--- a/src/HOL/ROOT Wed May 29 18:55:37 2013 +0200
+++ b/src/HOL/ROOT Wed May 29 23:11:21 2013 +0200
@@ -506,7 +506,6 @@
Higher_Order_Logic
Abstract_NAT
Guess
- Binary
Fundefs
Induction_Schema
LocaleTest2
--- a/src/HOL/ex/Binary.thy Wed May 29 18:55:37 2013 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,303 +0,0 @@
-(* Title: HOL/ex/Binary.thy
- Author: Makarius
-*)
-
-header {* Simple and efficient binary numerals *}
-
-theory Binary
-imports Main
-begin
-
-subsection {* Binary representation of natural numbers *}
-
-definition
- bit :: "nat \<Rightarrow> bool \<Rightarrow> nat" where
- "bit n b = (if b then 2 * n + 1 else 2 * n)"
-
-lemma bit_simps:
- "bit n False = 2 * n"
- "bit n True = 2 * n + 1"
- unfolding bit_def by simp_all
-
-ML {*
- fun dest_bit (Const (@{const_name False}, _)) = 0
- | dest_bit (Const (@{const_name True}, _)) = 1
- | dest_bit t = raise TERM ("dest_bit", [t]);
-
- fun dest_binary (Const (@{const_name Groups.zero}, Type (@{type_name nat}, _))) = 0
- | dest_binary (Const (@{const_name Groups.one}, Type (@{type_name nat}, _))) = 1
- | dest_binary (Const (@{const_name bit}, _) $ bs $ b) = 2 * dest_binary bs + dest_bit b
- | dest_binary t = raise TERM ("dest_binary", [t]);
-
- fun mk_bit 0 = @{term False}
- | mk_bit 1 = @{term True}
- | mk_bit _ = raise TERM ("mk_bit", []);
-
- fun mk_binary 0 = @{term "0::nat"}
- | mk_binary 1 = @{term "1::nat"}
- | mk_binary n =
- if n < 0 then raise TERM ("mk_binary", [])
- else
- let val (q, r) = Integer.div_mod n 2
- in @{term bit} $ mk_binary q $ mk_bit r end;
-*}
-
-
-subsection {* Direct operations -- plain normalization *}
-
-lemma binary_norm:
- "bit 0 False = 0"
- "bit 0 True = 1"
- unfolding bit_def by simp_all
-
-lemma binary_add:
- "n + 0 = n"
- "0 + n = n"
- "1 + 1 = bit 1 False"
- "bit n False + 1 = bit n True"
- "bit n True + 1 = bit (n + 1) False"
- "1 + bit n False = bit n True"
- "1 + bit n True = bit (n + 1) False"
- "bit m False + bit n False = bit (m + n) False"
- "bit m False + bit n True = bit (m + n) True"
- "bit m True + bit n False = bit (m + n) True"
- "bit m True + bit n True = bit ((m + n) + 1) False"
- by (simp_all add: bit_simps)
-
-lemma binary_mult:
- "n * 0 = 0"
- "0 * n = 0"
- "n * 1 = n"
- "1 * n = n"
- "bit m True * n = bit (m * n) False + n"
- "bit m False * n = bit (m * n) False"
- "n * bit m True = bit (m * n) False + n"
- "n * bit m False = bit (m * n) False"
- by (simp_all add: bit_simps)
-
-lemmas binary_simps = binary_norm binary_add binary_mult
-
-
-subsection {* Indirect operations -- ML will produce witnesses *}
-
-lemma binary_less_eq:
- fixes n :: nat
- shows "n \<equiv> m + k \<Longrightarrow> (m \<le> n) \<equiv> True"
- and "m \<equiv> n + k + 1 \<Longrightarrow> (m \<le> n) \<equiv> False"
- by simp_all
-
-lemma binary_less:
- fixes n :: nat
- shows "m \<equiv> n + k \<Longrightarrow> (m < n) \<equiv> False"
- and "n \<equiv> m + k + 1 \<Longrightarrow> (m < n) \<equiv> True"
- by simp_all
-
-lemma binary_diff:
- fixes n :: nat
- shows "m \<equiv> n + k \<Longrightarrow> m - n \<equiv> k"
- and "n \<equiv> m + k \<Longrightarrow> m - n \<equiv> 0"
- by simp_all
-
-lemma binary_divmod:
- fixes n :: nat
- assumes "m \<equiv> n * k + l" and "0 < n" and "l < n"
- shows "m div n \<equiv> k"
- and "m mod n \<equiv> l"
-proof -
- from `m \<equiv> n * k + l` have "m = l + k * n" by simp
- with `0 < n` and `l < n` show "m div n \<equiv> k" and "m mod n \<equiv> l" by simp_all
-qed
-
-ML {*
-local
- infix ==;
- val op == = Logic.mk_equals;
- fun plus m n = @{term "plus :: nat \<Rightarrow> nat \<Rightarrow> nat"} $ m $ n;
- fun mult m n = @{term "times :: nat \<Rightarrow> nat \<Rightarrow> nat"} $ m $ n;
-
- val binary_ss =
- simpset_of (put_simpset HOL_basic_ss @{context} addsimps @{thms binary_simps});
- fun prove ctxt prop =
- Goal.prove ctxt [] [] prop
- (fn _ => ALLGOALS (full_simp_tac (put_simpset binary_ss ctxt)));
-
- fun binary_proc proc ctxt ct =
- (case Thm.term_of ct of
- _ $ t $ u =>
- (case try (pairself (`dest_binary)) (t, u) of
- SOME args => proc ctxt args
- | NONE => NONE)
- | _ => NONE);
-in
-
-val less_eq_proc = binary_proc (fn ctxt => fn ((m, t), (n, u)) =>
- let val k = n - m in
- if k >= 0 then
- SOME (@{thm binary_less_eq(1)} OF [prove ctxt (u == plus t (mk_binary k))])
- else
- SOME (@{thm binary_less_eq(2)} OF
- [prove ctxt (t == plus (plus u (mk_binary (~ k - 1))) (mk_binary 1))])
- end);
-
-val less_proc = binary_proc (fn ctxt => fn ((m, t), (n, u)) =>
- let val k = m - n in
- if k >= 0 then
- SOME (@{thm binary_less(1)} OF [prove ctxt (t == plus u (mk_binary k))])
- else
- SOME (@{thm binary_less(2)} OF
- [prove ctxt (u == plus (plus t (mk_binary (~ k - 1))) (mk_binary 1))])
- end);
-
-val diff_proc = binary_proc (fn ctxt => fn ((m, t), (n, u)) =>
- let val k = m - n in
- if k >= 0 then
- SOME (@{thm binary_diff(1)} OF [prove ctxt (t == plus u (mk_binary k))])
- else
- SOME (@{thm binary_diff(2)} OF [prove ctxt (u == plus t (mk_binary (~ k)))])
- end);
-
-fun divmod_proc rule = binary_proc (fn ctxt => fn ((m, t), (n, u)) =>
- if n = 0 then NONE
- else
- let val (k, l) = Integer.div_mod m n
- in SOME (rule OF [prove ctxt (t == plus (mult u (mk_binary k)) (mk_binary l))]) end);
-
-end;
-*}
-
-simproc_setup binary_nat_less_eq ("m <= (n::nat)") = {* K less_eq_proc *}
-simproc_setup binary_nat_less ("m < (n::nat)") = {* K less_proc *}
-simproc_setup binary_nat_diff ("m - (n::nat)") = {* K diff_proc *}
-simproc_setup binary_nat_div ("m div (n::nat)") = {* K (divmod_proc @{thm binary_divmod(1)}) *}
-simproc_setup binary_nat_mod ("m mod (n::nat)") = {* K (divmod_proc @{thm binary_divmod(2)}) *}
-
-method_setup binary_simp = {*
- Scan.succeed (fn ctxt => SIMPLE_METHOD'
- (full_simp_tac
- (put_simpset HOL_basic_ss ctxt
- addsimps @{thms binary_simps}
- addsimprocs
- [@{simproc binary_nat_less_eq},
- @{simproc binary_nat_less},
- @{simproc binary_nat_diff},
- @{simproc binary_nat_div},
- @{simproc binary_nat_mod}])))
-*}
-
-
-subsection {* Concrete syntax *}
-
-syntax
- "_Binary" :: "num_const \<Rightarrow> 'a" ("$_")
-
-parse_translation {*
- let
- val syntax_consts =
- map_aterms (fn Const (c, T) => Const (Lexicon.mark_const c, T) | a => a);
-
- fun binary_tr [(c as Const (@{syntax_const "_constrain"}, _)) $ t $ u] = c $ binary_tr [t] $ u
- | binary_tr [Const (num, _)] =
- let
- val {leading_zeros = z, value = n, ...} = Lexicon.read_xnum num;
- val _ = z = 0 andalso n >= 0 orelse error ("Bad binary number: " ^ num);
- in syntax_consts (mk_binary n) end
- | binary_tr ts = raise TERM ("binary_tr", ts);
-
- in [(@{syntax_const "_Binary"}, K binary_tr)] end
-*}
-
-
-subsection {* Examples *}
-
-lemma "$6 = 6"
- by (simp add: bit_simps)
-
-lemma "bit (bit (bit 0 False) False) True = 1"
- by (simp add: bit_simps)
-
-lemma "bit (bit (bit 0 False) False) True = bit 0 True"
- by (simp add: bit_simps)
-
-lemma "$5 + $3 = $8"
- by binary_simp
-
-lemma "$5 * $3 = $15"
- by binary_simp
-
-lemma "$5 - $3 = $2"
- by binary_simp
-
-lemma "$3 - $5 = 0"
- by binary_simp
-
-lemma "$123456789 - $123 = $123456666"
- by binary_simp
-
-lemma "$1111111111222222222233333333334444444444 - $998877665544332211 =
- $1111111111222222222232334455668900112233"
- by binary_simp
-
-lemma "(1111111111222222222233333333334444444444::nat) - 998877665544332211 =
- 1111111111222222222232334455668900112233"
- by simp
-
-lemma "(1111111111222222222233333333334444444444::int) - 998877665544332211 =
- 1111111111222222222232334455668900112233"
- by simp
-
-lemma "$1111111111222222222233333333334444444444 * $998877665544332211 =
- $1109864072938022197293802219729380221972383090160869185684"
- by binary_simp
-
-lemma "$1111111111222222222233333333334444444444 * $998877665544332211 -
- $5555555555666666666677777777778888888888 =
- $1109864072938022191738246664062713555294605312381980296796"
- by binary_simp
-
-lemma "$42 < $4 = False"
- by binary_simp
-
-lemma "$4 < $42 = True"
- by binary_simp
-
-lemma "$42 <= $4 = False"
- by binary_simp
-
-lemma "$4 <= $42 = True"
- by binary_simp
-
-lemma "$1111111111222222222233333333334444444444 < $998877665544332211 = False"
- by binary_simp
-
-lemma "$998877665544332211 < $1111111111222222222233333333334444444444 = True"
- by binary_simp
-
-lemma "$1111111111222222222233333333334444444444 <= $998877665544332211 = False"
- by binary_simp
-
-lemma "$998877665544332211 <= $1111111111222222222233333333334444444444 = True"
- by binary_simp
-
-lemma "$1234 div $23 = $53"
- by binary_simp
-
-lemma "$1234 mod $23 = $15"
- by binary_simp
-
-lemma "$1111111111222222222233333333334444444444 div $998877665544332211 =
- $1112359550673033707875"
- by binary_simp
-
-lemma "$1111111111222222222233333333334444444444 mod $998877665544332211 =
- $42245174317582819"
- by binary_simp
-
-lemma "(1111111111222222222233333333334444444444::int) div 998877665544332211 =
- 1112359550673033707875"
- by simp -- {* legacy numerals: 30 times slower *}
-
-lemma "(1111111111222222222233333333334444444444::int) mod 998877665544332211 =
- 42245174317582819"
- by simp -- {* legacy numerals: 30 times slower *}
-
-end