--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Integ/int.ML Wed Sep 25 07:54:33 2002 +0200
@@ -0,0 +1,564 @@
+(* Title: HOL/Integ/Int.ML
+ ID: $Id$
+ Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+ Copyright 1998 University of Cambridge
+
+Type "int" is a linear order
+
+And many further lemmas
+*)
+
+(* legacy ML bindings *)
+
+structure Int =
+struct
+ val thy = the_context ();
+ val zabs_def = thm "zabs_def";
+ val nat_def = thm "nat_def";
+end;
+
+open Int;
+
+Goal "int 0 = (0::int)";
+by (simp_tac (simpset() addsimps [Zero_int_def]) 1);
+qed "int_0";
+
+Goal "int 1 = 1";
+by (simp_tac (simpset() addsimps [One_int_def]) 1);
+qed "int_1";
+
+Goal "int (Suc 0) = 1";
+by (simp_tac (simpset() addsimps [One_int_def, One_nat_def]) 1);
+qed "int_Suc0_eq_1";
+
+Goalw [zdiff_def,zless_def] "neg x = (x < 0)";
+by Auto_tac;
+qed "neg_eq_less_0";
+
+Goalw [zle_def] "(~neg x) = (0 <= x)";
+by (simp_tac (simpset() addsimps [neg_eq_less_0]) 1);
+qed "not_neg_eq_ge_0";
+
+(** Needed to simplify inequalities when Numeral1 can get simplified to 1 **)
+
+Goal "~ neg 0";
+by (simp_tac (simpset() addsimps [One_int_def, neg_eq_less_0]) 1);
+qed "not_neg_0";
+
+Goal "~ neg 1";
+by (simp_tac (simpset() addsimps [One_int_def, neg_eq_less_0]) 1);
+qed "not_neg_1";
+
+Goal "iszero 0";
+by (simp_tac (simpset() addsimps [iszero_def]) 1);
+qed "iszero_0";
+
+Goal "~ iszero 1";
+by (simp_tac (simpset() addsimps [Zero_int_def, One_int_def, One_nat_def,
+ iszero_def]) 1);
+qed "not_iszero_1";
+
+Goal "0 < (1::int)";
+by (simp_tac (simpset() addsimps [Zero_int_def, One_int_def, One_nat_def]) 1);
+qed "int_0_less_1";
+
+Goal "0 \\<noteq> (1::int)";
+by (simp_tac (simpset() addsimps [Zero_int_def, One_int_def, One_nat_def]) 1);
+qed "int_0_neq_1";
+
+Addsimps [int_0, int_1, int_0_neq_1];
+
+
+(*** Abel_Cancel simproc on the integers ***)
+
+(* Lemmas needed for the simprocs *)
+
+(*Deletion of other terms in the formula, seeking the -x at the front of z*)
+Goal "((x::int) + (y + z) = y + u) = ((x + z) = u)";
+by (stac zadd_left_commute 1);
+by (rtac zadd_left_cancel 1);
+qed "zadd_cancel_21";
+
+(*A further rule to deal with the case that
+ everything gets cancelled on the right.*)
+Goal "((x::int) + (y + z) = y) = (x = -z)";
+by (stac zadd_left_commute 1);
+by (res_inst_tac [("t", "y")] (zadd_0_right RS subst) 1
+ THEN stac zadd_left_cancel 1);
+by (simp_tac (simpset() addsimps [eq_zdiff_eq RS sym]) 1);
+qed "zadd_cancel_end";
+
+
+structure Int_Cancel_Data =
+struct
+ val ss = HOL_ss
+ val eq_reflection = eq_reflection
+
+ val sg_ref = Sign.self_ref (Theory.sign_of (the_context ()))
+ val T = HOLogic.intT
+ val zero = Const ("0", HOLogic.intT)
+ val restrict_to_left = restrict_to_left
+ val add_cancel_21 = zadd_cancel_21
+ val add_cancel_end = zadd_cancel_end
+ val add_left_cancel = zadd_left_cancel
+ val add_assoc = zadd_assoc
+ val add_commute = zadd_commute
+ val add_left_commute = zadd_left_commute
+ val add_0 = zadd_0
+ val add_0_right = zadd_0_right
+
+ val eq_diff_eq = eq_zdiff_eq
+ val eqI_rules = [zless_eqI, zeq_eqI, zle_eqI]
+ fun dest_eqI th =
+ #1 (HOLogic.dest_bin "op =" HOLogic.boolT
+ (HOLogic.dest_Trueprop (concl_of th)))
+
+ val diff_def = zdiff_def
+ val minus_add_distrib = zminus_zadd_distrib
+ val minus_minus = zminus_zminus
+ val minus_0 = zminus_0
+ val add_inverses = [zadd_zminus_inverse, zadd_zminus_inverse2]
+ val cancel_simps = [zadd_zminus_cancel, zminus_zadd_cancel]
+end;
+
+structure Int_Cancel = Abel_Cancel (Int_Cancel_Data);
+
+Addsimprocs [Int_Cancel.sum_conv, Int_Cancel.rel_conv];
+
+
+
+(*** misc ***)
+
+Goal "- (z - y) = y - (z::int)";
+by (Simp_tac 1);
+qed "zminus_zdiff_eq";
+Addsimps [zminus_zdiff_eq];
+
+Goal "(w<z) = neg(w-z)";
+by (simp_tac (simpset() addsimps [zless_def]) 1);
+qed "zless_eq_neg";
+
+Goal "(w=z) = iszero(w-z)";
+by (simp_tac (simpset() addsimps [iszero_def, zdiff_eq_eq]) 1);
+qed "eq_eq_iszero";
+
+Goal "(w<=z) = (~ neg(z-w))";
+by (simp_tac (simpset() addsimps [zle_def, zless_def]) 1);
+qed "zle_eq_not_neg";
+
+(** Inequality reasoning **)
+
+Goal "(w < z + (1::int)) = (w<z | w=z)";
+by (auto_tac (claset(),
+ simpset() addsimps [zless_iff_Suc_zadd, int_Suc,
+ gr0_conv_Suc, zero_reorient]));
+by (res_inst_tac [("x","Suc n")] exI 1);
+by (simp_tac (simpset() addsimps [int_Suc]) 1);
+qed "zless_add1_eq";
+
+Goal "(w + (1::int) <= z) = (w<z)";
+by (asm_full_simp_tac (simpset() addsimps [zle_def, zless_add1_eq]) 1);
+by (auto_tac (claset() addIs [zle_anti_sym],
+ simpset() addsimps [order_less_imp_le, symmetric zle_def]));
+qed "add1_zle_eq";
+
+Goal "((1::int) + w <= z) = (w<z)";
+by (stac zadd_commute 1);
+by (rtac add1_zle_eq 1);
+qed "add1_left_zle_eq";
+
+
+(*** Monotonicity results ***)
+
+Goal "(v+z < w+z) = (v < (w::int))";
+by (Simp_tac 1);
+qed "zadd_right_cancel_zless";
+
+Goal "(z+v < z+w) = (v < (w::int))";
+by (Simp_tac 1);
+qed "zadd_left_cancel_zless";
+
+Addsimps [zadd_right_cancel_zless, zadd_left_cancel_zless];
+
+Goal "(v+z <= w+z) = (v <= (w::int))";
+by (Simp_tac 1);
+qed "zadd_right_cancel_zle";
+
+Goal "(z+v <= z+w) = (v <= (w::int))";
+by (Simp_tac 1);
+qed "zadd_left_cancel_zle";
+
+Addsimps [zadd_right_cancel_zle, zadd_left_cancel_zle];
+
+(*"v<=w ==> v+z <= w+z"*)
+bind_thm ("zadd_zless_mono1", zadd_right_cancel_zless RS iffD2);
+
+(*"v<=w ==> z+v <= z+w"*)
+bind_thm ("zadd_zless_mono2", zadd_left_cancel_zless RS iffD2);
+
+(*"v<=w ==> v+z <= w+z"*)
+bind_thm ("zadd_zle_mono1", zadd_right_cancel_zle RS iffD2);
+
+(*"v<=w ==> z+v <= z+w"*)
+bind_thm ("zadd_zle_mono2", zadd_left_cancel_zle RS iffD2);
+
+Goal "[| w'<=w; z'<=z |] ==> w' + z' <= w + (z::int)";
+by (etac (zadd_zle_mono1 RS zle_trans) 1);
+by (Simp_tac 1);
+qed "zadd_zle_mono";
+
+Goal "[| w'<w; z'<=z |] ==> w' + z' < w + (z::int)";
+by (etac (zadd_zless_mono1 RS order_less_le_trans) 1);
+by (Simp_tac 1);
+qed "zadd_zless_mono";
+
+
+(*** Comparison laws ***)
+
+Goal "(- x < - y) = (y < (x::int))";
+by (simp_tac (simpset() addsimps [zless_def, zdiff_def] @ zadd_ac) 1);
+qed "zminus_zless_zminus";
+Addsimps [zminus_zless_zminus];
+
+Goal "(- x <= - y) = (y <= (x::int))";
+by (simp_tac (simpset() addsimps [zle_def]) 1);
+qed "zminus_zle_zminus";
+Addsimps [zminus_zle_zminus];
+
+(** The next several equations can make the simplifier loop! **)
+
+Goal "(x < - y) = (y < - (x::int))";
+by (simp_tac (simpset() addsimps [zless_def, zdiff_def] @ zadd_ac) 1);
+qed "zless_zminus";
+
+Goal "(- x < y) = (- y < (x::int))";
+by (simp_tac (simpset() addsimps [zless_def, zdiff_def] @ zadd_ac) 1);
+qed "zminus_zless";
+
+Goal "(x <= - y) = (y <= - (x::int))";
+by (simp_tac (simpset() addsimps [zle_def, zminus_zless]) 1);
+qed "zle_zminus";
+
+Goal "(- x <= y) = (- y <= (x::int))";
+by (simp_tac (simpset() addsimps [zle_def, zless_zminus]) 1);
+qed "zminus_zle";
+
+Goal "(x = - y) = (y = - (x::int))";
+by Auto_tac;
+qed "equation_zminus";
+
+Goal "(- x = y) = (- (y::int) = x)";
+by Auto_tac;
+qed "zminus_equation";
+
+
+(** Instances of the equations above, for zero **)
+
+(*instantiate a variable to zero and simplify*)
+fun zero_instance v th = simplify (simpset()) (inst v "0" th);
+
+Addsimps [zero_instance "x" zless_zminus,
+ zero_instance "y" zminus_zless,
+ zero_instance "x" zle_zminus,
+ zero_instance "y" zminus_zle,
+ zero_instance "x" equation_zminus,
+ zero_instance "y" zminus_equation];
+
+
+Goal "- (int (Suc n)) < 0";
+by (simp_tac (simpset() addsimps [zless_def]) 1);
+qed "negative_zless_0";
+
+Goal "- (int (Suc n)) < int m";
+by (rtac (negative_zless_0 RS order_less_le_trans) 1);
+by (Simp_tac 1);
+qed "negative_zless";
+AddIffs [negative_zless];
+
+Goal "- int n <= 0";
+by (simp_tac (simpset() addsimps [zminus_zle]) 1);
+qed "negative_zle_0";
+
+Goal "- int n <= int m";
+by (simp_tac (simpset() addsimps [zless_def, zle_def, zdiff_def, zadd_int]) 1);
+qed "negative_zle";
+AddIffs [negative_zle];
+
+Goal "~(0 <= - (int (Suc n)))";
+by (stac zle_zminus 1);
+by (Simp_tac 1);
+qed "not_zle_0_negative";
+Addsimps [not_zle_0_negative];
+
+Goal "(int n <= - int m) = (n = 0 & m = 0)";
+by Safe_tac;
+by (Simp_tac 3);
+by (dtac (zle_zminus RS iffD1) 2);
+by (ALLGOALS (dtac (negative_zle_0 RSN(2,zle_trans))));
+by (ALLGOALS Asm_full_simp_tac);
+qed "int_zle_neg";
+
+Goal "~(int n < - int m)";
+by (simp_tac (simpset() addsimps [symmetric zle_def]) 1);
+qed "not_int_zless_negative";
+
+Goal "(- int n = int m) = (n = 0 & m = 0)";
+by (rtac iffI 1);
+by (rtac (int_zle_neg RS iffD1) 1);
+by (dtac sym 1);
+by (ALLGOALS Asm_simp_tac);
+qed "negative_eq_positive";
+
+Addsimps [negative_eq_positive, not_int_zless_negative];
+
+
+Goal "(w <= z) = (EX n. z = w + int n)";
+by (auto_tac (claset() addIs [inst "x" "0::nat" exI]
+ addSIs [not_sym RS not0_implies_Suc],
+ simpset() addsimps [zless_iff_Suc_zadd, int_le_less]));
+qed "zle_iff_zadd";
+
+Goal "abs (int m) = int m";
+by (simp_tac (simpset() addsimps [zabs_def]) 1);
+qed "abs_int_eq";
+Addsimps [abs_int_eq];
+
+
+(**** nat: magnitide of an integer, as a natural number ****)
+
+Goalw [nat_def] "nat(int n) = n";
+by Auto_tac;
+qed "nat_int";
+Addsimps [nat_int];
+
+Goalw [nat_def] "nat(- (int n)) = 0";
+by (auto_tac (claset(),
+ simpset() addsimps [neg_eq_less_0, zero_reorient, zminus_zless]));
+qed "nat_zminus_int";
+Addsimps [nat_zminus_int];
+
+Goalw [Zero_int_def] "nat 0 = 0";
+by (rtac nat_int 1);
+qed "nat_zero";
+Addsimps [nat_zero];
+
+Goal "~ neg z ==> int (nat z) = z";
+by (dtac (not_neg_eq_ge_0 RS iffD1) 1);
+by (dtac zle_imp_zless_or_eq 1);
+by (auto_tac (claset(), simpset() addsimps [zless_iff_Suc_zadd]));
+qed "not_neg_nat";
+
+Goal "neg x ==> EX n. x = - (int (Suc n))";
+by (auto_tac (claset(),
+ simpset() addsimps [neg_eq_less_0, zless_iff_Suc_zadd,
+ zdiff_eq_eq RS sym, zdiff_def]));
+qed "negD";
+
+Goalw [nat_def] "neg z ==> nat z = 0";
+by Auto_tac;
+qed "neg_nat";
+
+Goal "(m < nat z) = (int m < z)";
+by (case_tac "neg z" 1);
+by (etac (not_neg_nat RS subst) 2);
+by (auto_tac (claset(), simpset() addsimps [neg_nat]));
+by (auto_tac (claset() addDs [order_less_trans],
+ simpset() addsimps [neg_eq_less_0]));
+qed "zless_nat_eq_int_zless";
+
+Goal "0 <= z ==> int (nat z) = z";
+by (asm_full_simp_tac
+ (simpset() addsimps [neg_eq_less_0, zle_def, not_neg_nat]) 1);
+qed "nat_0_le";
+
+Goal "z <= 0 ==> nat z = 0";
+by (auto_tac (claset(),
+ simpset() addsimps [order_le_less, neg_eq_less_0,
+ zle_def, neg_nat]));
+qed "nat_le_0";
+Addsimps [nat_0_le, nat_le_0];
+
+(*An alternative condition is 0 <= w *)
+Goal "0 < z ==> (nat w < nat z) = (w < z)";
+by (stac (zless_int RS sym) 1);
+by (asm_simp_tac (simpset() addsimps [not_neg_nat, not_neg_eq_ge_0,
+ order_le_less]) 1);
+by (case_tac "neg w" 1);
+by (asm_simp_tac (simpset() addsimps [not_neg_nat]) 2);
+by (asm_full_simp_tac (simpset() addsimps [neg_eq_less_0, neg_nat]) 1);
+by (blast_tac (claset() addIs [order_less_trans]) 1);
+val lemma = result();
+
+Goal "(nat w < nat z) = (0 < z & w < z)";
+by (case_tac "0 < z" 1);
+by (auto_tac (claset(), simpset() addsimps [lemma, linorder_not_less]));
+qed "zless_nat_conj";
+
+
+(* a case theorem distinguishing non-negative and negative int *)
+
+val prems = Goal
+ "[|!! n. z = int n ==> P; !! n. z = - (int (Suc n)) ==> P |] ==> P";
+by (case_tac "neg z" 1);
+by (fast_tac (claset() addSDs [negD] addSEs prems) 1);
+by (dtac (not_neg_nat RS sym) 1);
+by (eresolve_tac prems 1);
+qed "int_cases";
+
+fun int_case_tac x = res_inst_tac [("z",x)] int_cases;
+
+
+(*** Monotonicity of Multiplication ***)
+
+Goal "i <= (j::int) ==> i * int k <= j * int k";
+by (induct_tac "k" 1);
+by (stac int_Suc 2);
+by (ALLGOALS
+ (asm_simp_tac (simpset() addsimps [zadd_zmult_distrib2, zadd_zle_mono,
+ int_Suc0_eq_1])));
+val lemma = result();
+
+Goal "[| i <= j; (0::int) <= k |] ==> i*k <= j*k";
+by (res_inst_tac [("t", "k")] (not_neg_nat RS subst) 1);
+by (etac lemma 2);
+by (full_simp_tac (simpset() addsimps [not_neg_eq_ge_0]) 1);
+qed "zmult_zle_mono1";
+
+Goal "[| i <= j; k <= (0::int) |] ==> j*k <= i*k";
+by (rtac (zminus_zle_zminus RS iffD1) 1);
+by (asm_simp_tac (simpset() addsimps [zmult_zminus_right RS sym,
+ zmult_zle_mono1, zle_zminus]) 1);
+qed "zmult_zle_mono1_neg";
+
+Goal "[| i <= j; (0::int) <= k |] ==> k*i <= k*j";
+by (dtac zmult_zle_mono1 1);
+by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [zmult_commute])));
+qed "zmult_zle_mono2";
+
+Goal "[| i <= j; k <= (0::int) |] ==> k*j <= k*i";
+by (dtac zmult_zle_mono1_neg 1);
+by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [zmult_commute])));
+qed "zmult_zle_mono2_neg";
+
+(* <= monotonicity, BOTH arguments*)
+Goal "[| i <= j; k <= l; (0::int) <= j; (0::int) <= k |] ==> i*k <= j*l";
+by (etac (zmult_zle_mono1 RS order_trans) 1);
+by (assume_tac 1);
+by (etac zmult_zle_mono2 1);
+by (assume_tac 1);
+qed "zmult_zle_mono";
+
+
+(** strict, in 1st argument; proof is by induction on k>0 **)
+
+Goal "i<j ==> 0<k --> int k * i < int k * j";
+by (induct_tac "k" 1);
+by (stac int_Suc 2);
+by (case_tac "n=0" 2);
+by (ALLGOALS (asm_full_simp_tac
+ (simpset() addsimps [zadd_zmult_distrib, zadd_zless_mono,
+ int_Suc0_eq_1, order_le_less])));
+val lemma = result();
+
+Goal "[| i<j; (0::int) < k |] ==> k*i < k*j";
+by (res_inst_tac [("t", "k")] (not_neg_nat RS subst) 1);
+by (etac (lemma RS mp) 2);
+by (asm_simp_tac (simpset() addsimps [not_neg_eq_ge_0,
+ order_le_less]) 1);
+by (forward_tac [conjI RS (zless_nat_conj RS iffD2)] 1);
+by Auto_tac;
+qed "zmult_zless_mono2";
+
+Goal "[| i<j; (0::int) < k |] ==> i*k < j*k";
+by (dtac zmult_zless_mono2 1);
+by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [zmult_commute])));
+qed "zmult_zless_mono1";
+
+(* < monotonicity, BOTH arguments*)
+Goal "[| i < j; k < l; (0::int) < j; (0::int) < k |] ==> i*k < j*l";
+by (etac (zmult_zless_mono1 RS order_less_trans) 1);
+by (assume_tac 1);
+by (etac zmult_zless_mono2 1);
+by (assume_tac 1);
+qed "zmult_zless_mono";
+
+Goal "[| i<j; k < (0::int) |] ==> j*k < i*k";
+by (rtac (zminus_zless_zminus RS iffD1) 1);
+by (asm_simp_tac (simpset() addsimps [zmult_zminus_right RS sym,
+ zmult_zless_mono1, zless_zminus]) 1);
+qed "zmult_zless_mono1_neg";
+
+Goal "[| i<j; k < (0::int) |] ==> k*j < k*i";
+by (rtac (zminus_zless_zminus RS iffD1) 1);
+by (asm_simp_tac (simpset() addsimps [zmult_zminus RS sym,
+ zmult_zless_mono2, zless_zminus]) 1);
+qed "zmult_zless_mono2_neg";
+
+
+Goal "(m*n = (0::int)) = (m = 0 | n = 0)";
+by (case_tac "m < (0::int)" 1);
+by (auto_tac (claset(),
+ simpset() addsimps [linorder_not_less, order_le_less,
+ linorder_neq_iff]));
+by (REPEAT
+ (force_tac (claset() addDs [zmult_zless_mono1_neg, zmult_zless_mono1],
+ simpset()) 1));
+qed "zmult_eq_0_iff";
+AddIffs [zmult_eq_0_iff];
+
+
+(** Cancellation laws for k*m < k*n and m*k < n*k, also for <= and =,
+ but not (yet?) for k*m < n*k. **)
+
+Goal "(m*k < n*k) = (((0::int) < k & m<n) | (k < 0 & n<m))";
+by (case_tac "k = (0::int)" 1);
+by (auto_tac (claset(), simpset() addsimps [linorder_neq_iff,
+ zmult_zless_mono1, zmult_zless_mono1_neg]));
+by (auto_tac (claset(),
+ simpset() addsimps [linorder_not_less,
+ inst "y1" "m*k" (linorder_not_le RS sym),
+ inst "y1" "m" (linorder_not_le RS sym)]));
+by (ALLGOALS (etac notE));
+by (auto_tac (claset(), simpset() addsimps [order_less_imp_le, zmult_zle_mono1,
+ zmult_zle_mono1_neg]));
+qed "zmult_zless_cancel2";
+
+
+Goal "(k*m < k*n) = (((0::int) < k & m<n) | (k < 0 & n<m))";
+by (simp_tac (simpset() addsimps [inst "z" "k" zmult_commute,
+ zmult_zless_cancel2]) 1);
+qed "zmult_zless_cancel1";
+
+Goal "(m*k <= n*k) = (((0::int) < k --> m<=n) & (k < 0 --> n<=m))";
+by (simp_tac (simpset() addsimps [linorder_not_less RS sym,
+ zmult_zless_cancel2]) 1);
+qed "zmult_zle_cancel2";
+
+Goal "(k*m <= k*n) = (((0::int) < k --> m<=n) & (k < 0 --> n<=m))";
+by (simp_tac (simpset() addsimps [linorder_not_less RS sym,
+ zmult_zless_cancel1]) 1);
+qed "zmult_zle_cancel1";
+
+Goal "(m*k = n*k) = (k = (0::int) | m=n)";
+by (cut_facts_tac [linorder_less_linear] 1);
+by Safe_tac;
+by Auto_tac;
+by (REPEAT
+ (force_tac (claset() addD2 ("mono_neg", zmult_zless_mono1_neg)
+ addD2 ("mono_pos", zmult_zless_mono1),
+ simpset() addsimps [linorder_neq_iff]) 1));
+
+qed "zmult_cancel2";
+
+Goal "(k*m = k*n) = (k = (0::int) | m=n)";
+by (simp_tac (simpset() addsimps [inst "z" "k" zmult_commute,
+ zmult_cancel2]) 1);
+qed "zmult_cancel1";
+Addsimps [zmult_cancel1, zmult_cancel2];
+
+
+(*Analogous to zadd_int*)
+Goal "n<=m --> int m - int n = int (m-n)";
+by (induct_thm_tac diff_induct "m n" 1);
+by (auto_tac (claset(), simpset() addsimps [int_Suc, symmetric zdiff_def]));
+qed_spec_mp "zdiff_int";