diff -r db158e995bfc -r 9df6144e281b Quot/Examples/IntEx2.thy
--- a/Quot/Examples/IntEx2.thy	Thu Feb 25 07:48:57 2010 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,445 +0,0 @@
-theory IntEx2
-imports "../Quotient" "../Quotient_Product" Nat
-(*uses
-  ("Tools/numeral.ML")
-  ("Tools/numeral_syntax.ML")
-  ("Tools/int_arith.ML")*)
-begin
-
-fun
-  intrel :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat) \<Rightarrow> bool" (infix "\<approx>" 50)
-where
-  "intrel (x, y) (u, v) = (x + v = u + y)"
-
-quotient_type int = "nat \<times> nat" / intrel
-  unfolding equivp_def
-  by (auto simp add: mem_def expand_fun_eq)
-
-instantiation int :: "{zero, one, plus, uminus, minus, times, ord, abs, sgn}"
-begin
-
-ML {* @{term "0 \<Colon> int"} *}
-
-quotient_definition
-  "0 \<Colon> int" is "(0\<Colon>nat, 0\<Colon>nat)"
-
-quotient_definition
-  "1 \<Colon> int" is "(1\<Colon>nat, 0\<Colon>nat)"
-
-fun
-  plus_raw :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat) \<Rightarrow> (nat \<times> nat)"
-where
-  "plus_raw (x, y) (u, v) = (x + u, y + v)"
-
-quotient_definition
-  "(op +) \<Colon> (int \<Rightarrow> int \<Rightarrow> int)" is "plus_raw"
-
-fun
-  uminus_raw :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat)"
-where
-  "uminus_raw (x, y) = (y, x)"
-
-quotient_definition
-  "(uminus \<Colon> (int \<Rightarrow> int))" is "uminus_raw"
-
-definition
-  minus_int_def [code del]:  "z - w = z + (-w\<Colon>int)"
-
-fun
-  mult_raw :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat) \<Rightarrow> (nat \<times> nat)"
-where
-  "mult_raw (x, y) (u, v) = (x*u + y*v, x*v + y*u)"
-
-quotient_definition
-  mult_int_def: "(op *) :: (int \<Rightarrow> int \<Rightarrow> int)" is "mult_raw"
-
-fun
-  le_raw :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat) \<Rightarrow> bool"
-where
-  "le_raw (x, y) (u, v) = (x+v \<le> u+y)"
-
-quotient_definition
-  le_int_def: "(op \<le>) :: int \<Rightarrow> int \<Rightarrow> bool" is "le_raw"
-
-definition
-  less_int_def [code del]: "(z\<Colon>int) < w = (z \<le> w \<and> z \<noteq> w)"
-
-definition
-  zabs_def: "\<bar>i\<Colon>int\<bar> = (if i < 0 then - i else i)"
-
-definition
-  zsgn_def: "sgn (i\<Colon>int) = (if i = 0 then 0 else if 0 < i then 1 else - 1)"
-
-instance ..
-
-end
-
-lemma plus_raw_rsp[quot_respect]:
-  shows "(op \<approx> ===> op \<approx> ===> op \<approx>) plus_raw plus_raw"
-by auto
-
-lemma uminus_raw_rsp[quot_respect]:
-  shows "(op \<approx> ===> op \<approx>) uminus_raw uminus_raw"
-  by auto
-
-lemma mult_raw_fst:
-  assumes a: "x \<approx> z"
-  shows "mult_raw x y \<approx> mult_raw z y"
-using a
-apply(cases x, cases y, cases z)
-apply(auto simp add: mult_raw.simps intrel.simps)
-apply(rename_tac u v w x y z)
-apply(subgoal_tac "u*w + z*w = y*w + v*w  &  u*x + z*x = y*x + v*x")
-apply(simp add: mult_ac)
-apply(simp add: add_mult_distrib [symmetric])
-done
-
-lemma mult_raw_snd:
-  assumes a: "x \<approx> z"
-  shows "mult_raw y x \<approx> mult_raw y z"
-using a
-apply(cases x, cases y, cases z)
-apply(auto simp add: mult_raw.simps intrel.simps)
-apply(rename_tac u v w x y z)
-apply(subgoal_tac "u*w + z*w = y*w + v*w  &  u*x + z*x = y*x + v*x")
-apply(simp add: mult_ac)
-apply(simp add: add_mult_distrib [symmetric])
-done
-
-lemma mult_raw_rsp[quot_respect]:
-  shows "(op \<approx> ===> op \<approx> ===> op \<approx>) mult_raw mult_raw"
-apply(simp only: fun_rel_def)
-apply(rule allI | rule impI)+
-apply(rule equivp_transp[OF int_equivp])
-apply(rule mult_raw_fst)
-apply(assumption)
-apply(rule mult_raw_snd)
-apply(assumption)
-done
-
-lemma le_raw_rsp[quot_respect]:
-  shows "(op \<approx> ===> op \<approx> ===> op =) le_raw le_raw"
-by auto
-
-lemma plus_assoc_raw:
-  shows "plus_raw (plus_raw i j) k \<approx> plus_raw i (plus_raw j k)"
-by (cases i, cases j, cases k) (simp)
-
-lemma plus_sym_raw:
-  shows "plus_raw i j \<approx> plus_raw j i"
-by (cases i, cases j) (simp)
-
-lemma plus_zero_raw:
-  shows "plus_raw  (0, 0) i \<approx> i"
-by (cases i) (simp)
-
-lemma plus_minus_zero_raw:
-  shows "plus_raw (uminus_raw i) i \<approx> (0, 0)"
-by (cases i) (simp)
-
-lemma times_assoc_raw:
-  shows "mult_raw (mult_raw i j) k \<approx> mult_raw i (mult_raw j k)"
-by (cases i, cases j, cases k) 
-   (simp add: algebra_simps)
-
-lemma times_sym_raw:
-  shows "mult_raw i j \<approx> mult_raw j i"
-by (cases i, cases j) (simp add: algebra_simps)
-
-lemma times_one_raw:
-  shows "mult_raw  (1, 0) i \<approx> i"
-by (cases i) (simp)
-
-lemma times_plus_comm_raw:
-  shows "mult_raw (plus_raw i j) k \<approx> plus_raw (mult_raw i k) (mult_raw j k)"
-by (cases i, cases j, cases k) 
-   (simp add: algebra_simps)
-
-lemma one_zero_distinct:
-  shows "\<not> (0, 0) \<approx> ((1::nat), (0::nat))"
-  by simp
-
-text{* The integers form a @{text comm_ring_1}*}
-
-instance int :: comm_ring_1
-proof
-  fix i j k :: int
-  show "(i + j) + k = i + (j + k)"
-    by (lifting plus_assoc_raw)
-  show "i + j = j + i" 
-    by (lifting plus_sym_raw)
-  show "0 + i = (i::int)"
-    by (lifting plus_zero_raw)
-  show "- i + i = 0"
-    by (lifting plus_minus_zero_raw)
-  show "i - j = i + - j"
-    by (simp add: minus_int_def)
-  show "(i * j) * k = i * (j * k)"
-    by (lifting times_assoc_raw)
-  show "i * j = j * i"
-    by (lifting times_sym_raw)
-  show "1 * i = i"
-    by (lifting times_one_raw)
-  show "(i + j) * k = i * k + j * k"
-    by (lifting times_plus_comm_raw)
-  show "0 \<noteq> (1::int)"
-    by (lifting one_zero_distinct)
-qed
-
-lemma plus_raw_rsp_aux:
-  assumes a: "a \<approx> b" "c \<approx> d"
-  shows "plus_raw a c \<approx> plus_raw b d"
-using a
-by (cases a, cases b, cases c, cases d) 
-   (simp)
-
-lemma add:
-     "(abs_int (x,y)) + (abs_int (u,v)) =
-      (abs_int (x + u, y + v))"
-apply(simp add: plus_int_def id_simps)
-apply(fold plus_raw.simps)
-apply(rule Quotient_rel_abs[OF Quotient_int])
-apply(rule plus_raw_rsp_aux)
-apply(simp_all add: rep_abs_rsp_left[OF Quotient_int])
-done
-
-definition int_of_nat_raw: 
-  "int_of_nat_raw m = (m :: nat, 0 :: nat)"
-
-quotient_definition
-  "int_of_nat :: nat \<Rightarrow> int" is "int_of_nat_raw"
-
-lemma[quot_respect]: 
-  shows "(op = ===> op \<approx>) int_of_nat_raw int_of_nat_raw"
-by (simp add: equivp_reflp[OF int_equivp])
-
-lemma int_of_nat:
-  shows "of_nat m = int_of_nat m"
-apply (induct m)
-apply (simp_all add: zero_int_def one_int_def int_of_nat_def int_of_nat_raw add)
-done
-
-lemma le_antisym_raw:
-  shows "le_raw i j \<Longrightarrow> le_raw j i \<Longrightarrow> i \<approx> j"
-by (cases i, cases j) (simp)
-
-lemma le_refl_raw:
-  shows "le_raw i i"
-by (cases i) (simp)
-
-lemma le_trans_raw:
-  shows "le_raw i j \<Longrightarrow> le_raw j k \<Longrightarrow> le_raw i k"
-by (cases i, cases j, cases k) (simp)
-
-lemma le_cases_raw:
-  shows "le_raw i j \<or> le_raw j i"
-by (cases i, cases j) 
-   (simp add: linorder_linear)
-
-instance int :: linorder
-proof
-  fix i j k :: int
-  show antisym: "i \<le> j \<Longrightarrow> j \<le> i \<Longrightarrow> i = j"
-    by (lifting le_antisym_raw)
-  show "(i < j) = (i \<le> j \<and> \<not> j \<le> i)"
-    by (auto simp add: less_int_def dest: antisym) 
-  show "i \<le> i"
-    by (lifting le_refl_raw)
-  show "i \<le> j \<Longrightarrow> j \<le> k \<Longrightarrow> i \<le> k"
-    by (lifting le_trans_raw)
-  show "i \<le> j \<or> j \<le> i"
-    by (lifting le_cases_raw)
-qed
-
-instantiation int :: distrib_lattice
-begin
-
-definition
-  "(inf \<Colon> int \<Rightarrow> int \<Rightarrow> int) = min"
-
-definition
-  "(sup \<Colon> int \<Rightarrow> int \<Rightarrow> int) = max"
-
-instance
-  by intro_classes
-    (auto simp add: inf_int_def sup_int_def min_max.sup_inf_distrib1)
-
-end
-
-lemma le_plus_raw:
-  shows "le_raw i j \<Longrightarrow> le_raw (plus_raw k i) (plus_raw k j)"
-by (cases i, cases j, cases k) (simp)
-
-
-instance int :: ordered_cancel_ab_semigroup_add
-proof
-  fix i j k :: int
-  show "i \<le> j \<Longrightarrow> k + i \<le> k + j"
-    by (lifting le_plus_raw)
-qed
-
-abbreviation
-  "less_raw i j \<equiv> le_raw i j \<and> \<not>(i \<approx> j)"
-
-lemma zmult_zless_mono2_lemma:
-  fixes i j::int
-  and   k::nat
-  shows "i < j \<Longrightarrow> 0 < k \<Longrightarrow> of_nat k * i < of_nat k * j"
-apply(induct "k")
-apply(simp)
-apply(case_tac "k = 0")
-apply(simp_all add: left_distrib add_strict_mono)
-done
-
-lemma zero_le_imp_eq_int_raw:
-  fixes k::"(nat \<times> nat)"
-  shows "less_raw (0, 0) k \<Longrightarrow> (\<exists>n > 0. k \<approx> int_of_nat_raw n)"
-apply(cases k)
-apply(simp add:int_of_nat_raw)
-apply(auto)
-apply(rule_tac i="b" and j="a" in less_Suc_induct)
-apply(auto)
-done
-
-lemma zero_le_imp_eq_int:
-  fixes k::int
-  shows "0 < k \<Longrightarrow> \<exists>n > 0. k = of_nat n"
-  unfolding less_int_def int_of_nat
-  by (lifting zero_le_imp_eq_int_raw)
-
-lemma zmult_zless_mono2: 
-  fixes i j k::int
-  assumes a: "i < j" "0 < k"
-  shows "k * i < k * j"
-using a
-using a
-apply(drule_tac zero_le_imp_eq_int)
-apply(auto simp add: zmult_zless_mono2_lemma)
-done
-
-text{*The integers form an ordered integral domain*}
-instance int :: linordered_idom
-proof
-  fix i j k :: int
-  show "i < j \<Longrightarrow> 0 < k \<Longrightarrow> k * i < k * j"
-    by (rule zmult_zless_mono2)
-  show "\<bar>i\<bar> = (if i < 0 then -i else i)"
-    by (simp only: zabs_def)
-  show "sgn (i\<Colon>int) = (if i=0 then 0 else if 0<i then 1 else - 1)"
-    by (simp only: zsgn_def)
-qed
-
-lemmas int_distrib =
-  left_distrib [of "z1::int" "z2" "w", standard]
-  right_distrib [of "w::int" "z1" "z2", standard]
-  left_diff_distrib [of "z1::int" "z2" "w", standard]
-  right_diff_distrib [of "w::int" "z1" "z2", standard]
-
-
-subsection {* Embedding of the Integers into any @{text ring_1}: @{text of_int}*}
-
-(*
-context ring_1
-begin
-
- 
-definition 
-  of_int :: "int \<Rightarrow> 'a" 
-where
-  "of_int 
-*)
-
-
-subsection {* Binary representation *}
-
-text {*
-  This formalization defines binary arithmetic in terms of the integers
-  rather than using a datatype. This avoids multiple representations (leading
-  zeroes, etc.)  See @{text "ZF/Tools/twos-compl.ML"}, function @{text
-  int_of_binary}, for the numerical interpretation.
-
-  The representation expects that @{text "(m mod 2)"} is 0 or 1,
-  even if m is negative;
-  For instance, @{text "-5 div 2 = -3"} and @{text "-5 mod 2 = 1"}; thus
-  @{text "-5 = (-3)*2 + 1"}.
-  
-  This two's complement binary representation derives from the paper 
-  "An Efficient Representation of Arithmetic for Term Rewriting" by
-  Dave Cohen and Phil Watson, Rewriting Techniques and Applications,
-  Springer LNCS 488 (240-251), 1991.
-*}
-
-subsubsection {* The constructors @{term Bit0}, @{term Bit1}, @{term Pls} and @{term Min} *}
-
-definition
-  Pls :: int where
-  [code del]: "Pls = 0"
-
-definition
-  Min :: int where
-  [code del]: "Min = - 1"
-
-definition
-  Bit0 :: "int \<Rightarrow> int" where
-  [code del]: "Bit0 k = k + k"
-
-definition
-  Bit1 :: "int \<Rightarrow> int" where
-  [code del]: "Bit1 k = 1 + k + k"
-
-class number = -- {* for numeric types: nat, int, real, \dots *}
-  fixes number_of :: "int \<Rightarrow> 'a"
-
-(*use "~~/src/HOL/Tools/numeral.ML"
-
-syntax
-  "_Numeral" :: "num_const \<Rightarrow> 'a"    ("_")
-
-use "~~/src/HOL/Tools/numeral_syntax.ML"
-
-setup NumeralSyntax.setup
-
-abbreviation
-  "Numeral0 \<equiv> number_of Pls"
-
-abbreviation
-  "Numeral1 \<equiv> number_of (Bit1 Pls)"
-
-lemma Let_number_of [simp]: "Let (number_of v) f = f (number_of v)"
-  -- {* Unfold all @{text let}s involving constants *}
-  unfolding Let_def ..
-
-definition
-  succ :: "int \<Rightarrow> int" where
-  [code del]: "succ k = k + 1"
-
-definition
-  pred :: "int \<Rightarrow> int" where
-  [code del]: "pred k = k - 1"
-
-lemmas
-  max_number_of [simp] = max_def
-    [of "number_of u" "number_of v", standard, simp]
-and
-  min_number_of [simp] = min_def 
-    [of "number_of u" "number_of v", standard, simp]
-  -- {* unfolding @{text minx} and @{text max} on numerals *}
-
-lemmas numeral_simps = 
-  succ_def pred_def Pls_def Min_def Bit0_def Bit1_def
-
-text {* Removal of leading zeroes *}
-
-lemma Bit0_Pls [simp, code_post]:
-  "Bit0 Pls = Pls"
-  unfolding numeral_simps by simp
-
-lemma Bit1_Min [simp, code_post]:
-  "Bit1 Min = Min"
-  unfolding numeral_simps by simp
-
-lemmas normalize_bin_simps =
-  Bit0_Pls Bit1_Min
-*)
-
-end