--- a/src/HOL/Real/RealAbs.ML Wed Jul 28 22:01:58 1999 +0200
+++ b/src/HOL/Real/RealAbs.ML Thu Jul 29 12:44:57 1999 +0200
@@ -31,8 +31,8 @@
by (simp_tac (simpset() addsimps [(prem RS real_less_imp_le),rabs_eqI1]) 1);
qed "rabs_eqI2";
-val [prem] = goalw thy [rabs_def,real_le_def] "x<0r ==> rabs x = -x";
-by (simp_tac (simpset() addsimps [prem,if_not_P]) 1);
+Goalw [rabs_def,real_le_def] "x<0r ==> rabs x = -x";
+by (Asm_simp_tac 1);
qed "rabs_minus_eqI2";
Goal "x<=0r ==> rabs x = -x";
@@ -71,39 +71,38 @@
(* case splits nightmare *)
Goalw [rabs_def] "rabs(x*y) = (rabs x)*(rabs y)";
-by (auto_tac (claset(),simpset() addsimps [real_minus_mult_eq1,
- real_minus_mult_commute,real_minus_mult_eq2]));
+by (auto_tac (claset(),
+ simpset() addsimps [real_minus_mult_eq1, real_minus_mult_commute,
+ real_minus_mult_eq2]));
by (blast_tac (claset() addDs [real_le_mult_order]) 1);
by (auto_tac (claset() addSDs [not_real_leE],simpset()));
by (EVERY1[dtac real_mult_le_zero, assume_tac, dtac real_le_anti_sym]);
by (EVERY[dtac real_mult_le_zero 3, assume_tac 3, dtac real_le_anti_sym 3]);
by (dtac real_mult_less_zero1 5 THEN assume_tac 5);
by (auto_tac (claset() addDs [real_less_asym,sym],
- simpset() addsimps [real_minus_mult_eq2 RS sym] @real_mult_ac));
+ simpset() addsimps [real_minus_mult_eq2 RS sym] @real_mult_ac));
qed "rabs_mult";
Goalw [rabs_def] "x~= 0r ==> rabs(rinv(x)) = rinv(rabs(x))";
-by (auto_tac (claset(),simpset() addsimps [real_minus_rinv]
- ));
+by (auto_tac (claset(), simpset() addsimps [real_minus_rinv]));
by (ALLGOALS(dtac not_real_leE));
by (etac real_less_asym 1);
-by (blast_tac (claset() addDs [real_le_imp_less_or_eq,
- real_rinv_gt_zero]) 1);
+by (blast_tac (claset() addDs [real_le_imp_less_or_eq, real_rinv_gt_zero]) 1);
by (dtac (rinv_not_zero RS not_sym) 1);
by (rtac (real_rinv_less_zero RSN (2,real_less_asym)) 1);
by (assume_tac 2);
by (blast_tac (claset() addSDs [real_le_imp_less_or_eq]) 1);
qed "rabs_rinv";
-val [prem] = goal thy "y ~= 0r ==> rabs(x*rinv(y)) = rabs(x)*rinv(rabs(y))";
+Goal "y ~= 0r ==> rabs(x*rinv(y)) = rabs(x)*rinv(rabs(y))";
by (res_inst_tac [("c1","rabs y")] (real_mult_left_cancel RS subst) 1);
-by (simp_tac (simpset() addsimps [(rabs_not_zero_iff RS sym), prem]) 1);
-by (simp_tac (simpset() addsimps [(rabs_mult RS sym) ,real_mult_inv_right,
- prem,rabs_not_zero_iff RS sym] @ real_mult_ac) 1);
+by (asm_simp_tac (simpset() addsimps [rabs_not_zero_iff RS sym]) 1);
+by (asm_simp_tac (simpset() addsimps [rabs_mult RS sym, real_mult_inv_right,
+ rabs_not_zero_iff RS sym] @ real_mult_ac) 1);
qed "rabs_mult_rinv";
Goal "rabs(x+y) <= rabs x + rabs y";
-by (EVERY1 [res_inst_tac [("Q1","0r<=x+y")] (expand_if RS ssubst), rtac conjI]);
+by (case_tac "0r<=x+y" 1);
by (asm_simp_tac
(simpset() addsimps [rabs_eqI1,real_add_le_mono,rabs_ge_self]) 1);
by (asm_simp_tac
@@ -113,38 +112,38 @@
Goal "rabs(w + x + y + z) <= rabs(w) + rabs(x) + rabs(y) + rabs(z)";
by (full_simp_tac (simpset() addsimps [real_add_assoc]) 1);
-by (blast_tac (claset() addSIs [(rabs_triangle_ineq RS real_le_trans),
+by (blast_tac (claset() addSIs [rabs_triangle_ineq RS real_le_trans,
real_add_left_le_mono1]) 1);
qed "rabs_triangle_ineq_four";
Goalw [rabs_def] "rabs(-x)=rabs(x)";
-by (auto_tac (claset() addSDs [not_real_leE,real_less_asym] addIs [real_le_anti_sym],
- simpset() addsimps [real_ge_zero_iff]));
+by (auto_tac (claset() addSDs [not_real_leE,real_less_asym]
+ addIs [real_le_anti_sym],
+ simpset() addsimps [real_ge_zero_iff]));
qed "rabs_minus_cancel";
-Goal "rabs(x + -y) = rabs (y + -x)";
+Goal "rabs(x + (-y)) = rabs (y + (-x))";
by (rtac (rabs_minus_cancel RS subst) 1);
-by (simp_tac (simpset() addsimps
- [real_add_commute]) 1);
+by (simp_tac (simpset() addsimps [real_add_commute]) 1);
qed "rabs_minus_add_cancel";
-Goal "rabs(x + -y) <= rabs x + rabs y";
+Goal "rabs(x + (-y)) <= rabs x + rabs y";
by (res_inst_tac [("x1","y")] (rabs_minus_cancel RS subst) 1);
by (rtac rabs_triangle_ineq 1);
qed "rabs_triangle_minus_ineq";
-Goal "rabs (x + y + (-l + -m)) <= rabs(x + -l) + rabs(y + -m)";
+Goal "rabs (x + y + ((-l) + (-m))) <= rabs(x + (-l)) + rabs(y + (-m))";
by (full_simp_tac (simpset() addsimps [real_add_assoc]) 1);
by (res_inst_tac [("x1","y")] (real_add_left_commute RS ssubst) 1);
by (rtac (real_add_assoc RS subst) 1);
by (rtac rabs_triangle_ineq 1);
qed "rabs_sum_triangle_ineq";
-Goal "rabs(x) <= rabs(x + -y) + rabs(y)";
-by (res_inst_tac [("j","rabs(x + -y + y)")] real_le_trans 1);
+Goal "rabs(x) <= rabs(x + (-y)) + rabs(y)";
+by (res_inst_tac [("j","rabs(x + (-y) + y)")] real_le_trans 1);
by (simp_tac (simpset() addsimps [real_add_assoc]) 1);
by (simp_tac (simpset() addsimps [real_add_assoc RS sym,
- rabs_triangle_ineq]) 1);
+ rabs_triangle_ineq]) 1);
qed "rabs_triangle_ineq_minus_cancel";
Goal "[| rabs x < r; rabs y < s |] ==> rabs(x+y) < r+s";
@@ -153,7 +152,7 @@
by (REPEAT (ares_tac [real_add_less_mono] 1));
qed "rabs_add_less";
-Goal "[| rabs x < r; rabs y < s |] ==> rabs(x+ -y) < r+s";
+Goal "[| rabs x < r; rabs y < s |] ==> rabs(x+ (-y)) < r+s";
by (rotate_tac 1 1);
by (dtac (rabs_minus_cancel RS ssubst) 1);
by (asm_simp_tac (simpset() addsimps [rabs_add_less]) 1);
@@ -169,7 +168,7 @@
addIs [real_less_trans]) 1);
qed "real_mult_less_trans";
-Goal "!!(x::real) y.[| 0r<=y; x<r; y*r<t*s; 0r<t*s|] ==> y*x<t*s";
+Goal "[| 0r<=y; x<r; y*r<t*s; 0r<t*s|] ==> y*x<t*s";
by (dtac real_le_imp_less_or_eq 1);
by (fast_tac (HOL_cs addEs [real_mult_0_less RS iffD2,
real_mult_less_trans]) 1);
@@ -196,11 +195,11 @@
by (cut_inst_tac [("x1","y")] (rabs_ge_zero RS real_le_imp_less_or_eq) 1);
by (EVERY1[etac disjE,rtac real_less_imp_le]);
by (dres_inst_tac [("W","1r")] real_less_sum_gt_zero 1);
-by (forw_inst_tac [("y","rabs x + -1r")] real_mult_order 1);
+by (forw_inst_tac [("y","rabs x + (-1r)")] real_mult_order 1);
by (assume_tac 1);
by (rtac real_sum_gt_zero_less 1);
by (asm_full_simp_tac (simpset() addsimps [real_add_mult_distrib2,
- real_mult_commute, rabs_mult]) 1);
+ real_mult_commute, rabs_mult]) 1);
by (dtac sym 1);
by (asm_full_simp_tac (simpset() addsimps [rabs_mult]) 1);
qed "rabs_mult_le";
@@ -215,7 +214,7 @@
Goalw [rabs_def] "rabs 1r = 1r";
by (auto_tac (claset() addSDs [not_real_leE RS real_less_asym],
- simpset() addsimps [real_zero_less_one]));
+ simpset() addsimps [real_zero_less_one]));
qed "rabs_one";
Goal "[| 0r < x ; x < r |] ==> rabs x < r";
@@ -225,7 +224,7 @@
Goal "rabs x =x | rabs x = -x";
by (cut_inst_tac [("R1.0","0r"),("R2.0","x")] real_linear 1);
by (blast_tac (claset() addIs [rabs_eqI2,rabs_minus_eqI2,
- rabs_zero,rabs_minus_zero]) 1);
+ rabs_zero,rabs_minus_zero]) 1);
qed "rabs_disj";
Goal "rabs x = y ==> x = y | -x = y";
@@ -239,9 +238,8 @@
by Safe_tac;
by (rtac (real_less_swap_iff RS iffD2) 1);
by (asm_simp_tac (simpset() addsimps [(rabs_ge_minus_self
- RS real_le_less_trans)]) 1);
-by (asm_simp_tac (simpset() addsimps [(rabs_ge_self
- RS real_le_less_trans)]) 1);
+ RS real_le_less_trans)]) 1);
+by (asm_simp_tac (simpset() addsimps [rabs_ge_self RS real_le_less_trans]) 1);
by (EVERY1 [dtac (real_less_swap_iff RS iffD1), rotate_tac 1,
dtac (real_minus_minus RS subst),
cut_inst_tac [("x","x")] rabs_disj, dtac disjE ]);
@@ -252,57 +250,57 @@
by (auto_tac (claset(),simpset() addsimps [rabs_interval_iff]));
by (dtac (real_less_swap_iff RS iffD1) 1);
by (dtac (real_less_swap_iff RS iffD1) 2);
-by (Auto_tac);
+by Auto_tac;
qed "rabs_interval_iff2";
-Goal "!!x. rabs x <= r ==> -r<=x & x<=r";
+Goal "rabs x <= r ==> -r<=x & x<=r";
by (dtac real_le_imp_less_or_eq 1);
by (auto_tac (claset() addSDs [real_less_imp_le],
- simpset() addsimps [rabs_interval_iff,rabs_ge_self]));
+ simpset() addsimps [rabs_interval_iff,rabs_ge_self]));
by (auto_tac (claset(),simpset() addsimps [real_le_def]));
by (dtac (real_less_swap_iff RS iffD1) 1);
-by (auto_tac (claset() addSDs [rabs_ge_minus_self
- RS real_le_less_trans],simpset() addsimps [real_less_not_refl]));
+by (auto_tac (claset() addSDs [rabs_ge_minus_self RS real_le_less_trans],
+ simpset() addsimps [real_less_not_refl]));
qed "rabs_leD";
-Goal "!!x. [| -r<x; x<=r |] ==> rabs x <= r";
+Goal "[| -r<x; x<=r |] ==> rabs x <= r";
by (dtac real_le_imp_less_or_eq 1);
by (Step_tac 1);
by (blast_tac (claset() addIs
- [rabs_interval_iff RS iffD2 RS real_less_imp_le]) 1);
-by (auto_tac (claset() addSDs [rabs_eqI2],simpset() addsimps
- [real_le_def,real_gt_zero_iff RS sym,real_less_not_refl]));
+ [rabs_interval_iff RS iffD2 RS real_less_imp_le]) 1);
+by (auto_tac (claset() addSDs [rabs_eqI2],
+ simpset() addsimps
+ [real_le_def,real_gt_zero_iff RS sym,real_less_not_refl]));
qed "rabs_leI1";
-Goal "!!x. [| -r<=x; x<=r |] ==> rabs x <= r";
+Goal "[| -r<=x; x<=r |] ==> rabs x <= r";
by (REPEAT(dtac real_le_imp_less_or_eq 1));
by (Step_tac 1);
-by (blast_tac (claset() addIs
- [rabs_interval_iff RS iffD2 RS real_less_imp_le]) 1);
-by (auto_tac (claset() addSDs [rabs_eqI2],simpset() addsimps
- [real_le_def,real_gt_zero_iff RS sym,real_less_not_refl,
- rabs_minus_cancel]));
+by (blast_tac (claset()
+ addIs [rabs_interval_iff RS iffD2 RS real_less_imp_le]) 1);
+by (auto_tac (claset() addSDs [rabs_eqI2],
+ simpset() addsimps
+ [real_le_def,real_gt_zero_iff RS sym,real_less_not_refl,
+ rabs_minus_cancel]));
by (cut_inst_tac [("x","r")] rabs_disj 1);
by (rotate_tac 1 1);
-by (auto_tac (claset(),simpset() addsimps [real_less_not_refl]));
+by (auto_tac (claset(), simpset() addsimps [real_less_not_refl]));
qed "rabs_leI";
Goal "(rabs x <= r) = (-r<=x & x<=r)";
by (blast_tac (claset() addSIs [rabs_leD,rabs_leI]) 1);
qed "rabs_le_interval_iff";
-Goal
- "(rabs (x + -y) < r) = (y + -r < x & x < y + r)";
-by (auto_tac (claset(),simpset() addsimps
- [rabs_interval_iff]));
+Goal "(rabs (x + (-y)) < r) = (y + (-r) < x & x < y + r)";
+by (auto_tac (claset(), simpset() addsimps [rabs_interval_iff]));
by (ALLGOALS(dtac real_less_sum_gt_zero));
by (ALLGOALS(dtac real_less_sum_gt_zero));
by (ALLGOALS(rtac real_sum_gt_zero_less));
-by (auto_tac (claset(),simpset() addsimps
- [real_minus_add_distrib] addsimps real_add_ac));
+by (auto_tac (claset(),
+ simpset() addsimps [real_minus_add_distrib] @ real_add_ac));
qed "rabs_add_minus_interval_iff";
-Goal "!!k. 0r < k ==> 0r < k + rabs(x)";
+Goal "0r < k ==> 0r < k + rabs(x)";
by (res_inst_tac [("t","0r")] (real_add_zero_right RS subst) 1);
by (etac (rabs_ge_zero RSN (2,real_add_less_le_mono)) 1);
qed "rabs_add_pos_gt_zero";