--- a/src/ZF/Integ/IntDiv.ML Thu Sep 07 15:31:09 2000 +0200
+++ b/src/ZF/Integ/IntDiv.ML Thu Sep 07 17:36:37 2000 +0200
@@ -150,9 +150,8 @@
by (res_inst_tac [("b", "intify(k)")] (not_zneg_mag RS subst) 1);
by (rtac lemma 3);
by Auto_tac;
-by (dtac znegative_imp_zless_0 1);
-by (dtac zless_zle_trans 2);
-by Auto_tac;
+by (asm_full_simp_tac (simpset() addsimps [znegative_iff_zless_0,
+ not_zless_iff_zle RS iff_sym]) 1);
qed "zmult_zle_mono1";
Goal "[| i $<= j; k $<= #0 |] ==> j$*k $<= i$*k";
@@ -204,8 +203,8 @@
by (res_inst_tac [("b", "intify(k)")] (not_zneg_mag RS subst) 1);
by (rtac lemma 3);
by Auto_tac;
-by (dtac znegative_imp_zless_0 1);
-by (dtac zless_trans 2 THEN assume_tac 2);
+by (asm_full_simp_tac (simpset() addsimps [znegative_iff_zless_0]) 1);
+by (dtac zless_trans 1 THEN assume_tac 1);
by (auto_tac (claset(), simpset() addsimps [zero_lt_zmagnitude]));
qed "zmult_zless_mono2";
@@ -373,897 +372,3 @@
qed "unique_remainder";
-(*** THE REST TO PORT LATER. The division algorithm can wait; most properties
- of division flow from the uniqueness properties proved above...
-
-
- (*** Correctness of posDivAlg, the division algorithm for a>=0 and b>0 ***)
-
-
- Goal "adjust a b <q,r> = (let diff = r$-b in \
- \ if #0 $<= diff then <#2$*q $+ #1,diff> \
- \ else <#2$*q,r>)";
- by (simp_tac (simpset() addsimps [Let_def,adjust_def]) 1);
- qed "adjust_eq";
- Addsimps [adjust_eq];
-
- (*Proving posDivAlg's termination condition*)
- val [tc] = posDivAlg.tcs;
- goalw_cterm [] (cterm_of (sign_of (the_context ())) (HOLogic.mk_Trueprop tc));
- by Auto_tac;
- val lemma = result();
-
- (* removing the termination condition from the generated theorems *)
-
- bind_thm ("posDivAlg_raw_eqn", lemma RS hd posDivAlg.simps);
-
- (**use with simproc to avoid re-proving the premise*)
- Goal "#0 $< b ==> \
- \ posDivAlg <a,b> = \
- \ (if a$<b then <#0,a> else adjust a b (posDivAlg<a,#2$*b>))";
- by (rtac (posDivAlg_raw_eqn RS trans) 1);
- by (Asm_simp_tac 1);
- qed "posDivAlg_eqn";
-
- bind_thm ("posDivAlg_induct", lemma RS posDivAlg.induct);
-
-
- (*Correctness of posDivAlg: it computes quotients correctly*)
- Goal "#0 $<= a --> #0 $< b --> quorem (<a,b>, posDivAlg <a,b>)";
- by (res_inst_tac [("u", "a"), ("v", "b")] posDivAlg_induct 1);
- by Auto_tac;
- by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [quorem_def])));
- (*base case: a<b*)
- by (asm_full_simp_tac (simpset() addsimps [posDivAlg_eqn]) 1);
- (*main argument*)
- by (stac posDivAlg_eqn 1);
- by (ALLGOALS Asm_simp_tac);
- by (etac splitE 1);
- by (auto_tac (claset(), simpset() addsimps [zadd_zmult_distrib2, Let_def]));
- qed_spec_mp "posDivAlg_correct";
-
-
- (*** Correctness of negDivAlg, the division algorithm for a<0 and b>0 ***)
-
- (*Proving negDivAlg's termination condition*)
- val [tc] = negDivAlg.tcs;
- goalw_cterm [] (cterm_of (sign_of (the_context ())) (HOLogic.mk_Trueprop tc));
- by Auto_tac;
- val lemma = result();
-
- (* removing the termination condition from the generated theorems *)
-
- bind_thm ("negDivAlg_raw_eqn", lemma RS hd negDivAlg.simps);
-
- (**use with simproc to avoid re-proving the premise*)
- Goal "#0 $< b ==> \
- \ negDivAlg <a,b> = \
- \ (if #0$<=a$+b then <#-1,a$+b> else adjust a b (negDivAlg<a,#2$*b>))";
- by (rtac (negDivAlg_raw_eqn RS trans) 1);
- by (Asm_simp_tac 1);
- qed "negDivAlg_eqn";
-
- bind_thm ("negDivAlg_induct", lemma RS negDivAlg.induct);
-
-
- (*Correctness of negDivAlg: it computes quotients correctly
- It doesn't work if a=0 because the 0/b=0 rather than -1*)
- Goal "a $< #0 --> #0 $< b --> quorem (<a,b>, negDivAlg <a,b>)";
- by (res_inst_tac [("u", "a"), ("v", "b")] negDivAlg_induct 1);
- by Auto_tac;
- by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [quorem_def])));
- (*base case: 0$<=a$+b*)
- by (asm_full_simp_tac (simpset() addsimps [negDivAlg_eqn]) 1);
- (*main argument*)
- by (stac negDivAlg_eqn 1);
- by (ALLGOALS Asm_simp_tac);
- by (etac splitE 1);
- by (auto_tac (claset(), simpset() addsimps [zadd_zmult_distrib2, Let_def]));
- qed_spec_mp "negDivAlg_correct";
-
-
- (*** Existence shown by proving the division algorithm to be correct ***)
-
- (*the case a=0*)
- Goal "b ~= #0 ==> quorem (<#0,b>, <#0,#0>)";
- by (auto_tac (claset(),
- simpset() addsimps [quorem_def, neq_iff_zless]));
- qed "quorem_0";
-
- Goal "posDivAlg <#0,b> = <#0,#0>";
- by (stac posDivAlg_raw_eqn 1);
- by Auto_tac;
- qed "posDivAlg_0";
- Addsimps [posDivAlg_0];
-
- Goal "negDivAlg <#-1,b> = <#-1,b-#1>";
- by (stac negDivAlg_raw_eqn 1);
- by Auto_tac;
- qed "negDivAlg_minus1";
- Addsimps [negDivAlg_minus1];
-
- Goalw [negateSnd_def] "negateSnd<q,r> = <q,-r>";
- by Auto_tac;
- qed "negateSnd_eq";
- Addsimps [negateSnd_eq];
-
- Goal "quorem (<-a,-b>, qr) ==> quorem (<a,b>, negateSnd qr)";
- by (auto_tac (claset(), simpset() addsimps split_ifs@[quorem_def]));
- qed "quorem_neg";
-
- Goal "b ~= #0 ==> quorem (<a,b>, divAlg<a,b>)";
- by (auto_tac (claset(),
- simpset() addsimps [quorem_0, divAlg_def]));
- by (REPEAT_FIRST (resolve_tac [quorem_neg, posDivAlg_correct,
- negDivAlg_correct]));
- by (auto_tac (claset(),
- simpset() addsimps [quorem_def, neq_iff_zless]));
- qed "divAlg_correct";
-
- (** Aribtrary definitions for division by zero. Useful to simplify
- certain equations **)
-
- Goal "a div (#0::int) = #0";
- by (simp_tac (simpset() addsimps [div_def, divAlg_def, posDivAlg_raw_eqn]) 1);
- qed "DIVISION_BY_ZERO_ZDIV"; (*NOT for adding to default simpset*)
-
- Goal "a mod (#0::int) = a";
- by (simp_tac (simpset() addsimps [mod_def, divAlg_def, posDivAlg_raw_eqn]) 1);
- qed "DIVISION_BY_ZERO_ZMOD"; (*NOT for adding to default simpset*)
-
- fun zdiv_undefined_case_tac s i =
- case_tac s i THEN
- asm_simp_tac (simpset() addsimps [DIVISION_BY_ZERO_ZDIV,
- DIVISION_BY_ZERO_ZMOD]) i;
-
- (** Basic laws about division and remainder **)
-
- Goal "a = b $* (a div b) $+ (a mod b)";
- by (zdiv_undefined_case_tac "b = #0" 1);
- by (cut_inst_tac [("a","a"),("b","b")] divAlg_correct 1);
- by (auto_tac (claset(),
- simpset() addsimps [quorem_def, div_def, mod_def]));
- qed "zmod_zdiv_equality";
-
- Goal "(#0::int) $< b ==> #0 $<= a mod b & a mod b $< b";
- by (cut_inst_tac [("a","a"),("b","b")] divAlg_correct 1);
- by (auto_tac (claset(),
- simpset() addsimps [quorem_def, mod_def]));
- bind_thm ("pos_mod_sign", result() RS conjunct1);
- bind_thm ("pos_mod_bound", result() RS conjunct2);
-
- Goal "b $< (#0::int) ==> a mod b $<= #0 & b $< a mod b";
- by (cut_inst_tac [("a","a"),("b","b")] divAlg_correct 1);
- by (auto_tac (claset(),
- simpset() addsimps [quorem_def, div_def, mod_def]));
- bind_thm ("neg_mod_sign", result() RS conjunct1);
- bind_thm ("neg_mod_bound", result() RS conjunct2);
-
-
- (** proving general properties of div and mod **)
-
- Goal "b ~= #0 ==> quorem (<a,b>, <a div b,a mod b>)";
- by (cut_inst_tac [("a","a"),("b","b")] zmod_zdiv_equality 1);
- by (auto_tac
- (claset(),
- simpset() addsimps [quorem_def, neq_iff_zless,
- pos_mod_sign,pos_mod_bound,
- neg_mod_sign, neg_mod_bound]));
- qed "quorem_div_mod";
-
- Goal "[| quorem(<a,b>,<q,r>); b ~= #0 |] ==> a div b = q";
- by (asm_simp_tac (simpset() addsimps [quorem_div_mod RS unique_quotient]) 1);
- qed "quorem_div";
-
- Goal "[| quorem(<a,b>,<q,r>); b ~= #0 |] ==> a mod b = r";
- by (asm_simp_tac (simpset() addsimps [quorem_div_mod RS unique_remainder]) 1);
- qed "quorem_mod";
-
- Goal "[| (#0::int) $<= a; a $< b |] ==> a div b = #0";
- by (rtac quorem_div 1);
- by (auto_tac (claset(), simpset() addsimps [quorem_def]));
- qed "div_pos_pos_trivial";
-
- Goal "[| a $<= (#0::int); b $< a |] ==> a div b = #0";
- by (rtac quorem_div 1);
- by (auto_tac (claset(), simpset() addsimps [quorem_def]));
- qed "div_neg_neg_trivial";
-
- Goal "[| (#0::int) $< a; a$+b $<= #0 |] ==> a div b = #-1";
- by (rtac quorem_div 1);
- by (auto_tac (claset(), simpset() addsimps [quorem_def]));
- qed "div_pos_neg_trivial";
-
- (*There is no div_neg_pos_trivial because #0 div b = #0 would supersede it*)
-
- Goal "[| (#0::int) $<= a; a $< b |] ==> a mod b = a";
- by (res_inst_tac [("q","#0")] quorem_mod 1);
- by (auto_tac (claset(), simpset() addsimps [quorem_def]));
- qed "mod_pos_pos_trivial";
-
- Goal "[| a $<= (#0::int); b $< a |] ==> a mod b = a";
- by (res_inst_tac [("q","#0")] quorem_mod 1);
- by (auto_tac (claset(), simpset() addsimps [quorem_def]));
- qed "mod_neg_neg_trivial";
-
- Goal "[| (#0::int) $< a; a$+b $<= #0 |] ==> a mod b = a$+b";
- by (res_inst_tac [("q","#-1")] quorem_mod 1);
- by (auto_tac (claset(), simpset() addsimps [quorem_def]));
- qed "mod_pos_neg_trivial";
-
- (*There is no mod_neg_pos_trivial...*)
-
-
- (*Simpler laws such as -a div b = -(a div b) FAIL*)
- Goal "(-a) div (-b) = a div b";
- by (zdiv_undefined_case_tac "b = #0" 1);
- by (stac ((simplify(simpset()) (quorem_div_mod RS quorem_neg))
- RS quorem_div) 1);
- by Auto_tac;
- qed "zdiv_zminus_zminus";
- Addsimps [zdiv_zminus_zminus];
-
- (*Simpler laws such as -a mod b = -(a mod b) FAIL*)
- Goal "(-a) mod (-b) = - (a mod b)";
- by (zdiv_undefined_case_tac "b = #0" 1);
- by (stac ((simplify(simpset()) (quorem_div_mod RS quorem_neg))
- RS quorem_mod) 1);
- by Auto_tac;
- qed "zmod_zminus_zminus";
- Addsimps [zmod_zminus_zminus];
-
-
- (*** division of a number by itself ***)
-
- Goal "[| (#0::int) $< a; a = r $+ a$*q; r $< a |] ==> #1 $<= q";
- by (subgoal_tac "#0 $< a$*q" 1);
- by (arith_tac 2);
- by (asm_full_simp_tac (simpset() addsimps [int_0_less_mult_iff]) 1);
- val lemma1 = result();
-
- Goal "[| (#0::int) $< a; a = r $+ a$*q; #0 $<= r |] ==> q $<= #1";
- by (subgoal_tac "#0 $<= a$*(#1$-q)" 1);
- by (asm_simp_tac (simpset() addsimps [zdiff_zmult_distrib2]) 2);
- by (asm_full_simp_tac (simpset() addsimps [int_0_le_mult_iff]) 1);
- val lemma2 = result();
-
- Goal "[| quorem(<a,a>,<q,r>); a ~= (#0::int) |] ==> q = #1";
- by (asm_full_simp_tac
- (simpset() addsimps split_ifs@[quorem_def, neq_iff_zless]) 1);
- by (rtac order_antisym 1);
- by Safe_tac;
- by Auto_tac;
- by (res_inst_tac [("a", "-a"),("r", "-r")] lemma1 3);
- by (res_inst_tac [("a", "-a"),("r", "-r")] lemma2 1);
- by (REPEAT (force_tac (claset() addIs [lemma1,lemma2],
- simpset() addsimps [zadd_commute, zmult_zminus]) 1));
- qed "self_quotient";
-
- Goal "[| quorem(<a,a>,<q,r>); a ~= (#0::int) |] ==> r = #0";
- by (ftac self_quotient 1);
- by (assume_tac 1);
- by (asm_full_simp_tac (simpset() addsimps [quorem_def]) 1);
- qed "self_remainder";
-
- Goal "a ~= #0 ==> a div a = (#1::int)";
- by (asm_simp_tac (simpset() addsimps [quorem_div_mod RS self_quotient]) 1);
- qed "zdiv_self";
- Addsimps [zdiv_self];
-
- (*Here we have 0 mod 0 = 0, also assumed by Knuth (who puts m mod 0 = 0) *)
- Goal "a mod a = (#0::int)";
- by (zdiv_undefined_case_tac "a = #0" 1);
- by (asm_simp_tac (simpset() addsimps [quorem_div_mod RS self_remainder]) 1);
- qed "zmod_self";
- Addsimps [zmod_self];
-
-
- (*** Computation of division and remainder ***)
-
- Goal "(#0::int) div b = #0";
- by (simp_tac (simpset() addsimps [div_def, divAlg_def]) 1);
- qed "zdiv_zero";
-
- Goal "(#0::int) $< b ==> #-1 div b = #-1";
- by (asm_simp_tac (simpset() addsimps [div_def, divAlg_def]) 1);
- qed "div_eq_minus1";
-
- Goal "(#0::int) mod b = #0";
- by (simp_tac (simpset() addsimps [mod_def, divAlg_def]) 1);
- qed "zmod_zero";
-
- Addsimps [zdiv_zero, zmod_zero];
-
- Goal "(#0::int) $< b ==> #-1 div b = #-1";
- by (asm_simp_tac (simpset() addsimps [div_def, divAlg_def]) 1);
- qed "zdiv_minus1";
-
- Goal "(#0::int) $< b ==> #-1 mod b = b-#1";
- by (asm_simp_tac (simpset() addsimps [mod_def, divAlg_def]) 1);
- qed "zmod_minus1";
-
- (** a positive, b positive **)
-
- Goal "[| #0 $< a; #0 $<= b |] ==> a div b = fst (posDivAlg<a,b>)";
- by (asm_simp_tac (simpset() addsimps [div_def, divAlg_def]) 1);
- qed "div_pos_pos";
-
- Goal "[| #0 $< a; #0 $<= b |] ==> a mod b = snd (posDivAlg<a,b>)";
- by (asm_simp_tac (simpset() addsimps [mod_def, divAlg_def]) 1);
- qed "mod_pos_pos";
-
- (** a negative, b positive **)
-
- Goal "[| a $< #0; #0 $< b |] ==> a div b = fst (negDivAlg<a,b>)";
- by (asm_simp_tac (simpset() addsimps [div_def, divAlg_def]) 1);
- qed "div_neg_pos";
-
- Goal "[| a $< #0; #0 $< b |] ==> a mod b = snd (negDivAlg<a,b>)";
- by (asm_simp_tac (simpset() addsimps [mod_def, divAlg_def]) 1);
- qed "mod_neg_pos";
-
- (** a positive, b negative **)
-
- Goal "[| #0 $< a; b $< #0 |] ==> a div b = fst (negateSnd(negDivAlg<-a,-b>))";
- by (asm_simp_tac (simpset() addsimps [div_def, divAlg_def]) 1);
- qed "div_pos_neg";
-
- Goal "[| #0 $< a; b $< #0 |] ==> a mod b = snd (negateSnd(negDivAlg<-a,-b>))";
- by (asm_simp_tac (simpset() addsimps [mod_def, divAlg_def]) 1);
- qed "mod_pos_neg";
-
- (** a negative, b negative **)
-
- Goal "[| a $< #0; b $<= #0 |] ==> a div b = fst (negateSnd(posDivAlg<-a,-b>))";
- by (asm_simp_tac (simpset() addsimps [div_def, divAlg_def]) 1);
- qed "div_neg_neg";
-
- Goal "[| a $< #0; b $<= #0 |] ==> a mod b = snd (negateSnd(posDivAlg<-a,-b>))";
- by (asm_simp_tac (simpset() addsimps [mod_def, divAlg_def]) 1);
- qed "mod_neg_neg";
-
- Addsimps (map (read_instantiate_sg (sign_of (the_context ()))
- [("a", "number_of ?v"), ("b", "number_of ?w")])
- [div_pos_pos, div_neg_pos, div_pos_neg, div_neg_neg,
- mod_pos_pos, mod_neg_pos, mod_pos_neg, mod_neg_neg,
- posDivAlg_eqn, negDivAlg_eqn]);
-
-
- (** Special-case simplification **)
-
- Goal "a mod (#1::int) = #0";
- by (cut_inst_tac [("a","a"),("b","#1")] pos_mod_sign 1);
- by (cut_inst_tac [("a","a"),("b","#1")] pos_mod_bound 2);
- by Auto_tac;
- qed "zmod_1";
- Addsimps [zmod_1];
-
- Goal "a div (#1::int) = a";
- by (cut_inst_tac [("a","a"),("b","#1")] zmod_zdiv_equality 1);
- by Auto_tac;
- qed "zdiv_1";
- Addsimps [zdiv_1];
-
- Goal "a mod (#-1::int) = #0";
- by (cut_inst_tac [("a","a"),("b","#-1")] neg_mod_sign 1);
- by (cut_inst_tac [("a","a"),("b","#-1")] neg_mod_bound 2);
- by Auto_tac;
- qed "zmod_minus1_right";
- Addsimps [zmod_minus1_right];
-
- Goal "a div (#-1::int) = -a";
- by (cut_inst_tac [("a","a"),("b","#-1")] zmod_zdiv_equality 1);
- by Auto_tac;
- qed "zdiv_minus1_right";
- Addsimps [zdiv_minus1_right];
-
-
- (*** Monotonicity in the first argument (divisor) ***)
-
- Goal "[| a $<= a'; #0 $< b |] ==> a div b $<= a' div b";
- by (cut_inst_tac [("a","a"),("b","b")] zmod_zdiv_equality 1);
- by (cut_inst_tac [("a","a'"),("b","b")] zmod_zdiv_equality 1);
- by (rtac unique_quotient_lemma 1);
- by (etac subst 1);
- by (etac subst 1);
- by (ALLGOALS (asm_simp_tac (simpset() addsimps [pos_mod_sign,pos_mod_bound])));
- qed "zdiv_mono1";
-
- Goal "[| a $<= a'; b $< #0 |] ==> a' div b $<= a div b";
- by (cut_inst_tac [("a","a"),("b","b")] zmod_zdiv_equality 1);
- by (cut_inst_tac [("a","a'"),("b","b")] zmod_zdiv_equality 1);
- by (rtac unique_quotient_lemma_neg 1);
- by (etac subst 1);
- by (etac subst 1);
- by (ALLGOALS (asm_simp_tac (simpset() addsimps [neg_mod_sign,neg_mod_bound])));
- qed "zdiv_mono1_neg";
-
-
- (*** Monotonicity in the second argument (dividend) ***)
-
- Goal "[| b$*q $+ r = b'$*q' $+ r'; #0 $<= b'$*q' $+ r'; \
- \ r' $< b'; #0 $<= r; #0 $< b'; b' $<= b |] \
- \ ==> q $<= q'";
- by (subgoal_tac "#0 $<= q'" 1);
- by (subgoal_tac "#0 $< b'$*(q' $+ #1)" 2);
- by (asm_simp_tac (simpset() addsimps [zadd_zmult_distrib2]) 3);
- by (asm_full_simp_tac (simpset() addsimps [int_0_less_mult_iff]) 2);
- by (subgoal_tac "b$*q $< b$*(q' $+ #1)" 1);
- by (Asm_full_simp_tac 1);
- by (subgoal_tac "b$*q = r' $- r $+ b'$*q'" 1);
- by (Simp_tac 2);
- by (asm_simp_tac (simpset() addsimps [zadd_zmult_distrib2]) 1);
- by (stac zadd_commute 1 THEN rtac zadd_zless_mono 1 THEN arith_tac 1);
- by (rtac zmult_zle_mono1 1);
- by Auto_tac;
- qed "zdiv_mono2_lemma";
-
- Goal "[| (#0::int) $<= a; #0 $< b'; b' $<= b |] \
- \ ==> a div b $<= a div b'";
- by (subgoal_tac "b ~= #0" 1);
- by (arith_tac 2);
- by (cut_inst_tac [("a","a"),("b","b")] zmod_zdiv_equality 1);
- by (cut_inst_tac [("a","a"),("b","b'")] zmod_zdiv_equality 1);
- by (rtac zdiv_mono2_lemma 1);
- by (etac subst 1);
- by (etac subst 1);
- by (ALLGOALS (asm_simp_tac (simpset() addsimps [pos_mod_sign,pos_mod_bound])));
- qed "zdiv_mono2";
-
- Goal "[| b$*q $+ r = b'$*q' $+ r'; b'$*q' $+ r' $< #0; \
- \ r $< b; #0 $<= r'; #0 $< b'; b' $<= b |] \
- \ ==> q' $<= q";
- by (subgoal_tac "q' $< #0" 1);
- by (subgoal_tac "b'$*q' $< #0" 2);
- by (arith_tac 3);
- by (asm_full_simp_tac (simpset() addsimps [zmult_less_0_iff]) 2);
- by (subgoal_tac "b$*q' $< b$*(q $+ #1)" 1);
- by (Asm_full_simp_tac 1);
- by (asm_simp_tac (simpset() addsimps [zadd_zmult_distrib2]) 1);
- by (subgoal_tac "b$*q' $<= b'$*q'" 1);
- by (asm_simp_tac (simpset() addsimps [zmult_zle_mono1_neg]) 2);
- by (subgoal_tac "b'$*q' $< b $+ b$*q" 1);
- by (Asm_simp_tac 2);
- by (arith_tac 1);
- qed "zdiv_mono2_neg_lemma";
-
- Goal "[| a $< (#0::int); #0 $< b'; b' $<= b |] \
- \ ==> a div b' $<= a div b";
- by (cut_inst_tac [("a","a"),("b","b")] zmod_zdiv_equality 1);
- by (cut_inst_tac [("a","a"),("b","b'")] zmod_zdiv_equality 1);
- by (rtac zdiv_mono2_neg_lemma 1);
- by (etac subst 1);
- by (etac subst 1);
- by (ALLGOALS (asm_simp_tac (simpset() addsimps [pos_mod_sign,pos_mod_bound])));
- qed "zdiv_mono2_neg";
-
-
- (*** More algebraic laws for div and mod ***)
-
- (** proving (a*b) div c = a $* (b div c) $+ a * (b mod c) **)
-
- Goal "[| quorem(<b,c>,<q,r>); c ~= #0 |] \
- \ ==> quorem (<a$*b,c>, <a$*q $+ a$*r div c,a$*r mod c>)";
- by (auto_tac
- (claset(),
- simpset() addsimps split_ifs@
- [quorem_def, neq_iff_zless,
- zadd_zmult_distrib2,
- pos_mod_sign,pos_mod_bound,
- neg_mod_sign, neg_mod_bound]));
- by (ALLGOALS(rtac zmod_zdiv_equality));
- val lemma = result();
-
- Goal "(a$*b) div c = a$*(b div c) $+ a$*(b mod c) div c";
- by (zdiv_undefined_case_tac "c = #0" 1);
- by (blast_tac (claset() addIs [quorem_div_mod RS lemma RS quorem_div]) 1);
- qed "zdiv_zmult1_eq";
-
- Goal "(a$*b) mod c = a$*(b mod c) mod c";
- by (zdiv_undefined_case_tac "c = #0" 1);
- by (blast_tac (claset() addIs [quorem_div_mod RS lemma RS quorem_mod]) 1);
- qed "zmod_zmult1_eq";
-
- Goal "(a$*b) mod c = ((a mod c) $* b) mod c";
- by (rtac trans 1);
- by (res_inst_tac [("s","b$*a mod c")] trans 1);
- by (rtac zmod_zmult1_eq 2);
- by (ALLGOALS (simp_tac (simpset() addsimps [zmult_commute])));
- qed "zmod_zmult1_eq'";
-
- Goal "(a$*b) mod c = ((a mod c) $* (b mod c)) mod c";
- by (rtac (zmod_zmult1_eq' RS trans) 1);
- by (rtac zmod_zmult1_eq 1);
- qed "zmod_zmult_distrib";
-
- Goal "b ~= (#0::int) ==> (a$*b) div b = a";
- by (asm_simp_tac (simpset() addsimps [zdiv_zmult1_eq]) 1);
- qed "zdiv_zmult_self1";
-
- Goal "b ~= (#0::int) ==> (b$*a) div b = a";
- by (stac zmult_commute 1 THEN etac zdiv_zmult_self1 1);
- qed "zdiv_zmult_self2";
-
- Addsimps [zdiv_zmult_self1, zdiv_zmult_self2];
-
- Goal "(a$*b) mod b = (#0::int)";
- by (simp_tac (simpset() addsimps [zmod_zmult1_eq]) 1);
- qed "zmod_zmult_self1";
-
- Goal "(b$*a) mod b = (#0::int)";
- by (simp_tac (simpset() addsimps [zmult_commute, zmod_zmult1_eq]) 1);
- qed "zmod_zmult_self2";
-
- Addsimps [zmod_zmult_self1, zmod_zmult_self2];
-
-
- (** proving (a$+b) div c = a div c $+ b div c $+ ((a mod c $+ b mod c) div c) **)
-
- Goal "[| quorem(<a,c>,<aq,ar>); quorem(<b,c>,<bq,br>); c ~= #0 |] \
- \ ==> quorem (<a$+b,c>, (aq $+ bq $+ (ar$+br) div c, (ar$+br) mod c))";
- by (auto_tac
- (claset(),
- simpset() addsimps split_ifs@
- [quorem_def, neq_iff_zless,
- zadd_zmult_distrib2,
- pos_mod_sign,pos_mod_bound,
- neg_mod_sign, neg_mod_bound]));
- by (ALLGOALS(rtac zmod_zdiv_equality));
- val lemma = result();
-
- (*NOT suitable for rewriting: the RHS has an instance of the LHS*)
- Goal "(a$+b) div c = a div c $+ b div c $+ ((a mod c $+ b mod c) div c)";
- by (zdiv_undefined_case_tac "c = #0" 1);
- by (blast_tac (claset() addIs [[quorem_div_mod,quorem_div_mod]
- MRS lemma RS quorem_div]) 1);
- qed "zdiv_zadd1_eq";
-
- Goal "(a$+b) mod c = (a mod c $+ b mod c) mod c";
- by (zdiv_undefined_case_tac "c = #0" 1);
- by (blast_tac (claset() addIs [[quorem_div_mod,quorem_div_mod]
- MRS lemma RS quorem_mod]) 1);
- qed "zmod_zadd1_eq";
-
- Goal "(a mod b) div b = (#0::int)";
- by (zdiv_undefined_case_tac "b = #0" 1);
- by (auto_tac (claset(),
- simpset() addsimps [neq_iff_zless,
- pos_mod_sign, pos_mod_bound, div_pos_pos_trivial,
- neg_mod_sign, neg_mod_bound, div_neg_neg_trivial]));
- qed "mod_div_trivial";
- Addsimps [mod_div_trivial];
-
- Goal "(a mod b) mod b = a mod b";
- by (zdiv_undefined_case_tac "b = #0" 1);
- by (auto_tac (claset(),
- simpset() addsimps [neq_iff_zless,
- pos_mod_sign, pos_mod_bound, mod_pos_pos_trivial,
- neg_mod_sign, neg_mod_bound, mod_neg_neg_trivial]));
- qed "mod_mod_trivial";
- Addsimps [mod_mod_trivial];
-
- Goal "(a$+b) mod c = ((a mod c) $+ b) mod c";
- by (rtac (trans RS sym) 1);
- by (rtac zmod_zadd1_eq 1);
- by (Simp_tac 1);
- by (rtac (zmod_zadd1_eq RS sym) 1);
- qed "zmod_zadd_left_eq";
-
- Goal "(a$+b) mod c = (a $+ (b mod c)) mod c";
- by (rtac (trans RS sym) 1);
- by (rtac zmod_zadd1_eq 1);
- by (Simp_tac 1);
- by (rtac (zmod_zadd1_eq RS sym) 1);
- qed "zmod_zadd_right_eq";
-
-
- Goal "a ~= (#0::int) ==> (a$+b) div a = b div a $+ #1";
- by (asm_simp_tac (simpset() addsimps [zdiv_zadd1_eq]) 1);
- qed "zdiv_zadd_self1";
-
- Goal "a ~= (#0::int) ==> (b$+a) div a = b div a $+ #1";
- by (asm_simp_tac (simpset() addsimps [zdiv_zadd1_eq]) 1);
- qed "zdiv_zadd_self2";
- Addsimps [zdiv_zadd_self1, zdiv_zadd_self2];
-
- Goal "(a$+b) mod a = b mod a";
- by (zdiv_undefined_case_tac "a = #0" 1);
- by (asm_simp_tac (simpset() addsimps [zmod_zadd1_eq]) 1);
- qed "zmod_zadd_self1";
-
- Goal "(b$+a) mod a = b mod a";
- by (zdiv_undefined_case_tac "a = #0" 1);
- by (asm_simp_tac (simpset() addsimps [zmod_zadd1_eq]) 1);
- qed "zmod_zadd_self2";
- Addsimps [zmod_zadd_self1, zmod_zadd_self2];
-
-
- (*** proving a div (b*c) = (a div b) div c ***)
-
- (*The condition c>0 seems necessary. Consider that 7 div ~6 = ~2 but
- 7 div 2 div ~3 = 3 div ~3 = ~1. The subcase (a div b) mod c = 0 seems
- to cause particular problems.*)
-
- (** first, four lemmas to bound the remainder for the cases b<0 and b>0 **)
-
- Goal "[| (#0::int) $< c; b $< r; r $<= #0 |] ==> b$*c $< b$*(q mod c) $+ r";
- by (subgoal_tac "b $* (c $- q mod c) $< r $* #1" 1);
- by (asm_full_simp_tac (simpset() addsimps [zdiff_zmult_distrib2]) 1);
- by (rtac order_le_less_trans 1);
- by (etac zmult_zless_mono1 2);
- by (rtac zmult_zle_mono2_neg 1);
- by (auto_tac
- (claset(),
- simpset() addsimps zcompare_rls@
- [zadd_commute, add1_zle_eq, pos_mod_bound]));
- val lemma1 = result();
-
- Goal "[| (#0::int) $< c; b $< r; r $<= #0 |] ==> b $* (q mod c) $+ r $<= #0";
- by (subgoal_tac "b $* (q mod c) $<= #0" 1);
- by (arith_tac 1);
- by (asm_simp_tac (simpset() addsimps [zmult_le_0_iff, pos_mod_sign]) 1);
- val lemma2 = result();
-
- Goal "[| (#0::int) $< c; #0 $<= r; r $< b |] ==> #0 $<= b $* (q mod c) $+ r";
- by (subgoal_tac "#0 $<= b $* (q mod c)" 1);
- by (arith_tac 1);
- by (asm_simp_tac (simpset() addsimps [int_0_le_mult_iff, pos_mod_sign]) 1);
- val lemma3 = result();
-
- Goal "[| (#0::int) $< c; #0 $<= r; r $< b |] ==> b $* (q mod c) $+ r $< b $* c";
- by (subgoal_tac "r $* #1 $< b $* (c $- q mod c)" 1);
- by (asm_full_simp_tac (simpset() addsimps [zdiff_zmult_distrib2]) 1);
- by (rtac order_less_le_trans 1);
- by (etac zmult_zless_mono1 1);
- by (rtac zmult_zle_mono2 2);
- by (auto_tac
- (claset(),
- simpset() addsimps zcompare_rls@
- [zadd_commute, add1_zle_eq, pos_mod_bound]));
- val lemma4 = result();
-
- Goal "[| quorem (<a,b>, <q,r>); b ~= #0; #0 $< c |] \
- \ ==> quorem (<a,b$*c>, (q div c, b$*(q mod c) $+ r))";
- by (auto_tac
- (claset(),
- simpset() addsimps zmult_ac@
- [zmod_zdiv_equality, quorem_def, neq_iff_zless,
- int_0_less_mult_iff,
- zadd_zmult_distrib2 RS sym,
- lemma1, lemma2, lemma3, lemma4]));
- val lemma = result();
-
- Goal "(#0::int) $< c ==> a div (b$*c) = (a div b) div c";
- by (zdiv_undefined_case_tac "b = #0" 1);
- by (force_tac (claset(),
- simpset() addsimps [quorem_div_mod RS lemma RS quorem_div,
- zmult_eq_0_iff]) 1);
- qed "zdiv_zmult2_eq";
-
- Goal "(#0::int) $< c ==> a mod (b$*c) = b$*(a div b mod c) $+ a mod b";
- by (zdiv_undefined_case_tac "b = #0" 1);
- by (force_tac (claset(),
- simpset() addsimps [quorem_div_mod RS lemma RS quorem_mod,
- zmult_eq_0_iff]) 1);
- qed "zmod_zmult2_eq";
-
-
- (*** Cancellation of common factors in "div" ***)
-
- Goal "[| (#0::int) $< b; c ~= #0 |] ==> (c$*a) div (c$*b) = a div b";
- by (stac zdiv_zmult2_eq 1);
- by Auto_tac;
- val lemma1 = result();
-
- Goal "[| b $< (#0::int); c ~= #0 |] ==> (c$*a) div (c$*b) = a div b";
- by (subgoal_tac "(c $* (-a)) div (c $* (-b)) = (-a) div (-b)" 1);
- by (rtac lemma1 2);
- by Auto_tac;
- val lemma2 = result();
-
- Goal "c ~= (#0::int) ==> (c$*a) div (c$*b) = a div b";
- by (zdiv_undefined_case_tac "b = #0" 1);
- by (auto_tac
- (claset(),
- simpset() addsimps [read_instantiate [("x", "b")] neq_iff_zless,
- lemma1, lemma2]));
- qed "zdiv_zmult_zmult1";
-
- Goal "c ~= (#0::int) ==> (a$*c) div (b$*c) = a div b";
- by (dtac zdiv_zmult_zmult1 1);
- by (auto_tac (claset(), simpset() addsimps [zmult_commute]));
- qed "zdiv_zmult_zmult2";
-
-
-
- (*** Distribution of factors over "mod" ***)
-
- Goal "[| (#0::int) $< b; c ~= #0 |] ==> (c$*a) mod (c$*b) = c $* (a mod b)";
- by (stac zmod_zmult2_eq 1);
- by Auto_tac;
- val lemma1 = result();
-
- Goal "[| b $< (#0::int); c ~= #0 |] ==> (c$*a) mod (c$*b) = c $* (a mod b)";
- by (subgoal_tac "(c $* (-a)) mod (c $* (-b)) = c $* ((-a) mod (-b))" 1);
- by (rtac lemma1 2);
- by Auto_tac;
- val lemma2 = result();
-
- Goal "(c$*a) mod (c$*b) = c $* (a mod b)";
- by (zdiv_undefined_case_tac "b = #0" 1);
- by (zdiv_undefined_case_tac "c = #0" 1);
- by (auto_tac
- (claset(),
- simpset() addsimps [read_instantiate [("x", "b")] neq_iff_zless,
- lemma1, lemma2]));
- qed "zmod_zmult_zmult1";
-
- Goal "(a$*c) mod (b$*c) = (a mod b) $* c";
- by (cut_inst_tac [("c","c")] zmod_zmult_zmult1 1);
- by (auto_tac (claset(), simpset() addsimps [zmult_commute]));
- qed "zmod_zmult_zmult2";
-
-
- (*** Speeding up the division algorithm with shifting ***)
-
- (** computing "div" by shifting **)
-
- Goal "(#0::int) $<= a ==> (#1 $+ #2$*b) div (#2$*a) = b div a";
- by (zdiv_undefined_case_tac "a = #0" 1);
- by (subgoal_tac "#1 $<= a" 1);
- by (arith_tac 2);
- by (subgoal_tac "#1 $< a $* #2" 1);
- by (arith_tac 2);
- by (subgoal_tac "#2$*(#1 $+ b mod a) $<= #2$*a" 1);
- by (rtac zmult_zle_mono2 2);
- by (auto_tac (claset(),
- simpset() addsimps [zadd_commute, zmult_commute,
- add1_zle_eq, pos_mod_bound]));
- by (stac zdiv_zadd1_eq 1);
- by (asm_simp_tac (simpset() addsimps [zdiv_zmult_zmult2, zmod_zmult_zmult2,
- div_pos_pos_trivial]) 1);
- by (stac div_pos_pos_trivial 1);
- by (asm_simp_tac (simpset()
- addsimps [mod_pos_pos_trivial,
- pos_mod_sign RS zadd_zle_mono1 RSN (2,order_trans)]) 1);
- by (auto_tac (claset(),
- simpset() addsimps [mod_pos_pos_trivial]));
- by (subgoal_tac "#0 $<= b mod a" 1);
- by (asm_simp_tac (simpset() addsimps [pos_mod_sign]) 2);
- by (arith_tac 1);
- qed "pos_zdiv_mult_2";
-
-
- Goal "a $<= (#0::int) ==> (#1 $+ #2$*b) div (#2$*a) = (b$+#1) div a";
- by (subgoal_tac "(#1 $+ #2$*(-b-#1)) div (#2 $* (-a)) = (-b-#1) div (-a)" 1);
- by (rtac pos_zdiv_mult_2 2);
- by (auto_tac (claset(),
- simpset() addsimps [zmult_zminus_right]));
- by (subgoal_tac "(#-1 - (#2 $* b)) = - (#1 $+ (#2 $* b))" 1);
- by (Simp_tac 2);
- by (asm_full_simp_tac (HOL_ss
- addsimps [zdiv_zminus_zminus, zdiff_def,
- zminus_zadd_distrib RS sym]) 1);
- qed "neg_zdiv_mult_2";
-
-
- (*Not clear why this must be proved separately; probably number_of causes
- simplification problems*)
- Goal "~ #0 $<= x ==> x $<= (#0::int)";
- by Auto_tac;
- val lemma = result();
-
- Goal "number_of (v BIT b) div number_of (w BIT False) = \
- \ (if ~b | (#0::int) $<= number_of w \
- \ then number_of v div (number_of w) \
- \ else (number_of v $+ (#1::int)) div (number_of w))";
- by (simp_tac (simpset_of Int.thy addsimps [zadd_assoc, number_of_BIT]) 1);
- by (asm_simp_tac (simpset()
- delsimps bin_arith_extra_simps@bin_rel_simps
- addsimps [zdiv_zmult_zmult1,
- pos_zdiv_mult_2, lemma, neg_zdiv_mult_2]) 1);
- qed "zdiv_number_of_BIT";
-
- Addsimps [zdiv_number_of_BIT];
-
-
- (** computing "mod" by shifting (proofs resemble those for "div") **)
-
- Goal "(#0::int) $<= a ==> (#1 $+ #2$*b) mod (#2$*a) = #1 $+ #2 $* (b mod a)";
- by (zdiv_undefined_case_tac "a = #0" 1);
- by (subgoal_tac "#1 $<= a" 1);
- by (arith_tac 2);
- by (subgoal_tac "#1 $< a $* #2" 1);
- by (arith_tac 2);
- by (subgoal_tac "#2$*(#1 $+ b mod a) $<= #2$*a" 1);
- by (rtac zmult_zle_mono2 2);
- by (auto_tac (claset(),
- simpset() addsimps [zadd_commute, zmult_commute,
- add1_zle_eq, pos_mod_bound]));
- by (stac zmod_zadd1_eq 1);
- by (asm_simp_tac (simpset() addsimps [zmod_zmult_zmult2,
- mod_pos_pos_trivial]) 1);
- by (rtac mod_pos_pos_trivial 1);
- by (asm_simp_tac (simpset()
- addsimps [mod_pos_pos_trivial,
- pos_mod_sign RS zadd_zle_mono1 RSN (2,order_trans)]) 1);
- by (auto_tac (claset(),
- simpset() addsimps [mod_pos_pos_trivial]));
- by (subgoal_tac "#0 $<= b mod a" 1);
- by (asm_simp_tac (simpset() addsimps [pos_mod_sign]) 2);
- by (arith_tac 1);
- qed "pos_zmod_mult_2";
-
-
- Goal "a $<= (#0::int) ==> (#1 $+ #2$*b) mod (#2$*a) = #2 $* ((b$+#1) mod a) - #1";
- by (subgoal_tac
- "(#1 $+ #2$*(-b-#1)) mod (#2$*(-a)) = #1 $+ #2$*((-b-#1) mod (-a))" 1);
- by (rtac pos_zmod_mult_2 2);
- by (auto_tac (claset(),
- simpset() addsimps [zmult_zminus_right]));
- by (subgoal_tac "(#-1 - (#2 $* b)) = - (#1 $+ (#2 $* b))" 1);
- by (Simp_tac 2);
- by (asm_full_simp_tac (HOL_ss
- addsimps [zmod_zminus_zminus, zdiff_def,
- zminus_zadd_distrib RS sym]) 1);
- by (dtac (zminus_equation RS iffD1 RS sym) 1);
- by Auto_tac;
- qed "neg_zmod_mult_2";
-
- Goal "number_of (v BIT b) mod number_of (w BIT False) = \
- \ (if b then \
- \ if (#0::int) $<= number_of w \
- \ then #2 $* (number_of v mod number_of w) $+ #1 \
- \ else #2 $* ((number_of v $+ (#1::int)) mod number_of w) - #1 \
- \ else #2 $* (number_of v mod number_of w))";
- by (simp_tac (simpset_of Int.thy addsimps [zadd_assoc, number_of_BIT]) 1);
- by (asm_simp_tac (simpset()
- delsimps bin_arith_extra_simps@bin_rel_simps
- addsimps [zmod_zmult_zmult1,
- pos_zmod_mult_2, lemma, neg_zmod_mult_2]) 1);
- qed "zmod_number_of_BIT";
-
- Addsimps [zmod_number_of_BIT];
-
-
- (** Quotients of signs **)
-
- Goal "[| a $< (#0::int); #0 $< b |] ==> a div b $< #0";
- by (subgoal_tac "a div b $<= #-1" 1);
- by (Force_tac 1);
- by (rtac order_trans 1);
- by (res_inst_tac [("a'","#-1")] zdiv_mono1 1);
- by (auto_tac (claset(), simpset() addsimps [zdiv_minus1]));
- qed "div_neg_pos_less0";
-
- Goal "[| (#0::int) $<= a; b $< #0 |] ==> a div b $<= #0";
- by (dtac zdiv_mono1_neg 1);
- by Auto_tac;
- qed "div_nonneg_neg_le0";
-
- Goal "(#0::int) $< b ==> (#0 $<= a div b) = (#0 $<= a)";
- by Auto_tac;
- by (dtac zdiv_mono1 2);
- by (auto_tac (claset(), simpset() addsimps [neq_iff_zless]));
- by (full_simp_tac (simpset() addsimps [not_zless_iff_zle RS sym]) 1);
- by (blast_tac (claset() addIs [div_neg_pos_less0]) 1);
- qed "pos_imp_zdiv_nonneg_iff";
-
- Goal "b $< (#0::int) ==> (#0 $<= a div b) = (a $<= (#0::int))";
- by (stac (zdiv_zminus_zminus RS sym) 1);
- by (stac pos_imp_zdiv_nonneg_iff 1);
- by Auto_tac;
- qed "neg_imp_zdiv_nonneg_iff";
-
- (*But not (a div b $<= 0 iff a$<=0); consider a=1, b=2 when a div b = 0.*)
- Goal "(#0::int) $< b ==> (a div b $< #0) = (a $< #0)";
- by (asm_simp_tac (simpset() addsimps [linorder_not_le RS sym,
- pos_imp_zdiv_nonneg_iff]) 1);
- qed "pos_imp_zdiv_neg_iff";
-
- (*Again the law fails for $<=: consider a = -1, b = -2 when a div b = 0*)
- Goal "b $< (#0::int) ==> (a div b $< #0) = (#0 $< a)";
- by (asm_simp_tac (simpset() addsimps [linorder_not_le RS sym,
- neg_imp_zdiv_nonneg_iff]) 1);
- qed "neg_imp_zdiv_neg_iff";
-*)