--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Quot/Examples/IntEx2.thy Mon Dec 07 14:09:50 2009 +0100
@@ -0,0 +1,436 @@
+theory IntEx2
+imports "../QuotMain"
+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 int = "nat \<times> nat" / intrel
+ apply(unfold equivp_def)
+ apply(auto simp add: mem_def expand_fun_eq)
+ done
+
+instantiation int :: "{zero, one, plus, minus, uminus, times, ord, abs, sgn}"
+begin
+
+quotient_def
+ zero_qnt::"int"
+where
+ "zero_qnt \<equiv> (0::nat, 0::nat)"
+
+definition Zero_int_def[code del]:
+ "0 = zero_qnt"
+
+quotient_def
+ one_qnt::"int"
+where
+ "one_qnt \<equiv> (1::nat, 0::nat)"
+
+definition One_int_def[code del]:
+ "1 = one_qnt"
+
+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_def
+ plus_qnt::"int \<Rightarrow> int \<Rightarrow> int"
+where
+ "plus_qnt \<equiv> plus_raw"
+
+definition add_int_def[code del]:
+ "z + w = plus_qnt z w"
+
+fun
+ minus_raw :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat)"
+where
+ "minus_raw (x, y) = (y, x)"
+
+quotient_def
+ minus_qnt::"int \<Rightarrow> int"
+where
+ "minus_qnt \<equiv> minus_raw"
+
+definition minus_int_def [code del]:
+ "- z = minus_qnt z"
+
+definition
+ diff_int_def [code del]: "z - w = z + (-w::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_def
+ mult_qnt::"int \<Rightarrow> int \<Rightarrow> int"
+where
+ "mult_qnt \<equiv> mult_raw"
+
+definition
+ mult_int_def [code del]: "z * w = mult_qnt z w"
+
+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_def
+ le_qnt :: "int \<Rightarrow> int \<Rightarrow> bool"
+where
+ "le_qnt \<equiv> le_raw"
+
+definition
+ le_int_def [code del]:
+ "z \<le> w = le_qnt z w"
+
+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
+
+thm add_assoc
+
+lemma plus_raw_rsp[quotient_rsp]:
+ shows "(op \<approx> ===> op \<approx> ===> op \<approx>) plus_raw plus_raw"
+by auto
+
+lemma minus_raw_rsp[quotient_rsp]:
+ shows "(op \<approx> ===> op \<approx>) minus_raw minus_raw"
+ by auto
+
+lemma mult_raw_rsp[quotient_rsp]:
+ shows "(op \<approx> ===> op \<approx> ===> op \<approx>) mult_raw mult_raw"
+apply(auto)
+apply(simp add: mult algebra_simps)
+sorry
+
+lemma le_raw_rsp[quotient_rsp]:
+ 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 (minus_raw i) i \<approx> (0, 0)"
+by (cases i) (simp)
+
+lemma mult_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: mult algebra_simps)
+
+lemma mult_sym_raw:
+ shows "mult_raw i j \<approx> mult_raw j i"
+by (cases i, cases j) (simp)
+
+lemma mult_one_raw:
+ shows "mult_raw (1, 0) i \<approx> i"
+by (cases i) (simp)
+
+lemma mult_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: mult 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}*}
+
+
+ML {* val qty = @{typ "int"} *}
+ML {* val (rty, rel, rel_refl, rel_eqv) = lookup_quot_data @{context} qty *}
+ML {* val (trans2, reps_same, absrep, quot) = lookup_quot_thms @{context} "int" *}
+
+instance int :: comm_ring_1
+proof
+ fix i j k :: int
+ show "(i + j) + k = i + (j + k)"
+ unfolding add_int_def
+ apply(tactic {* lift_tac @{context} @{thm plus_assoc_raw} 1 *})
+ done
+ show "i + j = j + i"
+ unfolding add_int_def
+ apply(tactic {* lift_tac @{context} @{thm plus_sym_raw} 1 *})
+ done
+ show "0 + i = (i::int)"
+ unfolding add_int_def Zero_int_def
+ apply(tactic {* lift_tac @{context} @{thm plus_zero_raw} 1 *})
+ done
+ show "- i + i = 0"
+ unfolding add_int_def minus_int_def Zero_int_def
+ apply(tactic {* lift_tac @{context} @{thm plus_minus_zero_raw} 1 *})
+ done
+ show "i - j = i + - j"
+ by (simp add: diff_int_def)
+ show "(i * j) * k = i * (j * k)"
+ unfolding mult_int_def
+ apply(tactic {* lift_tac @{context} @{thm mult_assoc_raw} 1 *})
+ done
+ show "i * j = j * i"
+ unfolding mult_int_def
+ apply(tactic {* lift_tac @{context} @{thm mult_sym_raw} 1 *})
+ done
+ show "1 * i = i"
+ unfolding mult_int_def One_int_def
+ apply(tactic {* lift_tac @{context} @{thm mult_one_raw} 1 *})
+ done
+ show "(i + j) * k = i * k + j * k"
+ unfolding mult_int_def add_int_def
+ apply(tactic {* lift_tac @{context} @{thm mult_plus_comm_raw} 1 *})
+ done
+ show "0 \<noteq> (1::int)"
+ unfolding Zero_int_def One_int_def
+ apply(tactic {* lift_tac @{context} @{thm one_zero_distinct} 1 *})
+ done
+qed
+
+term of_nat
+thm of_nat_def
+
+lemma int_def: "of_nat m = ABS_int (m, 0)"
+apply(induct m)
+apply(simp add: Zero_int_def zero_qnt_def)
+apply(simp)
+apply(simp add: add_int_def One_int_def)
+apply(simp add: plus_qnt_def one_qnt_def)
+oops
+
+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"
+ unfolding le_int_def
+ apply(tactic {* lift_tac @{context} @{thm le_antisym_raw} 1 *})
+ done
+ show "(i < j) = (i \<le> j \<and> \<not> j \<le> i)"
+ by (auto simp add: less_int_def dest: antisym)
+ show "i \<le> i"
+ unfolding le_int_def
+ apply(tactic {* lift_tac @{context} @{thm le_refl_raw} 1 *})
+ done
+ show "i \<le> j \<Longrightarrow> j \<le> k \<Longrightarrow> i \<le> k"
+ unfolding le_int_def
+ apply(tactic {* lift_tac @{context} @{thm le_trans_raw} 1 *})
+ done
+ show "i \<le> j \<or> j \<le> i"
+ unfolding le_int_def
+ apply(tactic {* lift_tac @{context} @{thm le_cases_raw} 1 *})
+ done
+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 :: pordered_cancel_ab_semigroup_add
+proof
+ fix i j k :: int
+ show "i \<le> j \<Longrightarrow> k + i \<le> k + j"
+ unfolding le_int_def add_int_def
+ apply(tactic {* lift_tac @{context} @{thm le_plus_raw} 1 *})
+ done
+qed
+
+lemma test:
+ "\<lbrakk>le_raw i j \<and> \<not>i \<approx> j; le_raw (0, 0) k \<and> \<not>(0, 0) \<approx> k\<rbrakk>
+ \<Longrightarrow> le_raw (mult_raw k i) (mult_raw k j) \<and> \<not>mult_raw k i \<approx> mult_raw k j"
+apply(cases i, cases j, cases k)
+apply(auto simp add: mult algebra_simps)
+sorry
+
+
+text{*The integers form an ordered integral domain*}
+instance int :: ordered_idom
+proof
+ fix i j k :: int
+ show "i < j \<Longrightarrow> 0 < k \<Longrightarrow> k * i < k * j"
+ unfolding mult_int_def le_int_def less_int_def Zero_int_def
+ apply(tactic {* lift_tac @{context} @{thm test} 1 *})
+ done
+ 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
+
+instance int :: lordered_ring
+proof
+ fix k :: int
+ show "abs k = sup k (- k)"
+ by (auto simp add: sup_int_def zabs_def less_minus_self_iff [symmetric])
+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
+*)
\ No newline at end of file