(* Title: HOL/nat_bin.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1999 University of Cambridge
Binary arithmetic for the natural numbers
*)
val nat_number_of_def = thm "nat_number_of_def";
(** nat (coercion from int to nat) **)
Goal "nat (number_of w) = number_of w";
by (simp_tac (simpset() addsimps [nat_number_of_def]) 1);
qed "nat_number_of";
Addsimps [nat_number_of, nat_0, nat_1];
Goal "Numeral0 = (0::nat)";
by (simp_tac (simpset() addsimps [nat_number_of_def]) 1);
qed "numeral_0_eq_0";
Goal "Numeral1 = (1::nat)";
by (simp_tac (simpset() addsimps [nat_1, nat_number_of_def]) 1);
qed "numeral_1_eq_1";
Goal "Numeral1 = Suc 0";
by (simp_tac (simpset() addsimps [numeral_1_eq_1]) 1);
qed "numeral_1_eq_Suc_0";
Goalw [nat_number_of_def] "2 = Suc (Suc 0)";
by (rtac nat_2 1);
qed "numeral_2_eq_2";
(** int (coercion from nat to int) **)
(*"neg" is used in rewrite rules for binary comparisons*)
Goal "int (number_of v :: nat) = \
\ (if neg (number_of v) then 0 \
\ else (number_of v :: int))";
by (simp_tac
(simpset_of Int.thy addsimps [neg_nat, nat_number_of_def,
not_neg_nat, int_0]) 1);
qed "int_nat_number_of";
Addsimps [int_nat_number_of];
val nat_bin_arith_setup =
[Fast_Arith.map_data
(fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, simpset} =>
{add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms,
inj_thms = inj_thms,
lessD = lessD,
simpset = simpset addsimps [int_nat_number_of, not_neg_number_of_Pls,
neg_number_of_Min,neg_number_of_BIT]})];
(** Successor **)
Goal "(0::int) <= z ==> Suc (nat z) = nat (1 + z)";
by (rtac sym 1);
by (asm_simp_tac (simpset() addsimps [nat_eq_iff, int_Suc]) 1);
qed "Suc_nat_eq_nat_zadd1";
Goal "Suc (number_of v + n) = \
\ (if neg (number_of v) then 1+n else number_of (bin_succ v) + n)";
by (simp_tac
(simpset_of Int.thy addsimps [neg_nat, nat_1, not_neg_eq_ge_0,
nat_number_of_def, int_Suc,
Suc_nat_eq_nat_zadd1, number_of_succ]) 1);
qed "Suc_nat_number_of_add";
Goal "Suc (number_of v) = \
\ (if neg (number_of v) then 1 else number_of (bin_succ v))";
by (cut_inst_tac [("n","0")] Suc_nat_number_of_add 1);
by (asm_full_simp_tac (simpset() delcongs [if_weak_cong]) 1);
qed "Suc_nat_number_of";
Addsimps [Suc_nat_number_of];
(** Addition **)
Goal "[| (0::int) <= z; 0 <= z' |] ==> nat (z+z') = nat z + nat z'";
by (rtac (inj_int RS injD) 1);
by (asm_simp_tac (simpset() addsimps [zadd_int RS sym]) 1);
qed "nat_add_distrib";
(*"neg" is used in rewrite rules for binary comparisons*)
Goal "(number_of v :: nat) + number_of v' = \
\ (if neg (number_of v) then number_of v' \
\ else if neg (number_of v') then number_of v \
\ else number_of (bin_add v v'))";
by (simp_tac
(simpset_of Int.thy addsimps [neg_nat, not_neg_eq_ge_0, nat_number_of_def,
nat_add_distrib RS sym, number_of_add]) 1);
qed "add_nat_number_of";
Addsimps [add_nat_number_of];
(** Subtraction **)
Goal "[| (0::int) <= z'; z' <= z |] ==> nat (z-z') = nat z - nat z'";
by (rtac (inj_int RS injD) 1);
by (asm_simp_tac (simpset() addsimps [zdiff_int RS sym, nat_le_eq_zle]) 1);
qed "nat_diff_distrib";
Goal "nat z - nat z' = \
\ (if neg z' then nat z \
\ else let d = z-z' in \
\ if neg d then 0 else nat d)";
by (simp_tac (simpset() addsimps [Let_def, nat_diff_distrib RS sym,
neg_eq_less_0, not_neg_eq_ge_0]) 1);
by (simp_tac (simpset() addsimps [diff_is_0_eq, nat_le_eq_zle]) 1);
qed "diff_nat_eq_if";
Goalw [nat_number_of_def]
"(number_of v :: nat) - number_of v' = \
\ (if neg (number_of v') then number_of v \
\ else let d = number_of (bin_add v (bin_minus v')) in \
\ if neg d then 0 else nat d)";
by (simp_tac
(simpset_of Int.thy delcongs [if_weak_cong]
addsimps [not_neg_eq_ge_0, nat_0,
diff_nat_eq_if, diff_number_of_eq]) 1);
qed "diff_nat_number_of";
Addsimps [diff_nat_number_of];
(** Multiplication **)
Goal "(0::int) <= z ==> nat (z*z') = nat z * nat z'";
by (case_tac "0 <= z'" 1);
by (asm_full_simp_tac (simpset() addsimps [zmult_le_0_iff]) 2);
by (rtac (inj_int RS injD) 1);
by (asm_simp_tac (simpset() addsimps [zmult_int RS sym,
int_0_le_mult_iff]) 1);
qed "nat_mult_distrib";
Goal "z <= (0::int) ==> nat(z*z') = nat(-z) * nat(-z')";
by (rtac trans 1);
by (rtac nat_mult_distrib 2);
by Auto_tac;
qed "nat_mult_distrib_neg";
Goal "(number_of v :: nat) * number_of v' = \
\ (if neg (number_of v) then 0 else number_of (bin_mult v v'))";
by (simp_tac
(simpset_of Int.thy addsimps [neg_nat, not_neg_eq_ge_0, nat_number_of_def,
nat_mult_distrib RS sym, number_of_mult,
nat_0]) 1);
qed "mult_nat_number_of";
Addsimps [mult_nat_number_of];
(** Quotient **)
Goal "(0::int) <= z ==> nat (z div z') = nat z div nat z'";
by (case_tac "0 <= z'" 1);
by (auto_tac (claset(),
simpset() addsimps [div_nonneg_neg_le0, DIVISION_BY_ZERO_DIV]));
by (case_tac "z' = 0" 1 THEN asm_simp_tac (simpset() addsimps [DIVISION_BY_ZERO]) 1);
by (simp_tac (simpset() addsimps [numeral_0_eq_0, DIVISION_BY_ZERO_DIV]) 1);
by (auto_tac (claset() addSEs [nonneg_eq_int], simpset()));
by (rename_tac "m m'" 1);
by (subgoal_tac "0 <= int m div int m'" 1);
by (asm_full_simp_tac
(simpset() addsimps [numeral_0_eq_0, pos_imp_zdiv_nonneg_iff]) 2);
by (rtac (inj_int RS injD) 1);
by (Asm_simp_tac 1);
by (res_inst_tac [("r", "int (m mod m')")] quorem_div 1);
by (Force_tac 2);
by (asm_full_simp_tac
(simpset() addsimps [nat_less_iff RS sym, quorem_def,
numeral_0_eq_0, zadd_int, zmult_int]) 1);
by (rtac (mod_div_equality RS sym RS trans) 1);
by (asm_simp_tac (simpset() addsimps add_ac@mult_ac) 1);
qed "nat_div_distrib";
Goal "(number_of v :: nat) div number_of v' = \
\ (if neg (number_of v) then 0 \
\ else nat (number_of v div number_of v'))";
by (simp_tac
(simpset_of Int.thy addsimps [not_neg_eq_ge_0, nat_number_of_def, neg_nat,
nat_div_distrib RS sym, nat_0]) 1);
qed "div_nat_number_of";
Addsimps [div_nat_number_of];
(** Remainder **)
(*Fails if z'<0: the LHS collapses to (nat z) but the RHS doesn't*)
Goal "[| (0::int) <= z; 0 <= z' |] ==> nat (z mod z') = nat z mod nat z'";
by (case_tac "z' = 0" 1 THEN asm_simp_tac (simpset() addsimps [DIVISION_BY_ZERO]) 1);
by (simp_tac (simpset() addsimps [numeral_0_eq_0, DIVISION_BY_ZERO_MOD]) 1);
by (auto_tac (claset() addSEs [nonneg_eq_int], simpset()));
by (rename_tac "m m'" 1);
by (subgoal_tac "0 <= int m mod int m'" 1);
by (asm_full_simp_tac
(simpset() addsimps [nat_less_iff, numeral_0_eq_0, pos_mod_sign]) 2);
by (rtac (inj_int RS injD) 1);
by (Asm_simp_tac 1);
by (res_inst_tac [("q", "int (m div m')")] quorem_mod 1);
by (Force_tac 2);
by (asm_full_simp_tac
(simpset() addsimps [nat_less_iff RS sym, quorem_def,
numeral_0_eq_0, zadd_int, zmult_int]) 1);
by (rtac (mod_div_equality RS sym RS trans) 1);
by (asm_simp_tac (simpset() addsimps add_ac@mult_ac) 1);
qed "nat_mod_distrib";
Goal "(number_of v :: nat) mod number_of v' = \
\ (if neg (number_of v) then 0 \
\ else if neg (number_of v') then number_of v \
\ else nat (number_of v mod number_of v'))";
by (simp_tac
(simpset_of Int.thy addsimps [not_neg_eq_ge_0, nat_number_of_def,
neg_nat, nat_0, DIVISION_BY_ZERO_MOD,
nat_mod_distrib RS sym]) 1);
qed "mod_nat_number_of";
Addsimps [mod_nat_number_of];
structure NatAbstractNumeralsData =
struct
val dest_eq = HOLogic.dest_eq o HOLogic.dest_Trueprop o concl_of
val is_numeral = Bin_Simprocs.is_numeral
val numeral_0_eq_0 = numeral_0_eq_0
val numeral_1_eq_1 = numeral_1_eq_Suc_0
val prove_conv = Bin_Simprocs.prove_conv_nohyps "nat_abstract_numerals"
fun norm_tac simps = ALLGOALS (simp_tac (HOL_ss addsimps simps))
val simplify_meta_eq = Bin_Simprocs.simplify_meta_eq
end
structure NatAbstractNumerals = AbstractNumeralsFun (NatAbstractNumeralsData)
val nat_eval_numerals =
map Bin_Simprocs.prep_simproc
[("nat_div_eval_numerals",
Bin_Simprocs.prep_pats ["(Suc 0) div m"],
NatAbstractNumerals.proc div_nat_number_of),
("nat_mod_eval_numerals",
Bin_Simprocs.prep_pats ["(Suc 0) mod m"],
NatAbstractNumerals.proc mod_nat_number_of)];
Addsimprocs nat_eval_numerals;
(*** Comparisons ***)
(** Equals (=) **)
Goal "[| (0::int) <= z; 0 <= z' |] ==> (nat z = nat z') = (z=z')";
by (auto_tac (claset() addSEs [nonneg_eq_int], simpset()));
qed "eq_nat_nat_iff";
(*"neg" is used in rewrite rules for binary comparisons*)
Goal "((number_of v :: nat) = number_of v') = \
\ (if neg (number_of v) then (iszero (number_of v') | neg (number_of v')) \
\ else if neg (number_of v') then iszero (number_of v) \
\ else iszero (number_of (bin_add v (bin_minus v'))))";
by (simp_tac
(simpset_of Int.thy addsimps [neg_nat, not_neg_eq_ge_0, nat_number_of_def,
eq_nat_nat_iff, eq_number_of_eq, nat_0]) 1);
by (simp_tac (simpset_of Int.thy addsimps [nat_eq_iff, nat_eq_iff2,
iszero_def]) 1);
by (simp_tac (simpset () addsimps [not_neg_eq_ge_0 RS sym]) 1);
qed "eq_nat_number_of";
Addsimps [eq_nat_number_of];
(** Less-than (<) **)
(*"neg" is used in rewrite rules for binary comparisons*)
Goal "((number_of v :: nat) < number_of v') = \
\ (if neg (number_of v) then neg (number_of (bin_minus v')) \
\ else neg (number_of (bin_add v (bin_minus v'))))";
by (simp_tac
(simpset_of Int.thy addsimps [neg_nat, not_neg_eq_ge_0, nat_number_of_def,
nat_less_eq_zless, less_number_of_eq_neg,
nat_0]) 1);
by (simp_tac (simpset_of Int.thy addsimps [neg_eq_less_0, zminus_zless,
number_of_minus, zless_nat_eq_int_zless]) 1);
qed "less_nat_number_of";
Addsimps [less_nat_number_of];
(** Less-than-or-equals (<=) **)
Goal "(number_of x <= (number_of y::nat)) = \
\ (~ number_of y < (number_of x::nat))";
by (rtac (linorder_not_less RS sym) 1);
qed "le_nat_number_of_eq_not_less";
Addsimps [le_nat_number_of_eq_not_less];
(*Maps #n to n for n = 0, 1, 2*)
bind_thms ("numerals", [numeral_0_eq_0, numeral_1_eq_1, numeral_2_eq_2]);
val numeral_ss = simpset() addsimps numerals;
(** Nat **)
Goal "0 < n ==> n = Suc(n - 1)";
by (asm_full_simp_tac numeral_ss 1);
qed "Suc_pred'";
(*Expresses a natural number constant as the Suc of another one.
NOT suitable for rewriting because n recurs in the condition.*)
bind_thm ("expand_Suc", inst "n" "number_of ?v" Suc_pred');
(** Arith **)
Goal "Suc n = n + 1";
by (asm_simp_tac numeral_ss 1);
qed "Suc_eq_add_numeral_1";
(* These two can be useful when m = number_of... *)
Goal "(m::nat) + n = (if m=0 then n else Suc ((m - 1) + n))";
by (case_tac "m" 1);
by (ALLGOALS (asm_simp_tac numeral_ss));
qed "add_eq_if";
Goal "(m::nat) * n = (if m=0 then 0 else n + ((m - 1) * n))";
by (case_tac "m" 1);
by (ALLGOALS (asm_simp_tac numeral_ss));
qed "mult_eq_if";
Goal "(p ^ m :: nat) = (if m=0 then 1 else p * (p ^ (m - 1)))";
by (case_tac "m" 1);
by (ALLGOALS (asm_simp_tac numeral_ss));
qed "power_eq_if";
Goal "[| 0<n; 0<m |] ==> m - n < (m::nat)";
by (asm_full_simp_tac (numeral_ss addsimps [diff_less]) 1);
qed "diff_less'";
Addsimps [inst "n" "number_of ?v" diff_less'];
(** Power **)
Goal "(p::nat) ^ 2 = p*p";
by (simp_tac numeral_ss 1);
qed "power_two";
(*** Comparisons involving (0::nat) ***)
Goal "(number_of v = (0::nat)) = \
\ (if neg (number_of v) then True else iszero (number_of v))";
by (simp_tac (simpset() addsimps [numeral_0_eq_0 RS sym, iszero_0]) 1);
qed "eq_number_of_0";
Goal "((0::nat) = number_of v) = \
\ (if neg (number_of v) then True else iszero (number_of v))";
by (rtac ([eq_sym_conv, eq_number_of_0] MRS trans) 1);
qed "eq_0_number_of";
Goal "((0::nat) < number_of v) = neg (number_of (bin_minus v))";
by (simp_tac (simpset() addsimps [numeral_0_eq_0 RS sym]) 1);
qed "less_0_number_of";
(*Simplification already handles n<0, n<=0 and 0<=n.*)
Addsimps [eq_number_of_0, eq_0_number_of, less_0_number_of];
Goal "neg (number_of v) ==> number_of v = (0::nat)";
by (asm_simp_tac (simpset() addsimps [numeral_0_eq_0 RS sym, iszero_0]) 1);
qed "neg_imp_number_of_eq_0";
(*** Comparisons involving Suc ***)
Goal "(number_of v = Suc n) = \
\ (let pv = number_of (bin_pred v) in \
\ if neg pv then False else nat pv = n)";
by (simp_tac
(simpset_of Int.thy addsimps
[Let_def, neg_eq_less_0, linorder_not_less, number_of_pred,
nat_number_of_def, zadd_0] @ zadd_ac) 1);
by (res_inst_tac [("x", "number_of v")] spec 1);
by (auto_tac (claset(), simpset() addsimps [nat_eq_iff]));
qed "eq_number_of_Suc";
Goal "(Suc n = number_of v) = \
\ (let pv = number_of (bin_pred v) in \
\ if neg pv then False else nat pv = n)";
by (rtac ([eq_sym_conv, eq_number_of_Suc] MRS trans) 1);
qed "Suc_eq_number_of";
Goal "(number_of v < Suc n) = \
\ (let pv = number_of (bin_pred v) in \
\ if neg pv then True else nat pv < n)";
by (simp_tac
(simpset_of Int.thy addsimps
[Let_def, neg_eq_less_0, linorder_not_less, number_of_pred,
nat_number_of_def, zadd_0] @ zadd_ac) 1);
by (res_inst_tac [("x", "number_of v")] spec 1);
by (auto_tac (claset(), simpset() addsimps [nat_less_iff]));
qed "less_number_of_Suc";
Goal "(Suc n < number_of v) = \
\ (let pv = number_of (bin_pred v) in \
\ if neg pv then False else n < nat pv)";
by (simp_tac
(simpset_of Int.thy addsimps
[Let_def, neg_eq_less_0, linorder_not_less, number_of_pred,
nat_number_of_def, zadd_0] @ zadd_ac) 1);
by (res_inst_tac [("x", "number_of v")] spec 1);
by (auto_tac (claset(), simpset() addsimps [zless_nat_eq_int_zless]));
qed "less_Suc_number_of";
Goal "(number_of v <= Suc n) = \
\ (let pv = number_of (bin_pred v) in \
\ if neg pv then True else nat pv <= n)";
by (simp_tac
(simpset () addsimps
[Let_def, less_Suc_number_of, linorder_not_less RS sym]) 1);
qed "le_number_of_Suc";
Goal "(Suc n <= number_of v) = \
\ (let pv = number_of (bin_pred v) in \
\ if neg pv then False else n <= nat pv)";
by (simp_tac
(simpset () addsimps
[Let_def, less_number_of_Suc, linorder_not_less RS sym]) 1);
qed "le_Suc_number_of";
Addsimps [eq_number_of_Suc, Suc_eq_number_of,
less_number_of_Suc, less_Suc_number_of,
le_number_of_Suc, le_Suc_number_of];
(* Push int(.) inwards: *)
Addsimps [zadd_int RS sym];
Goal "(m+m = n+n) = (m = (n::int))";
by Auto_tac;
val lemma1 = result();
Goal "m+m ~= (1::int) + n + n";
by Auto_tac;
by (dres_inst_tac [("f", "%x. x mod 2")] arg_cong 1);
by (full_simp_tac (simpset() addsimps [zmod_zadd1_eq]) 1);
val lemma2 = result();
Goal "((number_of (v BIT x) ::int) = number_of (w BIT y)) = \
\ (x=y & (((number_of v) ::int) = number_of w))";
by (simp_tac (simpset_of Int.thy addsimps
[number_of_BIT, lemma1, lemma2, eq_commute]) 1);
qed "eq_number_of_BIT_BIT";
Goal "((number_of (v BIT x) ::int) = number_of Pls) = \
\ (x=False & (((number_of v) ::int) = number_of Pls))";
by (simp_tac (simpset_of Int.thy addsimps
[number_of_BIT, number_of_Pls, eq_commute]) 1);
by (res_inst_tac [("x", "number_of v")] spec 1);
by Safe_tac;
by (ALLGOALS Full_simp_tac);
by (dres_inst_tac [("f", "%x. x mod 2")] arg_cong 1);
by (full_simp_tac (simpset() addsimps [zmod_zadd1_eq]) 1);
qed "eq_number_of_BIT_Pls";
Goal "((number_of (v BIT x) ::int) = number_of Min) = \
\ (x=True & (((number_of v) ::int) = number_of Min))";
by (simp_tac (simpset_of Int.thy addsimps
[number_of_BIT, number_of_Min, eq_commute]) 1);
by (res_inst_tac [("x", "number_of v")] spec 1);
by Auto_tac;
by (dres_inst_tac [("f", "%x. x mod 2")] arg_cong 1);
by Auto_tac;
qed "eq_number_of_BIT_Min";
Goal "(number_of Pls ::int) ~= number_of Min";
by Auto_tac;
qed "eq_number_of_Pls_Min";
(*** Further lemmas about "nat" ***)
Goal "nat (abs (w * z)) = nat (abs w) * nat (abs z)";
by (case_tac "z=0 | w=0" 1);
by Auto_tac;
by (simp_tac (simpset() addsimps [zabs_def, nat_mult_distrib RS sym,
nat_mult_distrib_neg RS sym, zmult_less_0_iff]) 1);
by (arith_tac 1);
qed "nat_abs_mult_distrib";
(*Distributive laws for literals*)
Addsimps (map (inst "k" "number_of ?v")
[add_mult_distrib, add_mult_distrib2,
diff_mult_distrib, diff_mult_distrib2]);
(*** Literal arithmetic involving powers, type nat ***)
Goal "(0::int) <= z ==> nat (z^n) = nat z ^ n";
by (induct_tac "n" 1);
by (ALLGOALS (asm_simp_tac (simpset() addsimps [nat_mult_distrib])));
qed "nat_power_eq";
Goal "(number_of v :: nat) ^ n = \
\ (if neg (number_of v) then 0^n else nat ((number_of v :: int) ^ n))";
by (simp_tac
(simpset_of Int.thy addsimps [neg_nat, not_neg_eq_ge_0, nat_number_of_def,
nat_power_eq]) 1);
qed "power_nat_number_of";
Addsimps [inst "n" "number_of ?w" power_nat_number_of];
(*** Literal arithmetic involving powers, type int ***)
Goal "(z::int) ^ (2*a) = (z^a)^2";
by (simp_tac (simpset() addsimps [zpower_zpower, mult_commute]) 1);
qed "zpower_even";
Goal "(p::int) ^ 2 = p*p";
by (simp_tac numeral_ss 1);
qed "zpower_two";
Goal "(z::int) ^ (2*a + 1) = z * (z^a)^2";
by (simp_tac (simpset() addsimps [zpower_even, zpower_zadd_distrib]) 1);
qed "zpower_odd";
Goal "(z::int) ^ number_of (w BIT False) = \
\ (let w = z ^ (number_of w) in w*w)";
by (simp_tac (simpset_of Int.thy addsimps [nat_number_of_def,
number_of_BIT, Let_def]) 1);
by (res_inst_tac [("x","number_of w")] spec 1 THEN Clarify_tac 1);
by (case_tac "(0::int) <= x" 1);
by (auto_tac (claset(),
simpset() addsimps [nat_mult_distrib, zpower_even, zpower_two]));
qed "zpower_number_of_even";
Goal "(z::int) ^ number_of (w BIT True) = \
\ (if (0::int) <= number_of w \
\ then (let w = z ^ (number_of w) in z*w*w) \
\ else 1)";
by (simp_tac (simpset_of Int.thy addsimps [nat_number_of_def,
number_of_BIT, Let_def]) 1);
by (res_inst_tac [("x","number_of w")] spec 1 THEN Clarify_tac 1);
by (case_tac "(0::int) <= x" 1);
by (auto_tac (claset(),
simpset() addsimps [nat_add_distrib, nat_mult_distrib,
zpower_even, zpower_two]));
qed "zpower_number_of_odd";
Addsimps (map (inst "z" "number_of ?v")
[zpower_number_of_even, zpower_number_of_odd]);