Nibble5 = 5"
haftmann@22799:   | "nat_of_nibble Nibble6 = 6"
haftmann@22799:   | "nat_of_nibble Nibble7 = 7"
haftmann@22799:   | "nat_of_nibble Nibble8 = 8"
haftmann@22799:   | "nat_of_nibble Nibble9 = 9"
haftmann@22799:   | "nat_of_nibble NibbleA = 10"
haftmann@22799:   | "nat_of_nibble NibbleB = 11"
haftmann@22799:   | "nat_of_nibble NibbleC = 12"
haftmann@22799:   | "nat_of_nibble NibbleD = 13"
haftmann@22799:   | "nat_of_nibble NibbleE = 14"
haftmann@22799:   | "nat_of_nibble NibbleF = 15"
haftmann@22799: 
haftmann@22799: definition
haftmann@22799:   nibble_of_nat :: "nat \<Rightarrow> nibble" where
haftmann@22799:   "nibble_of_nat x = (let y = x mod 16 in
haftmann@22799:     if y = 0 then Nibble0 else
haftmann@22799:     if y = 1 then Nibble1 else
haftmann@22799:     if y = 2 then Nibble2 else
haftmann@22799:     if y = 3 then Nibble3 else
haftmann@22799:     if y = 4 then Nibble4 else
haftmann@22799:     if y = 5 then Nibble5 else
haftmann@22799:     if y = 6 then Nibble6 else
haftmann@22799:     if y = 7 then Nibble7 else
haftmann@22799:     if y = 8 then Nibble8 else
haftmann@22799:     if y = 9 then Nibble9 else
haftmann@22799:     if y = 10 then NibbleA else
haftmann@22799:     if y = 11 then NibbleB else
haftmann@22799:     if y = 12 then NibbleC else
haftmann@22799:     if y = 13 then NibbleD else
haftmann@22799:     if y = 14 then NibbleE else
haftmann@22799:     NibbleF)"
haftmann@22799: 
haftmann@22799: lemma nibble_of_nat_norm:
haftmann@22799:   "nibble_of_nat (n mod 16) = nibble_of_nat n"
huffman@36843:   unfolding nibble_of_nat_def mod_mod_trivial ..
haftmann@22799: 
haftmann@22799: lemma nibble_of_nat_simps [simp]:
haftmann@22799:   "nibble_of_nat  0 = Nibble0"
haftmann@22799:   "nibble_of_nat  1 = Nibble1"
haftmann@22799:   "nibble_of_nat  2 = Nibble2"
haftmann@22799:   "nibble_of_nat  3 = Nibble3"
haftmann@22799:   "nibble_of_nat  4 = Nibble4"
haftmann@22799:   "nibble_of_nat  5 = Nibble5"
haftmann@22799:   "nibble_of_nat  6 = Nibble6"
haftmann@22799:   "nibble_of_nat  7 = Nibble7"
haftmann@22799:   "nibble_of_nat  8 = Nibble8"
haftmann@22799:   "nibble_of_nat  9 = Nibble9"
haftmann@22799:   "nibble_of_nat 10 = NibbleA"
haftmann@22799:   "nibble_of_nat 11 = NibbleB"
haftmann@22799:   "nibble_of_nat 12 = NibbleC"
haftmann@22799:   "nibble_of_nat 13 = NibbleD"
haftmann@22799:   "nibble_of_nat 14 = NibbleE"
haftmann@22799:   "nibble_of_nat 15 = NibbleF"
huffman@36843:   unfolding nibble_of_nat_def by auto
haftmann@22799: 
haftmann@22799: lemma nibble_of_nat_of_nibble: "nibble_of_nat (nat_of_nibble n) = n"
haftmann@22799:   by (cases n) (simp_all only: nat_of_nibble.simps nibble_of_nat_simps)
haftmann@22799: 
haftmann@22799: lemma nat_of_nibble_of_nat: "nat_of_nibble (nibble_of_nat n) = n mod 16"
haftmann@22799: proof -
wenzelm@23394:   have nibble_nat_enum:
huffman@36843:     "n mod 16 \<in> {15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0}"
haftmann@22799:   proof -
haftmann@22799:     have set_unfold: "\<And>n. {0..Suc n} = insert (Suc n) {0..n}" by auto
haftmann@22799:     have "(n\<Colon>nat) mod 16 \<in> {0..Suc (Suc (Suc (Suc (Suc (Suc (Suc (Suc (Suc
haftmann@22799:       (Suc (Suc (Suc (Suc (Suc (Suc 0))))))))))))))}" by simp
haftmann@22799:     from this [simplified set_unfold atLeastAtMost_singleton]
huffman@36843:     show ?thesis by (simp add: numeral_2_eq_2 [symmetric])
haftmann@22799:   qed
huffman@36843:   then show ?thesis unfolding nibble_of_nat_def
haftmann@22799:   by auto
haftmann@22799: qed
haftmann@22799: 
haftmann@22799: lemma inj_nat_of_nibble: "inj nat_of_nibble"
haftmann@22799:   by (rule inj_on_inverseI) (rule nibble_of_nat_of_nibble)
haftmann@22799: 
haftmann@22799: lemma nat_of_nibble_eq: "nat_of_nibble n = nat_of_nibble m \<longleftrightarrow> n = m"
haftmann@22799:   by (rule inj_eq) (rule inj_nat_of_nibble)
haftmann@22799: 
haftmann@22799: lemma nat_of_nibble_less_16: "nat_of_nibble n < 16"
haftmann@22799:   by (cases n) auto
haftmann@22799: 
haftmann@22799: lemma nat_of_nibble_div_16: "nat_of_nibble n div 16 = 0"
haftmann@22799:   by (cases n) auto
haftmann@22799: 
haftmann@22799: 
haftmann@22799: text {* Conversion between chars and nats. *}
haftmann@22799: 
haftmann@22799: definition
wenzelm@23394:   nibble_pair_of_nat :: "nat \<Rightarrow> nibble \<times> nibble" where
haftmann@22799:   "nibble_pair_of_nat n = (nibble_of_nat (n div 16), nibble_of_nat (n mod 16))"
haftmann@22799: 
haftmann@28562: lemma nibble_of_pair [code]:
haftmann@22799:   "nibble_pair_of_nat n = (nibble_of_nat (n div 16), nibble_of_nat n)"
haftmann@22799:   unfolding nibble_of_nat_norm [of n, symmetric] nibble_pair_of_nat_def ..
haftmann@22799: 
haftmann@26152: primrec
haftmann@22799:   nat_of_char :: "char \<Rightarrow> nat" where
haftmann@22799:   "nat_of_char (Char n m) = nat_of_nibble n * 16 + nat_of_nibble m"
haftmann@22799: 
haftmann@22799: lemmas [simp del] = nat_of_char.simps
haftmann@22799: 
haftmann@22799: definition
haftmann@22799:   char_of_nat :: "nat \<Rightarrow> char" where
haftmann@22799:   char_of_nat_def: "char_of_nat n = split Char (nibble_pair_of_nat n)"
haftmann@22799: 
haftmann@22799: lemma Char_char_of_nat:
haftmann@22799:   "Char n m = char_of_nat (nat_of_nibble n * 16 + nat_of_nibble m)"
haftmann@22799:   unfolding char_of_nat_def Let_def nibble_pair_of_nat_def
nipkow@30224:   by (auto simp add: div_add1_eq mod_add_eq nat_of_nibble_div_16 nibble_of_nat_norm nibble_of_nat_of_nibble)
haftmann@22799: 
haftmann@22799: lemma char_of_nat_of_char:
haftmann@22799:   "char_of_nat (nat_of_char c) = c"
haftmann@22799:   by (cases c) (simp add: nat_of_char.simps, simp add: Char_char_of_nat)
haftmann@22799: 
haftmann@22799: lemma nat_of_char_of_nat:
haftmann@22799:   "nat_of_char (char_of_nat n) = n mod 256"
haftmann@22799: proof -
haftmann@22799:   from mod_div_equality [of n, symmetric, of 16]
haftmann@22799:   have mod_mult_self3: "\<And>m k n \<Colon> nat. (k * n + m) mod n = m mod n"
haftmann@22799:   proof -
haftmann@22799:     fix m k n :: nat
haftmann@22799:     show "(k * n + m) mod n = m mod n"
wenzelm@23394:       by (simp only: mod_mult_self1 [symmetric, of m n k] add_commute)
haftmann@22799:   qed
haftmann@22799:   from mod_div_decomp [of n 256] obtain k l where n: "n = k * 256 + l"
haftmann@22799:     and k: "k = n div 256" and l: "l = n mod 256" by blast
haftmann@22799:   have 16: "(0::nat) < 16" by auto
haftmann@22799:   have 256: "(256 :: nat) = 16 * 16" by auto
haftmann@22799:   have l_256: "l mod 256 = l" using l by auto
haftmann@22799:   have l_div_256: "l div 16 * 16 mod 256 = l div 16 * 16"
haftmann@22799:     using l by auto
haftmann@22799:   have aux2: "(k * 256 mod 16 + l mod 16) div 16 = 0"
haftmann@27651:     unfolding 256 mult_assoc [symmetric] mod_mult_self2_is_0 by simp
haftmann@22799:   have aux3: "(k * 256 + l) div 16 = k * 16 + l div 16"
haftmann@22799:     unfolding div_add1_eq [of "k * 256" l 16] aux2 256
haftmann@22799:       mult_assoc [symmetric] div_mult_self_is_m [OF 16] by simp
haftmann@22799:   have aux4: "(k * 256 + l) mod 16 = l mod 16"
haftmann@22799:     unfolding 256 mult_assoc [symmetric] mod_mult_self3 ..
haftmann@22799:   show ?thesis
wenzelm@23394:     by (simp add: nat_of_char.simps char_of_nat_def nibble_of_pair
huffman@47159:       nat_of_nibble_of_nat mult_mod_left
wenzelm@46730:       n aux3 l_256 aux4 mod_add_eq [of "256 * k"] l_div_256)
haftmann@22799: qed
haftmann@22799: 
haftmann@22799: lemma nibble_pair_of_nat_char:
haftmann@22799:   "nibble_pair_of_nat (nat_of_char (Char n m)) = (n, m)"
haftmann@22799: proof -
haftmann@22799:   have nat_of_nibble_256:
wenzelm@23394:     "\<And>n m. (nat_of_nibble n * 16 + nat_of_nibble m) mod 256 =
wenzelm@23394:       nat_of_nibble n * 16 + nat_of_nibble m"
haftmann@22799:   proof -
haftmann@22799:     fix n m
haftmann@22799:     have nat_of_nibble_less_eq_15: "\<And>n. nat_of_nibble n \<le> 15"
nipkow@40077:       using Suc_leI [OF nat_of_nibble_less_16] by (auto simp add: eval_nat_numeral)
wenzelm@23394:     have less_eq_240: "nat_of_nibble n * 16 \<le> 240"
wenzelm@23394:       using nat_of_nibble_less_eq_15 by auto
haftmann@22799:     have "nat_of_nibble n * 16 + nat_of_nibble m \<le> 240 + 15"
wenzelm@23394:       by (rule add_le_mono [of _ 240 _ 15]) (auto intro: nat_of_nibble_less_eq_15 less_eq_240)
haftmann@22799:     then have "nat_of_nibble n * 16 + nat_of_nibble m < 256" (is "?rhs < _") by auto
haftmann@22799:     then show "?rhs mod 256 = ?rhs" by auto
haftmann@22799:   qed
haftmann@22799:   show ?thesis
wenzelm@23394:     unfolding nibble_pair_of_nat_def Char_char_of_nat nat_of_char_of_nat nat_of_nibble_256
wenzelm@23394:     by (simp add: add_commute nat_of_nibble_div_16 nibble_of_nat_norm nibble_of_nat_of_nibble)
haftmann@22799: qed
haftmann@22799: 
haftmann@22799: 
haftmann@22799: text {* Code generator setup *}
haftmann@22799: 
haftmann@22799: code_modulename SML
haftmann@37221:   Char_nat String
haftmann@22799: 
haftmann@22799: code_modulename OCaml
haftmann@37221:   Char_nat String
haftmann@22799: 
haftmann@22799: code_modulename Haskell
haftmann@37221:   Char_nat String
haftmann@22799: 
haftmann@22799: end



