author | paulson |
Wed, 21 Jul 1999 15:20:26 +0200 | |
changeset 7056 | 522a7013d7df |
parent 7032 | d6efb3b8e669 |
child 7075 | 5ba8d1e42ca6 |
permissions | -rw-r--r-- |
7032 | 1 |
(* Title: HOL/NatBin.ML |
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ID: $Id$ |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Copyright 1999 University of Cambridge |
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Binary arithmetic for the natural numbers |
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*) |
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(** nat (coercion from int to nat) **) |
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Goal "nat (number_of w) = number_of w"; |
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by (simp_tac (simpset() addsimps [nat_number_of_def]) 1); |
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qed "nat_number_of"; |
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Addsimps [nat_number_of]; |
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(*These rewrites should one day be re-oriented...*) |
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Goal "#0 = 0"; |
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by (simp_tac (simpset_of Int.thy addsimps [nat_0, nat_number_of_def]) 1); |
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qed "numeral_0_eq_0"; |
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Goal "#1 = 1"; |
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by (simp_tac (simpset_of Int.thy addsimps [nat_1, nat_number_of_def]) 1); |
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qed "numeral_1_eq_1"; |
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Goal "#2 = 2"; |
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by (simp_tac (simpset_of Int.thy addsimps [nat_2, nat_number_of_def]) 1); |
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qed "numeral_2_eq_2"; |
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(** int (coercion from nat to int) **) |
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(*"neg" is used in rewrite rules for binary comparisons*) |
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Goal "int (number_of v :: nat) = \ |
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\ (if neg (number_of v) then #0 \ |
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\ else (number_of v :: int))"; |
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by (simp_tac |
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(simpset_of Int.thy addsimps [neg_nat, nat_number_of_def, |
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not_neg_nat, int_0]) 1); |
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qed "int_nat_number_of"; |
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Addsimps [int_nat_number_of]; |
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(** Successor **) |
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Goal "(#0::int) <= z ==> Suc (nat z) = nat (#1 + z)"; |
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br sym 1; |
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by (asm_simp_tac (simpset() addsimps [nat_eq_iff]) 1); |
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qed "Suc_nat_eq_nat_zadd1"; |
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Goal "Suc (number_of v) = \ |
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\ (if neg (number_of v) then #1 else number_of (bin_succ v))"; |
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by (simp_tac |
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(simpset_of Int.thy addsimps [neg_nat, nat_1, not_neg_eq_ge_0, |
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nat_number_of_def, int_Suc, |
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Suc_nat_eq_nat_zadd1, number_of_succ]) 1); |
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qed "Suc_nat_number_of"; |
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Goal "Suc #0 = #1"; |
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by (simp_tac (simpset() addsimps [Suc_nat_number_of]) 1); |
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qed "Suc_numeral_0_eq_1"; |
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Goal "Suc #1 = #2"; |
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by (simp_tac (simpset() addsimps [Suc_nat_number_of]) 1); |
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qed "Suc_numeral_1_eq_2"; |
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(** Addition **) |
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Goal "[| (#0::int) <= z; #0 <= z' |] ==> nat z + nat z' = nat (z+z')"; |
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by (rtac (inj_int RS injD) 1); |
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by (asm_simp_tac (simpset() addsimps [zadd_int RS sym]) 1); |
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qed "add_nat_eq_nat_zadd"; |
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(*"neg" is used in rewrite rules for binary comparisons*) |
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Goal "(number_of v :: nat) + number_of v' = \ |
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\ (if neg (number_of v) then number_of v' \ |
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\ else if neg (number_of v') then number_of v \ |
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\ else number_of (bin_add v v'))"; |
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by (simp_tac |
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(simpset_of Int.thy addsimps [neg_nat, not_neg_eq_ge_0, nat_number_of_def, |
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add_nat_eq_nat_zadd, number_of_add]) 1); |
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qed "add_nat_number_of"; |
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Addsimps [add_nat_number_of]; |
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(** Subtraction **) |
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Goal "[| (#0::int) <= z'; z' <= z |] ==> nat z - nat z' = nat (z-z')"; |
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by (rtac (inj_int RS injD) 1); |
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by (asm_simp_tac (simpset() addsimps [zdiff_int RS sym, nat_le_eq_zle]) 1); |
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qed "diff_nat_eq_nat_zdiff"; |
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Goal "nat z - nat z' = \ |
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\ (if neg z' then nat z \ |
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\ else let d = z-z' in \ |
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\ if neg d then 0 else nat d)"; |
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by (simp_tac (simpset() addsimps [Let_def, diff_nat_eq_nat_zdiff, |
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neg_eq_less_0, not_neg_eq_ge_0]) 1); |
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by (simp_tac (simpset() addsimps zcompare_rls@ |
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[diff_is_0_eq, nat_le_eq_zle]) 1); |
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qed "diff_nat_eq_if"; |
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Goalw [nat_number_of_def] |
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"(number_of v :: nat) - number_of v' = \ |
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\ (if neg (number_of v') then number_of v \ |
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\ else let d = number_of (bin_add v (bin_minus v')) in \ |
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\ if neg d then #0 else nat d)"; |
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by (simp_tac |
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(simpset_of Int.thy delcongs [if_weak_cong] |
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addsimps [not_neg_eq_ge_0, nat_0, |
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diff_nat_eq_if, diff_number_of_eq]) 1); |
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qed "diff_nat_number_of"; |
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Addsimps [diff_nat_number_of]; |
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(** Multiplication **) |
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Goal "(#0::int) <= z ==> nat z * nat z' = nat (z*z')"; |
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by (case_tac "#0 <= z'" 1); |
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by (subgoal_tac "z'*z <= #0" 2); |
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by (rtac (neg_imp_zmult_nonpos_iff RS iffD2) 3); |
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by Auto_tac; |
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by (subgoal_tac "#0 <= z*z'" 1); |
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by (force_tac (claset() addDs [zmult_zle_mono1], simpset()) 2); |
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by (rtac (inj_int RS injD) 1); |
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by (asm_simp_tac (simpset() addsimps [zmult_int RS sym]) 1); |
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qed "mult_nat_eq_nat_zmult"; |
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Goal "(number_of v :: nat) * number_of v' = \ |
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\ (if neg (number_of v) then #0 else number_of (bin_mult v v'))"; |
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by (simp_tac |
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(simpset_of Int.thy addsimps [neg_nat, not_neg_eq_ge_0, nat_number_of_def, |
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mult_nat_eq_nat_zmult, number_of_mult, |
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nat_0]) 1); |
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qed "mult_nat_number_of"; |
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Addsimps [mult_nat_number_of]; |
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(** Quotient **) |
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Goal "(#0::int) <= z ==> nat z div nat z' = nat (z div z')"; |
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by (case_tac "#0 <= z'" 1); |
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by (auto_tac (claset(), |
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simpset() addsimps [div_nonneg_neg, DIVISION_BY_ZERO_DIV])); |
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by (zdiv_undefined_case_tac "z' = #0" 1); |
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by (simp_tac (simpset() addsimps [numeral_0_eq_0, DIVISION_BY_ZERO_DIV]) 1); |
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by (auto_tac (claset() addSEs [nonneg_eq_int], simpset())); |
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by (rename_tac "m m'" 1); |
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by (subgoal_tac "#0 <= int m div int m'" 1); |
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by (asm_simp_tac (simpset() addsimps [nat_less_iff RS sym, numeral_0_eq_0, |
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pos_imp_zdiv_nonneg_iff]) 2); |
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by (rtac (inj_int RS injD) 1); |
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by (Asm_simp_tac 1); |
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by (rtac sym 1); |
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by (res_inst_tac [("r", "int (m mod m')")] quorem_div 1); |
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by (Force_tac 2); |
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by (asm_simp_tac (simpset() addsimps [nat_less_iff RS sym, quorem_def, |
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numeral_0_eq_0, zadd_int, zmult_int, |
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mod_less_divisor]) 1); |
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by (rtac (mod_div_equality RS sym RS trans) 1); |
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by (asm_simp_tac (simpset() addsimps add_ac@mult_ac) 1); |
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qed "div_nat_eq_nat_zdiv"; |
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Goal "(number_of v :: nat) div number_of v' = \ |
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\ (if neg (number_of v) then #0 \ |
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\ else nat (number_of v div number_of v'))"; |
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by (simp_tac |
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(simpset_of Int.thy addsimps [not_neg_eq_ge_0, nat_number_of_def, neg_nat, |
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div_nat_eq_nat_zdiv, nat_0]) 1); |
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qed "div_nat_number_of"; |
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Addsimps [div_nat_number_of]; |
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(** Remainder **) |
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(*Fails if z'<0: the LHS collapses to (nat z) but the RHS doesn't*) |
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Goal "[| (#0::int) <= z; #0 <= z' |] ==> nat z mod nat z' = nat (z mod z')"; |
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by (zdiv_undefined_case_tac "z' = #0" 1); |
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by (simp_tac (simpset() addsimps [numeral_0_eq_0, DIVISION_BY_ZERO_MOD]) 1); |
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by (auto_tac (claset() addSEs [nonneg_eq_int], simpset())); |
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by (rename_tac "m m'" 1); |
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by (subgoal_tac "#0 <= int m mod int m'" 1); |
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by (asm_simp_tac (simpset() addsimps [nat_less_iff RS sym, numeral_0_eq_0, |
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pos_mod_sign]) 2); |
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by (rtac (inj_int RS injD) 1); |
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by (Asm_simp_tac 1); |
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by (rtac sym 1); |
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by (res_inst_tac [("q", "int (m div m')")] quorem_mod 1); |
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by (Force_tac 2); |
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by (asm_simp_tac (simpset() addsimps [nat_less_iff RS sym, quorem_def, |
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numeral_0_eq_0, zadd_int, zmult_int, |
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mod_less_divisor]) 1); |
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by (rtac (mod_div_equality RS sym RS trans) 1); |
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by (asm_simp_tac (simpset() addsimps add_ac@mult_ac) 1); |
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qed "mod_nat_eq_nat_zmod"; |
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Goal "(number_of v :: nat) mod number_of v' = \ |
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\ (if neg (number_of v) then #0 \ |
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\ else if neg (number_of v') then number_of v \ |
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\ else nat (number_of v mod number_of v'))"; |
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by (simp_tac |
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(simpset_of Int.thy addsimps [not_neg_eq_ge_0, nat_number_of_def, |
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neg_nat, nat_0, DIVISION_BY_ZERO_MOD, |
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mod_nat_eq_nat_zmod]) 1); |
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qed "mod_nat_number_of"; |
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Addsimps [mod_nat_number_of]; |
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(*** Comparisons ***) |
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(** Equals (=) **) |
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Goal "[| (#0::int) <= z; #0 <= z' |] ==> (nat z = nat z') = (z=z')"; |
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by (auto_tac (claset() addSEs [nonneg_eq_int], simpset())); |
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qed "eq_nat_nat_iff"; |
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(*"neg" is used in rewrite rules for binary comparisons*) |
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Goal "((number_of v :: nat) = number_of v') = \ |
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\ (if neg (number_of v) then ((#0::nat) = number_of v') \ |
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\ else if neg (number_of v') then iszero (number_of v) \ |
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\ else iszero (number_of (bin_add v (bin_minus v'))))"; |
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by (simp_tac |
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(simpset_of Int.thy addsimps [neg_nat, not_neg_eq_ge_0, nat_number_of_def, |
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eq_nat_nat_iff, eq_number_of_eq, nat_0]) 1); |
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by (simp_tac (simpset_of Int.thy addsimps [nat_eq_iff, iszero_def]) 1); |
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qed "eq_nat_number_of"; |
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Addsimps [eq_nat_number_of]; |
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(** Less-than (<) **) |
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(*"neg" is used in rewrite rules for binary comparisons*) |
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Goal "((number_of v :: nat) < number_of v') = \ |
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\ (if neg (number_of v) then neg (number_of (bin_minus v')) \ |
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\ else neg (number_of (bin_add v (bin_minus v'))))"; |
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by (simp_tac |
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(simpset_of Int.thy addsimps [neg_nat, not_neg_eq_ge_0, nat_number_of_def, |
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nat_less_eq_zless, less_number_of_eq_neg, |
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nat_0]) 1); |
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by (simp_tac (simpset_of Int.thy addsimps [neg_eq_less_int0, zminus_zless, |
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number_of_minus, zless_zero_nat]) 1); |
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qed "less_nat_number_of"; |
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Addsimps [less_nat_number_of]; |
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(** Less-than-or-equals (<=) **) |
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Goal "(number_of x <= (number_of y::nat)) = \ |
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\ (~ number_of y < (number_of x::nat))"; |
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by (rtac (linorder_not_less RS sym) 1); |
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qed "le_nat_number_of_eq_not_less"; |
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Addsimps [le_nat_number_of_eq_not_less]; |
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(*** New versions of existing theorems involving 0, 1, 2 ***) |
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fun change_theory thy th = |
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[th, read_instantiate_sg (sign_of thy) [("t","dummyVar")] refl] |
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MRS (conjI RS conjunct1) |> standard; |
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(*Maps n to #n for n = 0, 1, 2*) |
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val numeral_sym_ss = |
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HOL_ss addsimps [numeral_0_eq_0 RS sym, |
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numeral_1_eq_1 RS sym, |
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numeral_2_eq_2 RS sym, |
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Suc_numeral_1_eq_2, Suc_numeral_0_eq_1]; |
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fun rename_numerals thy th = simplify numeral_sym_ss (change_theory thy th); |
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(*Maps #n to n for n = 0, 1, 2*) |
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val numeral_ss = |
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simpset() addsimps [numeral_0_eq_0, numeral_1_eq_1, numeral_2_eq_2]; |
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(** Nat **) |
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Goal "#0 < n ==> n = Suc(n - #1)"; |
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by (asm_full_simp_tac numeral_ss 1); |
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qed "Suc_pred'"; |
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fun inst x t = read_instantiate_sg (sign_of NatBin.thy) [(x,t)]; |
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(*Expresses a natural number constant as the Suc of another one. |
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NOT suitable for rewriting because n recurs in the condition.*) |
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bind_thm ("expand_Suc", inst "n" "number_of ?v" Suc_pred'); |
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(** NatDef & Nat **) |
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Addsimps (map (rename_numerals thy) |
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[min_0L, min_0R, max_0L, max_0R]); |
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AddIffs (map (rename_numerals thy) |
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[Suc_not_Zero, Zero_not_Suc, zero_less_Suc, not_less0, less_one, |
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le0, le_0_eq, |
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neq0_conv, zero_neq_conv, not_gr0]); |
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7032 | 305 |
(** Arith **) |
306 |
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307 |
Addsimps (map (rename_numerals thy) |
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308 |
[diff_0_eq_0, add_0, add_0_right, add_pred, |
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diff_is_0_eq, zero_is_diff_eq, zero_less_diff, |
7032 | 310 |
mult_0, mult_0_right, mult_1, mult_1_right, |
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mult_is_0, zero_less_mult_iff, |
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mult_eq_1_iff]); |
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313 |
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AddIffs (map (rename_numerals thy) [add_is_0, zero_is_add, add_gr_0]); |
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Goal "Suc n = n + #1"; |
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317 |
by (asm_simp_tac numeral_ss 1); |
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qed "Suc_eq_add_numeral_1"; |
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7032 | 320 |
(* These two can be useful when m = number_of... *) |
321 |
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322 |
Goal "(m::nat) + n = (if m=#0 then n else Suc ((m - #1) + n))"; |
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by (exhaust_tac "m" 1); |
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by (ALLGOALS (asm_simp_tac numeral_ss)); |
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qed "add_eq_if"; |
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Goal "(m::nat) * n = (if m=#0 then #0 else n + ((m - #1) * n))"; |
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by (exhaust_tac "m" 1); |
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by (ALLGOALS (asm_simp_tac numeral_ss)); |
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qed "mult_eq_if"; |
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332 |
Goal "(p ^ m :: nat) = (if m=#0 then #1 else p * (p ^ (m - #1)))"; |
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by (exhaust_tac "m" 1); |
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by (ALLGOALS (asm_simp_tac numeral_ss)); |
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qed "power_eq_if"; |
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|
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Goal "[| #0<n; #0<m |] ==> m - n < (m::nat)"; |
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by (asm_full_simp_tac (numeral_ss addsimps [diff_less]) 1); |
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qed "diff_less'"; |
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Addsimps [inst "n" "number_of ?v" diff_less']; |
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(*various theorems that aren't in the default simpset*) |
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val add_is_one' = rename_numerals thy add_is_1; |
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val one_is_add' = rename_numerals thy one_is_add; |
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val zero_induct' = rename_numerals thy zero_induct; |
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val diff_self_eq_0' = rename_numerals thy diff_self_eq_0; |
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val mult_eq_self_implies_10' = rename_numerals thy mult_eq_self_implies_10; |
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val le_pred_eq' = rename_numerals thy le_pred_eq; |
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val less_pred_eq' = rename_numerals thy less_pred_eq; |
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||
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(** Divides **) |
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||
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Addsimps (map (rename_numerals thy) |
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[mod_1, mod_0, div_1, div_0, mod2_gr_0, mod2_add_self_eq_0, |
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mod2_add_self]); |
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||
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AddIffs (map (rename_numerals thy) |
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[dvd_1_left, dvd_0_right]); |
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||
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(*useful?*) |
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val mod_self' = rename_numerals thy mod_self; |
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val div_self' = rename_numerals thy div_self; |
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val div_less' = rename_numerals thy div_less; |
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val mod_mult_self_is_zero' = rename_numerals thy mod_mult_self_is_0; |
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||
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(** Power **) |
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||
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Goal "(p::nat) ^ #0 = #1"; |
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by (simp_tac numeral_ss 1); |
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qed "power_zero"; |
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Addsimps [power_zero]; |
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||
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val binomial_zero = rename_numerals thy binomial_0; |
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val binomial_Suc' = rename_numerals thy binomial_Suc; |
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val binomial_n_n' = rename_numerals thy binomial_n_n; |
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||
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(*binomial_0_Suc doesn't work well on numerals*) |
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Addsimps (map (rename_numerals thy) |
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[binomial_n_0, binomial_zero, binomial_1]); |
|
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