author | paulson |
Wed, 25 Nov 1998 15:54:41 +0100 | |
changeset 5971 | c5a7a7685826 |
parent 5588 | a3ab526bb891 |
child 7077 | 60b098bb8b8a |
permissions | -rw-r--r-- |
5078 | 1 |
(* Title : RealAbs.ML |
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Author : Jacques D. Fleuriot |
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Copyright : 1998 University of Cambridge |
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Description : Absolute value function for the reals |
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*) |
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(*---------------------------------------------------------------------------- |
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Properties of the absolute value function over the reals |
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(adapted version of previously proved theorems about abs) |
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----------------------------------------------------------------------------*) |
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Goalw [rabs_def] "rabs r = (if 0r<=r then r else -r)"; |
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by Auto_tac; |
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qed "rabs_iff"; |
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Goalw [rabs_def] "rabs 0r = 0r"; |
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by (rtac (real_le_refl RS if_P) 1); |
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qed "rabs_zero"; |
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||
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Addsimps [rabs_zero]; |
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||
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Goalw [rabs_def] "rabs 0r = -0r"; |
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by (stac real_minus_zero 1); |
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by (rtac if_cancel 1); |
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qed "rabs_minus_zero"; |
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||
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val [prem] = goalw thy [rabs_def] "0r<=x ==> rabs x = x"; |
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by (rtac (prem RS if_P) 1); |
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qed "rabs_eqI1"; |
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val [prem] = goalw thy [rabs_def] "0r<x ==> rabs x = x"; |
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by (simp_tac (simpset() addsimps [(prem RS real_less_imp_le),rabs_eqI1]) 1); |
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qed "rabs_eqI2"; |
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||
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val [prem] = goalw thy [rabs_def,real_le_def] "x<0r ==> rabs x = -x"; |
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by (simp_tac (simpset() addsimps [prem,if_not_P]) 1); |
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qed "rabs_minus_eqI2"; |
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Goal "x<=0r ==> rabs x = -x"; |
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by (dtac real_le_imp_less_or_eq 1); |
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by (blast_tac (HOL_cs addIs [rabs_minus_zero,rabs_minus_eqI2]) 1); |
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qed "rabs_minus_eqI1"; |
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Goalw [rabs_def,real_le_def] "0r<= rabs x"; |
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by (Full_simp_tac 1); |
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by (blast_tac (claset() addDs [real_minus_zero_less_iff RS iffD2, |
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real_less_asym]) 1); |
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qed "rabs_ge_zero"; |
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Goal "rabs(rabs x)=rabs x"; |
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by (res_inst_tac [("r1","rabs x")] (rabs_iff RS ssubst) 1); |
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by (blast_tac (claset() addIs [if_P,rabs_ge_zero]) 1); |
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qed "rabs_idempotent"; |
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Goalw [rabs_def] "(x=0r) = (rabs x = 0r)"; |
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by (Full_simp_tac 1); |
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qed "rabs_zero_iff"; |
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Goal "(x ~= 0r) = (rabs x ~= 0r)"; |
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by (full_simp_tac (simpset() addsimps [rabs_zero_iff RS sym]) 1); |
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qed "rabs_not_zero_iff"; |
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Goalw [rabs_def] "x<=rabs x"; |
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by (Full_simp_tac 1); |
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by (auto_tac (claset() addDs [not_real_leE RS real_less_imp_le], |
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simpset() addsimps [real_le_zero_iff])); |
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qed "rabs_ge_self"; |
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Goalw [rabs_def] "-x<=rabs x"; |
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by (full_simp_tac (simpset() addsimps [real_ge_zero_iff]) 1); |
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qed "rabs_ge_minus_self"; |
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(* case splits nightmare *) |
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Goalw [rabs_def] "rabs(x*y) = (rabs x)*(rabs y)"; |
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by (auto_tac (claset(),simpset() addsimps [real_minus_mult_eq1, |
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real_minus_mult_commute,real_minus_mult_eq2])); |
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by (blast_tac (claset() addDs [real_le_mult_order]) 1); |
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by (auto_tac (claset() addSDs [not_real_leE],simpset())); |
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by (EVERY1[dtac real_mult_le_zero, assume_tac, dtac real_le_anti_sym]); |
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by (EVERY[dtac real_mult_le_zero 3, assume_tac 3, dtac real_le_anti_sym 3]); |
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by (dtac real_mult_less_zero1 5 THEN assume_tac 5); |
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by (auto_tac (claset() addDs [real_less_asym,sym], |
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simpset() addsimps [real_minus_mult_eq2 RS sym] @real_mult_ac)); |
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qed "rabs_mult"; |
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||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
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Goalw [rabs_def] "x~= 0r ==> rabs(rinv(x)) = rinv(rabs(x))"; |
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by (auto_tac (claset(),simpset() addsimps [real_minus_rinv] |
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)); |
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by (ALLGOALS(dtac not_real_leE)); |
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by (etac real_less_asym 1); |
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by (blast_tac (claset() addDs [real_le_imp_less_or_eq, |
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real_rinv_gt_zero]) 1); |
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by (dtac (rinv_not_zero RS not_sym) 1); |
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by (rtac (real_rinv_less_zero RSN (2,real_less_asym)) 1); |
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by (assume_tac 2); |
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by (blast_tac (claset() addSDs [real_le_imp_less_or_eq]) 1); |
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qed "rabs_rinv"; |
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val [prem] = goal thy "y ~= 0r ==> rabs(x*rinv(y)) = rabs(x)*rinv(rabs(y))"; |
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by (res_inst_tac [("c1","rabs y")] (real_mult_left_cancel RS subst) 1); |
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by (simp_tac (simpset() addsimps [(rabs_not_zero_iff RS sym), prem]) 1); |
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by (simp_tac (simpset() addsimps [(rabs_mult RS sym) ,real_mult_inv_right, |
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prem,rabs_not_zero_iff RS sym] @ real_mult_ac) 1); |
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qed "rabs_mult_rinv"; |
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Goal "rabs(x+y) <= rabs x + rabs y"; |
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by (EVERY1 [res_inst_tac [("Q1","0r<=x+y")] (expand_if RS ssubst), rtac conjI]); |
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by (asm_simp_tac |
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(simpset() addsimps [rabs_eqI1,real_add_le_mono,rabs_ge_self]) 1); |
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by (asm_simp_tac |
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(simpset() addsimps [not_real_leE,rabs_minus_eqI2,real_add_le_mono, |
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rabs_ge_minus_self]) 1); |
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qed "rabs_triangle_ineq"; |
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Goal "rabs(w + x + y + z) <= rabs(w) + rabs(x) + rabs(y) + rabs(z)"; |
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by (full_simp_tac (simpset() addsimps [real_add_assoc]) 1); |
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by (blast_tac (claset() addSIs [(rabs_triangle_ineq RS real_le_trans), |
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real_add_left_le_mono1]) 1); |
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qed "rabs_triangle_ineq_four"; |
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Goalw [rabs_def] "rabs(-x)=rabs(x)"; |
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by (auto_tac (claset() addSDs [not_real_leE,real_less_asym] addIs [real_le_anti_sym], |
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simpset() addsimps [real_ge_zero_iff])); |
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qed "rabs_minus_cancel"; |
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Goal "rabs(x + -y) <= rabs x + rabs y"; |
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by (res_inst_tac [("x1","y")] (rabs_minus_cancel RS subst) 1); |
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by (rtac rabs_triangle_ineq 1); |
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qed "rabs_triangle_minus_ineq"; |
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Goal "rabs (x + y + (-l + -m)) <= rabs(x + -l) + rabs(y + -m)"; |
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by (full_simp_tac (simpset() addsimps [real_add_assoc]) 1); |
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by (res_inst_tac [("x1","y")] (real_add_left_commute RS ssubst) 1); |
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by (rtac (real_add_assoc RS subst) 1); |
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by (rtac rabs_triangle_ineq 1); |
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qed "rabs_sum_triangle_ineq"; |
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Goal "[| rabs x < r; rabs y < s |] ==> rabs(x+y) < r+s"; |
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by (rtac real_le_less_trans 1); |
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by (rtac rabs_triangle_ineq 1); |
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by (REPEAT (ares_tac [real_add_less_mono] 1)); |
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qed "rabs_add_less"; |
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Goal "[| rabs x < r; rabs y < s |] ==> rabs(x+ -y) < r+s"; |
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by (rotate_tac 1 1); |
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by (dtac (rabs_minus_cancel RS ssubst) 1); |
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by (asm_simp_tac (simpset() addsimps [rabs_add_less]) 1); |
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qed "rabs_add_minus_less"; |
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(* lemmas manipulating terms *) |
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Goal "(0r*x<r)=(0r<r)"; |
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by (Simp_tac 1); |
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qed "real_mult_0_less"; |
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Goal "[| 0r<y; x<r; y*r<t*s |] ==> y*x<t*s"; |
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by (blast_tac (claset() addSIs [real_mult_less_mono2] |
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addIs [real_less_trans]) 1); |
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qed "real_mult_less_trans"; |
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Goal "!!(x::real) y.[| 0r<=y; x<r; y*r<t*s; 0r<t*s|] ==> y*x<t*s"; |
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by (dtac real_le_imp_less_or_eq 1); |
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by (fast_tac (HOL_cs addEs [real_mult_0_less RS iffD2, |
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real_mult_less_trans]) 1); |
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qed "real_mult_le_less_trans"; |
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(* proofs lifted from previous older version *) |
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Goal "[| rabs x<r; rabs y<s |] ==> rabs(x*y)<r*s"; |
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by (simp_tac (simpset() addsimps [rabs_mult]) 1); |
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by (rtac real_mult_le_less_trans 1); |
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by (rtac rabs_ge_zero 1); |
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by (assume_tac 1); |
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by (blast_tac (HOL_cs addIs [rabs_ge_zero, real_mult_less_mono1, |
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real_le_less_trans]) 1); |
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by (blast_tac (HOL_cs addIs [rabs_ge_zero, real_mult_order, |
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real_le_less_trans]) 1); |
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qed "rabs_mult_less"; |
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Goal "[| rabs x < r; rabs y < s |] ==> rabs(x)*rabs(y)<r*s"; |
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by (auto_tac (claset() addIs [rabs_mult_less], |
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simpset() addsimps [rabs_mult RS sym])); |
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qed "rabs_mult_less2"; |
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||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
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Goal "1r < rabs x ==> rabs y <= rabs(x*y)"; |
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by (cut_inst_tac [("x1","y")] (rabs_ge_zero RS real_le_imp_less_or_eq) 1); |
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by (EVERY1[etac disjE,rtac real_less_imp_le]); |
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by (dres_inst_tac [("W","1r")] real_less_sum_gt_zero 1); |
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by (forw_inst_tac [("y","rabs x + -1r")] real_mult_order 1); |
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by (assume_tac 1); |
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by (rtac real_sum_gt_zero_less 1); |
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by (asm_full_simp_tac (simpset() addsimps [real_add_mult_distrib2, |
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real_mult_commute, rabs_mult]) 1); |
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by (dtac sym 1); |
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by (asm_full_simp_tac (simpset() addsimps [rabs_mult]) 1); |
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qed "rabs_mult_le"; |
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||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
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Goal "[| 1r < rabs x; r < rabs y|] ==> r < rabs(x*y)"; |
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by (blast_tac (HOL_cs addIs [rabs_mult_le, real_less_le_trans]) 1); |
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qed "rabs_mult_gt"; |
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||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
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Goal "rabs(x)<r ==> 0r<r"; |
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by (blast_tac (claset() addSIs [real_le_less_trans,rabs_ge_zero]) 1); |
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qed "rabs_less_gt_zero"; |
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Goalw [rabs_def] "rabs 1r = 1r"; |
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by (auto_tac (claset() addSDs [not_real_leE RS real_less_asym], |
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simpset() addsimps [real_zero_less_one])); |
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qed "rabs_one"; |
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Goal "[| 0r < x ; x < r |] ==> rabs x < r"; |
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by (asm_simp_tac (simpset() addsimps [rabs_eqI2]) 1); |
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qed "rabs_lessI"; |
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Goal "rabs x =x | rabs x = -x"; |
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by (cut_inst_tac [("R1.0","0r"),("R2.0","x")] real_linear 1); |
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by (blast_tac (claset() addIs [rabs_eqI2,rabs_minus_eqI2, |
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rabs_zero,rabs_minus_zero]) 1); |
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qed "rabs_disj"; |
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||
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Goal "rabs x = y ==> x = y | -x = y"; |
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by (dtac sym 1); |
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by (hyp_subst_tac 1); |
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by (res_inst_tac [("x1","x")] (rabs_disj RS disjE) 1); |
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by (REPEAT(Asm_simp_tac 1)); |
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qed "rabs_eq_disj"; |
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Goal "(rabs x < r) = (-r<x & x<r)"; |
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by Safe_tac; |
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by (rtac (real_less_swap_iff RS iffD2) 1); |
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by (asm_simp_tac (simpset() addsimps [(rabs_ge_minus_self |
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RS real_le_less_trans)]) 1); |
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by (asm_simp_tac (simpset() addsimps [(rabs_ge_self |
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RS real_le_less_trans)]) 1); |
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by (EVERY1 [dtac (real_less_swap_iff RS iffD1), rotate_tac 1, |
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dtac (real_minus_minus RS subst), |
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cut_inst_tac [("x","x")] rabs_disj, dtac disjE ]); |
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by (assume_tac 3 THEN Auto_tac); |
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qed "rabs_interval_iff"; |
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