author | berghofe |
Fri, 24 Jul 1998 13:19:38 +0200 | |
changeset 5184 | 9b8547a9496a |
parent 5148 | 74919e8f221c |
child 5459 | 1dbaf888f4e7 |
permissions | -rw-r--r-- |
5078 | 1 |
(* Title : Real.ML |
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Author : Jacques D. Fleuriot |
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Copyright : 1998 University of Cambridge |
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Description : The reals |
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*) |
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open Real; |
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(*** Proving that realrel is an equivalence relation ***) |
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5143
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Goal "[| (x1::preal) + y2 = x2 + y1; x2 + y3 = x3 + y2 |] \ |
5078 | 12 |
\ ==> x1 + y3 = x3 + y1"; |
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by (res_inst_tac [("C","y2")] preal_add_right_cancel 1); |
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by (rotate_tac 1 1 THEN dtac sym 1); |
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by (asm_full_simp_tac (simpset() addsimps preal_add_ac) 1); |
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by (rtac (preal_add_left_commute RS subst) 1); |
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by (res_inst_tac [("x1","x1")] (preal_add_assoc RS subst) 1); |
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by (asm_full_simp_tac (simpset() addsimps preal_add_ac) 1); |
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qed "preal_trans_lemma"; |
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(** Natural deduction for realrel **) |
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Goalw [realrel_def] |
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"(((x1,y1),(x2,y2)): realrel) = (x1 + y2 = x2 + y1)"; |
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by (Fast_tac 1); |
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qed "realrel_iff"; |
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Goalw [realrel_def] |
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"[| x1 + y2 = x2 + y1 |] ==> ((x1,y1),(x2,y2)): realrel"; |
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by (Fast_tac 1); |
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qed "realrelI"; |
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Goalw [realrel_def] |
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"p: realrel --> (EX x1 y1 x2 y2. \ |
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\ p = ((x1,y1),(x2,y2)) & x1 + y2 = x2 + y1)"; |
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by (Fast_tac 1); |
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qed "realrelE_lemma"; |
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val [major,minor] = goal thy |
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"[| p: realrel; \ |
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\ !!x1 y1 x2 y2. [| p = ((x1,y1),(x2,y2)); x1+y2 = x2+y1 \ |
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\ |] ==> Q |] ==> Q"; |
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by (cut_facts_tac [major RS (realrelE_lemma RS mp)] 1); |
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by (REPEAT (eresolve_tac [asm_rl,exE,conjE,minor] 1)); |
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qed "realrelE"; |
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AddSIs [realrelI]; |
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AddSEs [realrelE]; |
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Goal "(x,x): realrel"; |
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by (stac surjective_pairing 1 THEN rtac (refl RS realrelI) 1); |
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qed "realrel_refl"; |
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Goalw [equiv_def, refl_def, sym_def, trans_def] |
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"equiv {x::(preal*preal).True} realrel"; |
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by (fast_tac (claset() addSIs [realrel_refl] |
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addSEs [sym,preal_trans_lemma]) 1); |
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qed "equiv_realrel"; |
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val equiv_realrel_iff = |
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[TrueI, TrueI] MRS |
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([CollectI, CollectI] MRS |
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(equiv_realrel RS eq_equiv_class_iff)); |
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Goalw [real_def,realrel_def,quotient_def] "realrel^^{(x,y)}:real"; |
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by (Blast_tac 1); |
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qed "realrel_in_real"; |
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Goal "inj_on Abs_real real"; |
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by (rtac inj_on_inverseI 1); |
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by (etac Abs_real_inverse 1); |
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qed "inj_on_Abs_real"; |
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Addsimps [equiv_realrel_iff,inj_on_Abs_real RS inj_on_iff, |
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realrel_iff, realrel_in_real, Abs_real_inverse]; |
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Addsimps [equiv_realrel RS eq_equiv_class_iff]; |
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val eq_realrelD = equiv_realrel RSN (2,eq_equiv_class); |
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Goal "inj(Rep_real)"; |
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by (rtac inj_inverseI 1); |
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by (rtac Rep_real_inverse 1); |
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qed "inj_Rep_real"; |
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(** real_preal: the injection from preal to real **) |
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Goal "inj(real_preal)"; |
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by (rtac injI 1); |
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by (rewtac real_preal_def); |
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by (dtac (inj_on_Abs_real RS inj_onD) 1); |
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by (REPEAT (rtac realrel_in_real 1)); |
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by (dtac eq_equiv_class 1); |
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by (rtac equiv_realrel 1); |
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by (Fast_tac 1); |
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by Safe_tac; |
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by (Asm_full_simp_tac 1); |
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qed "inj_real_preal"; |
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val [prem] = goal thy |
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"(!!x y. z = Abs_real(realrel^^{(x,y)}) ==> P) ==> P"; |
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by (res_inst_tac [("x1","z")] |
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(rewrite_rule [real_def] Rep_real RS quotientE) 1); |
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by (dres_inst_tac [("f","Abs_real")] arg_cong 1); |
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by (res_inst_tac [("p","x")] PairE 1); |
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by (rtac prem 1); |
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by (asm_full_simp_tac (simpset() addsimps [Rep_real_inverse]) 1); |
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qed "eq_Abs_real"; |
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(**** real_minus: additive inverse on real ****) |
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Goalw [congruent_def] |
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"congruent realrel (%p. split (%x y. realrel^^{(y,x)}) p)"; |
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by Safe_tac; |
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by (asm_full_simp_tac (simpset() addsimps [preal_add_commute]) 1); |
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qed "real_minus_congruent"; |
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(*Resolve th against the corresponding facts for real_minus*) |
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val real_minus_ize = RSLIST [equiv_realrel, real_minus_congruent]; |
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Goalw [real_minus_def] |
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"%~ (Abs_real(realrel^^{(x,y)})) = Abs_real(realrel ^^ {(y,x)})"; |
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by (res_inst_tac [("f","Abs_real")] arg_cong 1); |
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by (simp_tac (simpset() addsimps |
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[realrel_in_real RS Abs_real_inverse,real_minus_ize UN_equiv_class]) 1); |
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qed "real_minus"; |
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Goal "%~ (%~ z) = z"; |
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by (res_inst_tac [("z","z")] eq_Abs_real 1); |
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by (asm_simp_tac (simpset() addsimps [real_minus]) 1); |
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qed "real_minus_minus"; |
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Addsimps [real_minus_minus]; |
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Goal "inj(real_minus)"; |
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by (rtac injI 1); |
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by (dres_inst_tac [("f","real_minus")] arg_cong 1); |
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by (asm_full_simp_tac (simpset() addsimps [real_minus_minus]) 1); |
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qed "inj_real_minus"; |
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Goalw [real_zero_def] "%~0r = 0r"; |
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by (simp_tac (simpset() addsimps [real_minus]) 1); |
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qed "real_minus_zero"; |
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Addsimps [real_minus_zero]; |
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Goal "(%~x = 0r) = (x = 0r)"; |
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by (res_inst_tac [("z","x")] eq_Abs_real 1); |
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by (auto_tac (claset(),simpset() addsimps [real_zero_def, |
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real_minus] @ preal_add_ac)); |
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qed "real_minus_zero_iff"; |
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Addsimps [real_minus_zero_iff]; |
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Goal "(%~x ~= 0r) = (x ~= 0r)"; |
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by Auto_tac; |
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qed "real_minus_not_zero_iff"; |
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(*** Congruence property for addition ***) |
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Goalw [congruent2_def] |
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"congruent2 realrel (%p1 p2. \ |
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\ split (%x1 y1. split (%x2 y2. realrel^^{(x1+x2, y1+y2)}) p2) p1)"; |
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by Safe_tac; |
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by (asm_simp_tac (simpset() addsimps [preal_add_assoc]) 1); |
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by (res_inst_tac [("z1.1","x1a")] (preal_add_left_commute RS ssubst) 1); |
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by (asm_simp_tac (simpset() addsimps [preal_add_assoc RS sym]) 1); |
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by (asm_simp_tac (simpset() addsimps preal_add_ac) 1); |
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qed "real_add_congruent2"; |
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(*Resolve th against the corresponding facts for real_add*) |
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val real_add_ize = RSLIST [equiv_realrel, real_add_congruent2]; |
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Goalw [real_add_def] |
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"Abs_real(realrel^^{(x1,y1)}) + Abs_real(realrel^^{(x2,y2)}) = \ |
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\ Abs_real(realrel^^{(x1+x2, y1+y2)})"; |
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by (asm_simp_tac |
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(simpset() addsimps [real_add_ize UN_equiv_class2]) 1); |
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qed "real_add"; |
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Goal "(z::real) + w = w + z"; |
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by (res_inst_tac [("z","z")] eq_Abs_real 1); |
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by (res_inst_tac [("z","w")] eq_Abs_real 1); |
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by (asm_simp_tac (simpset() addsimps (preal_add_ac @ [real_add])) 1); |
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qed "real_add_commute"; |
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Goal "((z1::real) + z2) + z3 = z1 + (z2 + z3)"; |
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by (res_inst_tac [("z","z1")] eq_Abs_real 1); |
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by (res_inst_tac [("z","z2")] eq_Abs_real 1); |
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by (res_inst_tac [("z","z3")] eq_Abs_real 1); |
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by (asm_simp_tac (simpset() addsimps [real_add, preal_add_assoc]) 1); |
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qed "real_add_assoc"; |
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(*For AC rewriting*) |
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Goal "(x::real)+(y+z)=y+(x+z)"; |
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by (rtac (real_add_commute RS trans) 1); |
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by (rtac (real_add_assoc RS trans) 1); |
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by (rtac (real_add_commute RS arg_cong) 1); |
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qed "real_add_left_commute"; |
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(* real addition is an AC operator *) |
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val real_add_ac = [real_add_assoc,real_add_commute,real_add_left_commute]; |
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Goalw [real_preal_def,real_zero_def] "0r + z = z"; |
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by (res_inst_tac [("z","z")] eq_Abs_real 1); |
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by (asm_full_simp_tac (simpset() addsimps [real_add] @ preal_add_ac) 1); |
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qed "real_add_zero_left"; |
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Goal "z + 0r = z"; |
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by (simp_tac (simpset() addsimps [real_add_zero_left,real_add_commute]) 1); |
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qed "real_add_zero_right"; |
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Goalw [real_zero_def] "z + %~z = 0r"; |
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by (res_inst_tac [("z","z")] eq_Abs_real 1); |
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by (asm_full_simp_tac (simpset() addsimps [real_minus, |
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real_add, preal_add_commute]) 1); |
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qed "real_add_minus"; |
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Goal "%~z + z = 0r"; |
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by (simp_tac (simpset() addsimps |
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[real_add_commute,real_add_minus]) 1); |
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qed "real_add_minus_left"; |
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Goal "? y. (x::real) + y = 0r"; |
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by (fast_tac (claset() addIs [real_add_minus]) 1); |
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qed "real_minus_ex"; |
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Goal "?! y. (x::real) + y = 0r"; |
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by (auto_tac (claset() addIs [real_add_minus],simpset())); |
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by (dres_inst_tac [("f","%x. ya+x")] arg_cong 1); |
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by (asm_full_simp_tac (simpset() addsimps [real_add_assoc RS sym]) 1); |
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by (asm_full_simp_tac (simpset() addsimps [real_add_commute, |
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real_add_zero_right,real_add_zero_left]) 1); |
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qed "real_minus_ex1"; |
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Goal "?! y. y + (x::real) = 0r"; |
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by (auto_tac (claset() addIs [real_add_minus_left],simpset())); |
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by (dres_inst_tac [("f","%x. x+ya")] arg_cong 1); |
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by (asm_full_simp_tac (simpset() addsimps [real_add_assoc]) 1); |
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by (asm_full_simp_tac (simpset() addsimps [real_add_commute, |
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real_add_zero_right,real_add_zero_left]) 1); |
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qed "real_minus_left_ex1"; |
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5143
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Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
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Goal "x + y = 0r ==> x = %~y"; |
5078 | 242 |
by (cut_inst_tac [("z","y")] real_add_minus_left 1); |
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by (res_inst_tac [("x1","y")] (real_minus_left_ex1 RS ex1E) 1); |
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by (Blast_tac 1); |
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qed "real_add_minus_eq_minus"; |
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Goal "? y. x = %~y"; |
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by (cut_inst_tac [("x","x")] real_minus_ex 1); |
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by (etac exE 1 THEN dtac real_add_minus_eq_minus 1); |
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by (Fast_tac 1); |
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qed "real_as_add_inverse_ex"; |
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(* real_minus_add_distrib *) |
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Goal "%~(x + y) = %~x + %~y"; |
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by (res_inst_tac [("z","x")] eq_Abs_real 1); |
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by (res_inst_tac [("z","y")] eq_Abs_real 1); |
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by (auto_tac (claset(),simpset() addsimps [real_minus,real_add])); |
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qed "real_minus_add_eq"; |
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val real_minus_add_distrib = real_minus_add_eq; |
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Goal "((x::real) + y = x + z) = (y = z)"; |
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by (Step_tac 1); |
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by (dres_inst_tac [("f","%t.%~x + t")] arg_cong 1); |
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by (asm_full_simp_tac (simpset() addsimps [real_add_minus_left, |
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real_add_assoc RS sym,real_add_zero_left]) 1); |
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qed "real_add_left_cancel"; |
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Goal "(y + (x::real)= z + x) = (y = z)"; |
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by (simp_tac (simpset() addsimps [real_add_commute,real_add_left_cancel]) 1); |
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qed "real_add_right_cancel"; |
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272 |
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(*** Congruence property for multiplication ***) |
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Goal "!!(x1::preal). [| x1 + y2 = x2 + y1 |] ==> \ |
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\ x * x1 + y * y1 + (x * y2 + x2 * y) = \ |
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\ x * x2 + y * y2 + (x * y1 + x1 * y)"; |
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by (asm_full_simp_tac (simpset() addsimps [preal_add_left_commute, |
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preal_add_assoc RS sym,preal_add_mult_distrib2 RS sym]) 1); |
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by (rtac (preal_mult_commute RS subst) 1); |
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by (res_inst_tac [("y1","x2")] (preal_mult_commute RS subst) 1); |
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by (asm_full_simp_tac (simpset() addsimps [preal_add_assoc, |
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preal_add_mult_distrib2 RS sym]) 1); |
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by (asm_full_simp_tac (simpset() addsimps [preal_add_commute]) 1); |
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qed "real_mult_congruent2_lemma"; |
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285 |
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Goal |
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"congruent2 realrel (%p1 p2. \ |
|
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\ split (%x1 y1. split (%x2 y2. realrel^^{(x1*x2 + y1*y2, x1*y2+x2*y1)}) p2) p1)"; |
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by (rtac (equiv_realrel RS congruent2_commuteI) 1); |
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by Safe_tac; |
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by (rewtac split_def); |
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by (asm_simp_tac (simpset() addsimps [preal_mult_commute,preal_add_commute]) 1); |
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by (auto_tac (claset(),simpset() addsimps [real_mult_congruent2_lemma])); |
|
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qed "real_mult_congruent2"; |
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295 |
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296 |
(*Resolve th against the corresponding facts for real_mult*) |
|
297 |
val real_mult_ize = RSLIST [equiv_realrel, real_mult_congruent2]; |
|
298 |
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299 |
Goalw [real_mult_def] |
|
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"Abs_real((realrel^^{(x1,y1)})) * Abs_real((realrel^^{(x2,y2)})) = \ |
|
301 |
\ Abs_real(realrel ^^ {(x1*x2+y1*y2,x1*y2+x2*y1)})"; |
|
302 |
by (simp_tac (simpset() addsimps [real_mult_ize UN_equiv_class2]) 1); |
|
303 |
qed "real_mult"; |
|
304 |
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305 |
Goal "(z::real) * w = w * z"; |
|
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by (res_inst_tac [("z","z")] eq_Abs_real 1); |
|
307 |
by (res_inst_tac [("z","w")] eq_Abs_real 1); |
|
308 |
by (asm_simp_tac (simpset() addsimps ([real_mult] @ preal_add_ac @ preal_mult_ac)) 1); |
|
309 |
qed "real_mult_commute"; |
|
310 |
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311 |
Goal "((z1::real) * z2) * z3 = z1 * (z2 * z3)"; |
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by (res_inst_tac [("z","z1")] eq_Abs_real 1); |
|
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by (res_inst_tac [("z","z2")] eq_Abs_real 1); |
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314 |
by (res_inst_tac [("z","z3")] eq_Abs_real 1); |
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315 |
by (asm_simp_tac (simpset() addsimps ([preal_add_mult_distrib2,real_mult] @ |
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preal_add_ac @ preal_mult_ac)) 1); |
|
317 |
qed "real_mult_assoc"; |
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318 |
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319 |
qed_goal "real_mult_left_commute" thy |
|
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"(z1::real) * (z2 * z3) = z2 * (z1 * z3)" |
|
321 |
(fn _ => [rtac (real_mult_commute RS trans) 1, rtac (real_mult_assoc RS trans) 1, |
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322 |
rtac (real_mult_commute RS arg_cong) 1]); |
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323 |
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324 |
(* real multiplication is an AC operator *) |
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val real_mult_ac = [real_mult_assoc, real_mult_commute, real_mult_left_commute]; |
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326 |
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327 |
Goalw [real_one_def,pnat_one_def] "1r * z = z"; |
|
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by (res_inst_tac [("z","z")] eq_Abs_real 1); |
|
329 |
by (asm_full_simp_tac (simpset() addsimps [real_mult, |
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330 |
preal_add_mult_distrib2,preal_mult_1_right] |
|
331 |
@ preal_mult_ac @ preal_add_ac) 1); |
|
332 |
qed "real_mult_1"; |
|
333 |
||
334 |
Goal "z * 1r = z"; |
|
335 |
by (simp_tac (simpset() addsimps [real_mult_commute, |
|
336 |
real_mult_1]) 1); |
|
337 |
qed "real_mult_1_right"; |
|
338 |
||
339 |
Goalw [real_zero_def,pnat_one_def] "0r * z = 0r"; |
|
340 |
by (res_inst_tac [("z","z")] eq_Abs_real 1); |
|
341 |
by (asm_full_simp_tac (simpset() addsimps [real_mult, |
|
342 |
preal_add_mult_distrib2,preal_mult_1_right] |
|
343 |
@ preal_mult_ac @ preal_add_ac) 1); |
|
344 |
qed "real_mult_0"; |
|
345 |
||
346 |
Goal "z * 0r = 0r"; |
|
347 |
by (simp_tac (simpset() addsimps [real_mult_commute, |
|
348 |
real_mult_0]) 1); |
|
349 |
qed "real_mult_0_right"; |
|
350 |
||
351 |
Addsimps [real_mult_0_right,real_mult_0]; |
|
352 |
||
353 |
Goal "%~(x * y) = %~x * y"; |
|
354 |
by (res_inst_tac [("z","x")] eq_Abs_real 1); |
|
355 |
by (res_inst_tac [("z","y")] eq_Abs_real 1); |
|
356 |
by (auto_tac (claset(),simpset() addsimps [real_minus,real_mult] |
|
357 |
@ preal_mult_ac @ preal_add_ac)); |
|
358 |
qed "real_minus_mult_eq1"; |
|
359 |
||
360 |
Goal "%~(x * y) = x * %~y"; |
|
361 |
by (res_inst_tac [("z","x")] eq_Abs_real 1); |
|
362 |
by (res_inst_tac [("z","y")] eq_Abs_real 1); |
|
363 |
by (auto_tac (claset(),simpset() addsimps [real_minus,real_mult] |
|
364 |
@ preal_mult_ac @ preal_add_ac)); |
|
365 |
qed "real_minus_mult_eq2"; |
|
366 |
||
367 |
Goal "%~x*%~y = x*y"; |
|
368 |
by (full_simp_tac (simpset() addsimps [real_minus_mult_eq2 RS sym, |
|
369 |
real_minus_mult_eq1 RS sym]) 1); |
|
370 |
qed "real_minus_mult_cancel"; |
|
371 |
||
372 |
Addsimps [real_minus_mult_cancel]; |
|
373 |
||
374 |
Goal "%~x*y = x*%~y"; |
|
375 |
by (full_simp_tac (simpset() addsimps [real_minus_mult_eq2 RS sym, |
|
376 |
real_minus_mult_eq1 RS sym]) 1); |
|
377 |
qed "real_minus_mult_commute"; |
|
378 |
||
379 |
(*----------------------------------------------------------------------------- |
|
380 |
||
381 |
-----------------------------------------------------------------------------*) |
|
382 |
||
383 |
(** Lemmas **) |
|
384 |
||
385 |
qed_goal "real_add_assoc_cong" thy |
|
386 |
"!!z. (z::real) + v = z' + v' ==> z + (v + w) = z' + (v' + w)" |
|
387 |
(fn _ => [(asm_simp_tac (simpset() addsimps [real_add_assoc RS sym]) 1)]); |
|
388 |
||
389 |
qed_goal "real_add_assoc_swap" thy "(z::real) + (v + w) = v + (z + w)" |
|
390 |
(fn _ => [(REPEAT (ares_tac [real_add_commute RS real_add_assoc_cong] 1))]); |
|
391 |
||
392 |
Goal "((z1::real) + z2) * w = (z1 * w) + (z2 * w)"; |
|
393 |
by (res_inst_tac [("z","z1")] eq_Abs_real 1); |
|
394 |
by (res_inst_tac [("z","z2")] eq_Abs_real 1); |
|
395 |
by (res_inst_tac [("z","w")] eq_Abs_real 1); |
|
396 |
by (asm_simp_tac |
|
397 |
(simpset() addsimps ([preal_add_mult_distrib2, real_add, real_mult] @ |
|
398 |
preal_add_ac @ preal_mult_ac)) 1); |
|
399 |
qed "real_add_mult_distrib"; |
|
400 |
||
401 |
val real_mult_commute'= read_instantiate [("z","w")] real_mult_commute; |
|
402 |
||
403 |
Goal "(w::real) * (z1 + z2) = (w * z1) + (w * z2)"; |
|
404 |
by (simp_tac (simpset() addsimps [real_mult_commute',real_add_mult_distrib]) 1); |
|
405 |
qed "real_add_mult_distrib2"; |
|
406 |
||
407 |
val real_mult_simps = [real_mult_1, real_mult_1_right]; |
|
408 |
Addsimps real_mult_simps; |
|
409 |
||
410 |
(*** one and zero are distinct ***) |
|
411 |
Goalw [real_zero_def,real_one_def] "0r ~= 1r"; |
|
412 |
by (auto_tac (claset(),simpset() addsimps |
|
413 |
[preal_self_less_add_left RS preal_not_refl2])); |
|
414 |
qed "real_zero_not_eq_one"; |
|
415 |
||
416 |
(*** existence of inverse ***) |
|
417 |
(** lemma -- alternative definition for 0r **) |
|
418 |
Goalw [real_zero_def] "0r = Abs_real (realrel ^^ {(x, x)})"; |
|
419 |
by (auto_tac (claset(),simpset() addsimps [preal_add_commute])); |
|
420 |
qed "real_zero_iff"; |
|
421 |
||
422 |
Goalw [real_zero_def,real_one_def] |
|
423 |
"!!(x::real). x ~= 0r ==> ? y. x*y = 1r"; |
|
424 |
by (res_inst_tac [("z","x")] eq_Abs_real 1); |
|
425 |
by (cut_inst_tac [("r1.0","xa"),("r2.0","y")] preal_linear 1); |
|
426 |
by (auto_tac (claset() addSDs [preal_less_add_left_Ex], |
|
427 |
simpset() addsimps [real_zero_iff RS sym])); |
|
428 |
by (res_inst_tac [("x","Abs_real (realrel ^^ {(@#$#1p,pinv(D)+@#$#1p)})")] exI 1); |
|
429 |
by (res_inst_tac [("x","Abs_real (realrel ^^ {(pinv(D)+@#$#1p,@#$#1p)})")] exI 2); |
|
430 |
by (auto_tac (claset(),simpset() addsimps [real_mult, |
|
431 |
pnat_one_def,preal_mult_1_right,preal_add_mult_distrib2, |
|
432 |
preal_add_mult_distrib,preal_mult_1,preal_mult_inv_right] |
|
433 |
@ preal_add_ac @ preal_mult_ac)); |
|
434 |
qed "real_mult_inv_right_ex"; |
|
435 |
||
436 |
Goal "!!(x::real). x ~= 0r ==> ? y. y*x = 1r"; |
|
437 |
by (asm_simp_tac (simpset() addsimps [real_mult_commute, |
|
438 |
real_mult_inv_right_ex]) 1); |
|
439 |
qed "real_mult_inv_left_ex"; |
|
440 |
||
441 |
Goalw [rinv_def] "!!(x::real). x ~= 0r ==> rinv(x)*x = 1r"; |
|
442 |
by (forward_tac [real_mult_inv_left_ex] 1); |
|
443 |
by (Step_tac 1); |
|
444 |
by (rtac selectI2 1); |
|
445 |
by Auto_tac; |
|
446 |
qed "real_mult_inv_left"; |
|
447 |
||
448 |
Goal "!!(x::real). x ~= 0r ==> x*rinv(x) = 1r"; |
|
449 |
by (auto_tac (claset() addIs [real_mult_commute RS subst], |
|
450 |
simpset() addsimps [real_mult_inv_left])); |
|
451 |
qed "real_mult_inv_right"; |
|
452 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
453 |
Goal "(c::real) ~= 0r ==> (c*a=c*b) = (a=b)"; |
5078 | 454 |
by Auto_tac; |
455 |
by (dres_inst_tac [("f","%x. x*rinv c")] arg_cong 1); |
|
456 |
by (asm_full_simp_tac (simpset() addsimps [real_mult_inv_right] @ real_mult_ac) 1); |
|
457 |
qed "real_mult_left_cancel"; |
|
458 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
459 |
Goal "(c::real) ~= 0r ==> (a*c=b*c) = (a=b)"; |
5078 | 460 |
by (Step_tac 1); |
461 |
by (dres_inst_tac [("f","%x. x*rinv c")] arg_cong 1); |
|
462 |
by (asm_full_simp_tac (simpset() addsimps [real_mult_inv_right] @ real_mult_ac) 1); |
|
463 |
qed "real_mult_right_cancel"; |
|
464 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
465 |
Goalw [rinv_def] "x ~= 0r ==> rinv(x) ~= 0r"; |
5078 | 466 |
by (forward_tac [real_mult_inv_left_ex] 1); |
467 |
by (etac exE 1); |
|
468 |
by (rtac selectI2 1); |
|
469 |
by (auto_tac (claset(),simpset() addsimps [real_mult_0, |
|
470 |
real_zero_not_eq_one])); |
|
471 |
qed "rinv_not_zero"; |
|
472 |
||
473 |
Addsimps [real_mult_inv_left,real_mult_inv_right]; |
|
474 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
475 |
Goal "x ~= 0r ==> rinv(rinv x) = x"; |
5078 | 476 |
by (res_inst_tac [("c1","rinv x")] (real_mult_right_cancel RS iffD1) 1); |
477 |
by (etac rinv_not_zero 1); |
|
478 |
by (auto_tac (claset() addDs [rinv_not_zero],simpset())); |
|
479 |
qed "real_rinv_rinv"; |
|
480 |
||
481 |
Goalw [rinv_def] "rinv(1r) = 1r"; |
|
482 |
by (cut_facts_tac [real_zero_not_eq_one RS |
|
483 |
not_sym RS real_mult_inv_left_ex] 1); |
|
484 |
by (etac exE 1); |
|
485 |
by (rtac selectI2 1); |
|
486 |
by (auto_tac (claset(),simpset() addsimps |
|
487 |
[real_zero_not_eq_one RS not_sym])); |
|
488 |
qed "real_rinv_1"; |
|
489 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
490 |
Goal "x ~= 0r ==> rinv(%~x) = %~rinv(x)"; |
5078 | 491 |
by (res_inst_tac [("c1","%~x")] (real_mult_right_cancel RS iffD1) 1); |
492 |
by Auto_tac; |
|
493 |
qed "real_minus_rinv"; |
|
494 |
||
495 |
(*** theorems for ordering ***) |
|
496 |
(* prove introduction and elimination rules for real_less *) |
|
497 |
||
498 |
Goalw [real_less_def] |
|
499 |
"P < (Q::real) = (EX x1 y1 x2 y2. x1 + y2 < x2 + y1 & \ |
|
500 |
\ (x1,y1::preal):Rep_real(P) & \ |
|
501 |
\ (x2,y2):Rep_real(Q))"; |
|
502 |
by (Fast_tac 1); |
|
503 |
qed "real_less_iff"; |
|
504 |
||
505 |
Goalw [real_less_def] |
|
5148
74919e8f221c
More tidying and removal of "\!\!... from Goal commands
paulson
parents:
5143
diff
changeset
|
506 |
"[| x1 + y2 < x2 + y1; (x1,y1::preal):Rep_real(P); \ |
5078 | 507 |
\ (x2,y2):Rep_real(Q) |] ==> P < (Q::real)"; |
508 |
by (Fast_tac 1); |
|
509 |
qed "real_lessI"; |
|
510 |
||
511 |
Goalw [real_less_def] |
|
5148
74919e8f221c
More tidying and removal of "\!\!... from Goal commands
paulson
parents:
5143
diff
changeset
|
512 |
"!!P. [| R1 < (R2::real); \ |
5078 | 513 |
\ !!x1 x2 y1 y2. x1 + y2 < x2 + y1 ==> P; \ |
514 |
\ !!x1 y1. (x1,y1::preal):Rep_real(R1) ==> P; \ |
|
515 |
\ !!x2 y2. (x2,y2::preal):Rep_real(R2) ==> P |] \ |
|
516 |
\ ==> P"; |
|
517 |
by Auto_tac; |
|
518 |
qed "real_lessE"; |
|
519 |
||
520 |
Goalw [real_less_def] |
|
5148
74919e8f221c
More tidying and removal of "\!\!... from Goal commands
paulson
parents:
5143
diff
changeset
|
521 |
"R1 < (R2::real) ==> (EX x1 y1 x2 y2. x1 + y2 < x2 + y1 & \ |
5078 | 522 |
\ (x1,y1::preal):Rep_real(R1) & \ |
523 |
\ (x2,y2):Rep_real(R2))"; |
|
524 |
by (Fast_tac 1); |
|
525 |
qed "real_lessD"; |
|
526 |
||
527 |
(* real_less is a strong order i.e nonreflexive and transitive *) |
|
528 |
(*** lemmas ***) |
|
529 |
Goal "!!(x::preal). [| x = y; x1 = y1 |] ==> x + y1 = x1 + y"; |
|
530 |
by (asm_simp_tac (simpset() addsimps [preal_add_commute]) 1); |
|
531 |
qed "preal_lemma_eq_rev_sum"; |
|
532 |
||
533 |
Goal "!!(b::preal). x + (b + y) = x1 + (b + y1) ==> x + y = x1 + y1"; |
|
534 |
by (asm_full_simp_tac (simpset() addsimps preal_add_ac) 1); |
|
535 |
qed "preal_add_left_commute_cancel"; |
|
536 |
||
537 |
Goal |
|
538 |
"!!(x::preal). [| x + y2a = x2a + y; \ |
|
539 |
\ x + y2b = x2b + y |] \ |
|
540 |
\ ==> x2a + y2b = x2b + y2a"; |
|
541 |
by (dtac preal_lemma_eq_rev_sum 1); |
|
542 |
by (assume_tac 1); |
|
543 |
by (thin_tac "x + y2b = x2b + y" 1); |
|
544 |
by (asm_full_simp_tac (simpset() addsimps preal_add_ac) 1); |
|
545 |
by (dtac preal_add_left_commute_cancel 1); |
|
546 |
by (asm_full_simp_tac (simpset() addsimps preal_add_ac) 1); |
|
547 |
qed "preal_lemma_for_not_refl"; |
|
548 |
||
549 |
Goal "~ (R::real) < R"; |
|
550 |
by (res_inst_tac [("z","R")] eq_Abs_real 1); |
|
551 |
by (auto_tac (claset(),simpset() addsimps [real_less_def])); |
|
552 |
by (dtac preal_lemma_for_not_refl 1); |
|
553 |
by (assume_tac 1 THEN rotate_tac 2 1); |
|
554 |
by (auto_tac (claset(),simpset() addsimps [preal_less_not_refl])); |
|
555 |
qed "real_less_not_refl"; |
|
556 |
||
557 |
(*** y < y ==> P ***) |
|
558 |
bind_thm("real_less_irrefl",real_less_not_refl RS notE); |
|
559 |
||
560 |
Goal "!!(x::real). x < y ==> x ~= y"; |
|
561 |
by (auto_tac (claset(),simpset() addsimps [real_less_not_refl])); |
|
562 |
qed "real_not_refl2"; |
|
563 |
||
564 |
(* lemma re-arranging and eliminating terms *) |
|
565 |
Goal "!! (a::preal). [| a + b = c + d; \ |
|
566 |
\ x2b + d + (c + y2e) < a + y2b + (x2e + b) |] \ |
|
567 |
\ ==> x2b + y2e < x2e + y2b"; |
|
568 |
by (asm_full_simp_tac (simpset() addsimps preal_add_ac) 1); |
|
569 |
by (res_inst_tac [("C","c+d")] preal_add_left_less_cancel 1); |
|
570 |
by (asm_full_simp_tac (simpset() addsimps [preal_add_assoc RS sym]) 1); |
|
571 |
qed "preal_lemma_trans"; |
|
572 |
||
573 |
(** heavy re-writing involved*) |
|
574 |
Goal "!!(R1::real). [| R1 < R2; R2 < R3 |] ==> R1 < R3"; |
|
575 |
by (res_inst_tac [("z","R1")] eq_Abs_real 1); |
|
576 |
by (res_inst_tac [("z","R2")] eq_Abs_real 1); |
|
577 |
by (res_inst_tac [("z","R3")] eq_Abs_real 1); |
|
578 |
by (auto_tac (claset(),simpset() addsimps [real_less_def])); |
|
579 |
by (REPEAT(rtac exI 1)); |
|
580 |
by (EVERY[rtac conjI 1, rtac conjI 2]); |
|
581 |
by (REPEAT(Blast_tac 2)); |
|
582 |
by (dtac preal_lemma_for_not_refl 1 THEN assume_tac 1); |
|
583 |
by (blast_tac (claset() addDs [preal_add_less_mono] |
|
584 |
addIs [preal_lemma_trans]) 1); |
|
585 |
qed "real_less_trans"; |
|
586 |
||
587 |
Goal "!! (R1::real). [| R1 < R2; R2 < R1 |] ==> P"; |
|
588 |
by (dtac real_less_trans 1 THEN assume_tac 1); |
|
589 |
by (asm_full_simp_tac (simpset() addsimps [real_less_not_refl]) 1); |
|
590 |
qed "real_less_asym"; |
|
591 |
||
592 |
(****)(****)(****)(****)(****)(****)(****)(****)(****)(****) |
|
593 |
(****** Map and more real_less ******) |
|
594 |
(*** mapping from preal into real ***) |
|
595 |
Goalw [real_preal_def] |
|
596 |
"%#((z1::preal) + z2) = %#z1 + %#z2"; |
|
597 |
by (asm_simp_tac (simpset() addsimps [real_add, |
|
598 |
preal_add_mult_distrib,preal_mult_1] addsimps preal_add_ac) 1); |
|
599 |
qed "real_preal_add"; |
|
600 |
||
601 |
Goalw [real_preal_def] |
|
602 |
"%#((z1::preal) * z2) = %#z1* %#z2"; |
|
603 |
by (full_simp_tac (simpset() addsimps [real_mult, |
|
604 |
preal_add_mult_distrib2,preal_mult_1, |
|
605 |
preal_mult_1_right,pnat_one_def] |
|
606 |
@ preal_add_ac @ preal_mult_ac) 1); |
|
607 |
qed "real_preal_mult"; |
|
608 |
||
609 |
Goalw [real_preal_def] |
|
610 |
"!!(x::preal). y < x ==> ? m. Abs_real (realrel ^^ {(x,y)}) = %#m"; |
|
611 |
by (auto_tac (claset() addSDs [preal_less_add_left_Ex], |
|
612 |
simpset() addsimps preal_add_ac)); |
|
613 |
qed "real_preal_ExI"; |
|
614 |
||
615 |
Goalw [real_preal_def] |
|
616 |
"!!(x::preal). ? m. Abs_real (realrel ^^ {(x,y)}) = %#m ==> y < x"; |
|
617 |
by (auto_tac (claset(),simpset() addsimps |
|
618 |
[preal_add_commute,preal_add_assoc])); |
|
619 |
by (asm_full_simp_tac (simpset() addsimps |
|
620 |
[preal_add_assoc RS sym,preal_self_less_add_left]) 1); |
|
621 |
qed "real_preal_ExD"; |
|
622 |
||
623 |
Goal "(? m. Abs_real (realrel ^^ {(x,y)}) = %#m) = (y < x)"; |
|
624 |
by (fast_tac (claset() addSIs [real_preal_ExI,real_preal_ExD]) 1); |
|
625 |
qed "real_preal_iff"; |
|
626 |
||
627 |
(*** Gleason prop 9-4.4 p 127 ***) |
|
628 |
Goalw [real_preal_def,real_zero_def] |
|
629 |
"? m. (x::real) = %#m | x = 0r | x = %~(%#m)"; |
|
630 |
by (res_inst_tac [("z","x")] eq_Abs_real 1); |
|
631 |
by (auto_tac (claset(),simpset() addsimps [real_minus] @ preal_add_ac)); |
|
632 |
by (cut_inst_tac [("r1.0","x"),("r2.0","y")] preal_linear 1); |
|
633 |
by (auto_tac (claset() addSDs [preal_less_add_left_Ex], |
|
634 |
simpset() addsimps [preal_add_assoc RS sym])); |
|
635 |
by (auto_tac (claset(),simpset() addsimps [preal_add_commute])); |
|
636 |
qed "real_preal_trichotomy"; |
|
637 |
||
5148
74919e8f221c
More tidying and removal of "\!\!... from Goal commands
paulson
parents:
5143
diff
changeset
|
638 |
Goal "!!P. [| !!m. x = %#m ==> P; \ |
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
639 |
\ x = 0r ==> P; \ |
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
640 |
\ !!m. x = %~(%#m) ==> P |] ==> P"; |
5078 | 641 |
by (cut_inst_tac [("x","x")] real_preal_trichotomy 1); |
642 |
by Auto_tac; |
|
643 |
qed "real_preal_trichotomyE"; |
|
644 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
645 |
Goalw [real_preal_def] "%#m1 < %#m2 ==> m1 < m2"; |
5078 | 646 |
by (auto_tac (claset(),simpset() addsimps [real_less_def] @ preal_add_ac)); |
647 |
by (auto_tac (claset(),simpset() addsimps [preal_add_assoc RS sym])); |
|
648 |
by (auto_tac (claset(),simpset() addsimps preal_add_ac)); |
|
649 |
qed "real_preal_lessD"; |
|
650 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
651 |
Goal "m1 < m2 ==> %#m1 < %#m2"; |
5078 | 652 |
by (dtac preal_less_add_left_Ex 1); |
653 |
by (auto_tac (claset(),simpset() addsimps [real_preal_add, |
|
654 |
real_preal_def,real_less_def])); |
|
655 |
by (REPEAT(rtac exI 1)); |
|
656 |
by (EVERY[rtac conjI 1, rtac conjI 2]); |
|
657 |
by (REPEAT(Fast_tac 2)); |
|
658 |
by (simp_tac (simpset() addsimps [preal_self_less_add_left] |
|
659 |
delsimps [preal_add_less_iff2]) 1); |
|
660 |
qed "real_preal_lessI"; |
|
661 |
||
662 |
Goal "(%#m1 < %#m2) = (m1 < m2)"; |
|
663 |
by (fast_tac (claset() addIs [real_preal_lessI,real_preal_lessD]) 1); |
|
664 |
qed "real_preal_less_iff1"; |
|
665 |
||
666 |
Addsimps [real_preal_less_iff1]; |
|
667 |
||
668 |
Goal "%~ %#m < %#m"; |
|
669 |
by (auto_tac (claset(),simpset() addsimps |
|
670 |
[real_preal_def,real_less_def,real_minus])); |
|
671 |
by (REPEAT(rtac exI 1)); |
|
672 |
by (EVERY[rtac conjI 1, rtac conjI 2]); |
|
673 |
by (REPEAT(Fast_tac 2)); |
|
674 |
by (full_simp_tac (simpset() addsimps preal_add_ac) 1); |
|
675 |
by (full_simp_tac (simpset() addsimps [preal_self_less_add_right, |
|
676 |
preal_add_assoc RS sym]) 1); |
|
677 |
qed "real_preal_minus_less_self"; |
|
678 |
||
679 |
Goalw [real_zero_def] "%~ %#m < 0r"; |
|
680 |
by (auto_tac (claset(),simpset() addsimps |
|
681 |
[real_preal_def,real_less_def,real_minus])); |
|
682 |
by (REPEAT(rtac exI 1)); |
|
683 |
by (EVERY[rtac conjI 1, rtac conjI 2]); |
|
684 |
by (REPEAT(Fast_tac 2)); |
|
685 |
by (full_simp_tac (simpset() addsimps |
|
686 |
[preal_self_less_add_right] @ preal_add_ac) 1); |
|
687 |
qed "real_preal_minus_less_zero"; |
|
688 |
||
689 |
Goal "~ 0r < %~ %#m"; |
|
690 |
by (cut_facts_tac [real_preal_minus_less_zero] 1); |
|
691 |
by (fast_tac (claset() addDs [real_less_trans] |
|
692 |
addEs [real_less_irrefl]) 1); |
|
693 |
qed "real_preal_not_minus_gt_zero"; |
|
694 |
||
695 |
Goalw [real_zero_def] " 0r < %#m"; |
|
696 |
by (auto_tac (claset(),simpset() addsimps |
|
697 |
[real_preal_def,real_less_def,real_minus])); |
|
698 |
by (REPEAT(rtac exI 1)); |
|
699 |
by (EVERY[rtac conjI 1, rtac conjI 2]); |
|
700 |
by (REPEAT(Fast_tac 2)); |
|
701 |
by (full_simp_tac (simpset() addsimps |
|
702 |
[preal_self_less_add_right] @ preal_add_ac) 1); |
|
703 |
qed "real_preal_zero_less"; |
|
704 |
||
705 |
Goal "~ %#m < 0r"; |
|
706 |
by (cut_facts_tac [real_preal_zero_less] 1); |
|
707 |
by (fast_tac (claset() addDs [real_less_trans] |
|
708 |
addEs [real_less_irrefl]) 1); |
|
709 |
qed "real_preal_not_less_zero"; |
|
710 |
||
711 |
Goal "0r < %~ %~ %#m"; |
|
712 |
by (simp_tac (simpset() addsimps |
|
713 |
[real_preal_zero_less]) 1); |
|
714 |
qed "real_minus_minus_zero_less"; |
|
715 |
||
716 |
(* another lemma *) |
|
717 |
Goalw [real_zero_def] " 0r < %#m + %#m1"; |
|
718 |
by (auto_tac (claset(),simpset() addsimps |
|
719 |
[real_preal_def,real_less_def,real_add])); |
|
720 |
by (REPEAT(rtac exI 1)); |
|
721 |
by (EVERY[rtac conjI 1, rtac conjI 2]); |
|
722 |
by (REPEAT(Fast_tac 2)); |
|
723 |
by (full_simp_tac (simpset() addsimps preal_add_ac) 1); |
|
724 |
by (full_simp_tac (simpset() addsimps [preal_self_less_add_right, |
|
725 |
preal_add_assoc RS sym]) 1); |
|
726 |
qed "real_preal_sum_zero_less"; |
|
727 |
||
728 |
Goal "%~ %#m < %#m1"; |
|
729 |
by (auto_tac (claset(),simpset() addsimps |
|
730 |
[real_preal_def,real_less_def,real_minus])); |
|
731 |
by (REPEAT(rtac exI 1)); |
|
732 |
by (EVERY[rtac conjI 1, rtac conjI 2]); |
|
733 |
by (REPEAT(Fast_tac 2)); |
|
734 |
by (full_simp_tac (simpset() addsimps preal_add_ac) 1); |
|
735 |
by (full_simp_tac (simpset() addsimps [preal_self_less_add_right, |
|
736 |
preal_add_assoc RS sym]) 1); |
|
737 |
qed "real_preal_minus_less_all"; |
|
738 |
||
739 |
Goal "~ %#m < %~ %#m1"; |
|
740 |
by (cut_facts_tac [real_preal_minus_less_all] 1); |
|
741 |
by (fast_tac (claset() addDs [real_less_trans] |
|
742 |
addEs [real_less_irrefl]) 1); |
|
743 |
qed "real_preal_not_minus_gt_all"; |
|
744 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
745 |
Goal "%~ %#m1 < %~ %#m2 ==> %#m2 < %#m1"; |
5078 | 746 |
by (auto_tac (claset(),simpset() addsimps |
747 |
[real_preal_def,real_less_def,real_minus])); |
|
748 |
by (REPEAT(rtac exI 1)); |
|
749 |
by (EVERY[rtac conjI 1, rtac conjI 2]); |
|
750 |
by (REPEAT(Fast_tac 2)); |
|
751 |
by (auto_tac (claset(),simpset() addsimps preal_add_ac)); |
|
752 |
by (asm_full_simp_tac (simpset() addsimps [preal_add_assoc RS sym]) 1); |
|
753 |
by (auto_tac (claset(),simpset() addsimps preal_add_ac)); |
|
754 |
qed "real_preal_minus_less_rev1"; |
|
755 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
756 |
Goal "%#m1 < %#m2 ==> %~ %#m2 < %~ %#m1"; |
5078 | 757 |
by (auto_tac (claset(),simpset() addsimps |
758 |
[real_preal_def,real_less_def,real_minus])); |
|
759 |
by (REPEAT(rtac exI 1)); |
|
760 |
by (EVERY[rtac conjI 1, rtac conjI 2]); |
|
761 |
by (REPEAT(Fast_tac 2)); |
|
762 |
by (auto_tac (claset(),simpset() addsimps preal_add_ac)); |
|
763 |
by (asm_full_simp_tac (simpset() addsimps [preal_add_assoc RS sym]) 1); |
|
764 |
by (auto_tac (claset(),simpset() addsimps preal_add_ac)); |
|
765 |
qed "real_preal_minus_less_rev2"; |
|
766 |
||
767 |
Goal "(%~ %#m1 < %~ %#m2) = (%#m2 < %#m1)"; |
|
768 |
by (blast_tac (claset() addSIs [real_preal_minus_less_rev1, |
|
769 |
real_preal_minus_less_rev2]) 1); |
|
770 |
qed "real_preal_minus_less_rev_iff"; |
|
771 |
||
772 |
Addsimps [real_preal_minus_less_rev_iff]; |
|
773 |
||
774 |
(*** linearity ***) |
|
775 |
Goal "(R1::real) < R2 | R1 = R2 | R2 < R1"; |
|
776 |
by (res_inst_tac [("x","R1")] real_preal_trichotomyE 1); |
|
777 |
by (ALLGOALS(res_inst_tac [("x","R2")] real_preal_trichotomyE)); |
|
778 |
by (auto_tac (claset() addSDs [preal_le_anti_sym], |
|
779 |
simpset() addsimps [preal_less_le_iff,real_preal_minus_less_zero, |
|
780 |
real_preal_zero_less,real_preal_minus_less_all])); |
|
781 |
qed "real_linear"; |
|
782 |
||
5148
74919e8f221c
More tidying and removal of "\!\!... from Goal commands
paulson
parents:
5143
diff
changeset
|
783 |
Goal "!!(R1::real). [| R1 < R2 ==> P; R1 = R2 ==> P; \ |
74919e8f221c
More tidying and removal of "\!\!... from Goal commands
paulson
parents:
5143
diff
changeset
|
784 |
\ R2 < R1 ==> P |] ==> P"; |
5078 | 785 |
by (cut_inst_tac [("R1.0","R1"),("R2.0","R2")] real_linear 1); |
786 |
by Auto_tac; |
|
787 |
qed "real_linear_less2"; |
|
788 |
||
789 |
(*** Properties of <= ***) |
|
790 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
791 |
Goalw [real_le_def] "~(w < z) ==> z <= (w::real)"; |
5078 | 792 |
by (assume_tac 1); |
793 |
qed "real_leI"; |
|
794 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
795 |
Goalw [real_le_def] "z<=w ==> ~(w<(z::real))"; |
5078 | 796 |
by (assume_tac 1); |
797 |
qed "real_leD"; |
|
798 |
||
799 |
val real_leE = make_elim real_leD; |
|
800 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
801 |
Goal "(~(w < z)) = (z <= (w::real))"; |
5078 | 802 |
by (fast_tac (claset() addSIs [real_leI,real_leD]) 1); |
803 |
qed "real_less_le_iff"; |
|
804 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
805 |
Goalw [real_le_def] "~ z <= w ==> w<(z::real)"; |
5078 | 806 |
by (Fast_tac 1); |
807 |
qed "not_real_leE"; |
|
808 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
809 |
Goalw [real_le_def] "z < w ==> z <= (w::real)"; |
5078 | 810 |
by (fast_tac (claset() addEs [real_less_asym]) 1); |
811 |
qed "real_less_imp_le"; |
|
812 |
||
813 |
Goalw [real_le_def] "!!(x::real). x <= y ==> x < y | x = y"; |
|
814 |
by (cut_facts_tac [real_linear] 1); |
|
815 |
by (fast_tac (claset() addEs [real_less_irrefl,real_less_asym]) 1); |
|
816 |
qed "real_le_imp_less_or_eq"; |
|
817 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
818 |
Goalw [real_le_def] "z<w | z=w ==> z <=(w::real)"; |
5078 | 819 |
by (cut_facts_tac [real_linear] 1); |
820 |
by (fast_tac (claset() addEs [real_less_irrefl,real_less_asym]) 1); |
|
821 |
qed "real_less_or_eq_imp_le"; |
|
822 |
||
823 |
Goal "(x <= (y::real)) = (x < y | x=y)"; |
|
824 |
by (REPEAT(ares_tac [iffI, real_less_or_eq_imp_le, real_le_imp_less_or_eq] 1)); |
|
825 |
qed "real_le_eq_less_or_eq"; |
|
826 |
||
827 |
Goal "w <= (w::real)"; |
|
828 |
by (simp_tac (simpset() addsimps [real_le_eq_less_or_eq]) 1); |
|
829 |
qed "real_le_refl"; |
|
830 |
||
831 |
val prems = goal Real.thy "!!i. [| i <= j; j < k |] ==> i < (k::real)"; |
|
832 |
by (dtac real_le_imp_less_or_eq 1); |
|
833 |
by (fast_tac (claset() addIs [real_less_trans]) 1); |
|
834 |
qed "real_le_less_trans"; |
|
835 |
||
836 |
Goal "!! (i::real). [| i < j; j <= k |] ==> i < k"; |
|
837 |
by (dtac real_le_imp_less_or_eq 1); |
|
838 |
by (fast_tac (claset() addIs [real_less_trans]) 1); |
|
839 |
qed "real_less_le_trans"; |
|
840 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
841 |
Goal "[| i <= j; j <= k |] ==> i <= (k::real)"; |
5078 | 842 |
by (EVERY1 [dtac real_le_imp_less_or_eq, dtac real_le_imp_less_or_eq, |
843 |
rtac real_less_or_eq_imp_le, fast_tac (claset() addIs [real_less_trans])]); |
|
844 |
qed "real_le_trans"; |
|
845 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
846 |
Goal "[| z <= w; w <= z |] ==> z = (w::real)"; |
5078 | 847 |
by (EVERY1 [dtac real_le_imp_less_or_eq, dtac real_le_imp_less_or_eq, |
848 |
fast_tac (claset() addEs [real_less_irrefl,real_less_asym])]); |
|
849 |
qed "real_le_anti_sym"; |
|
850 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
851 |
Goal "[| ~ y < x; y ~= x |] ==> x < (y::real)"; |
5078 | 852 |
by (rtac not_real_leE 1); |
853 |
by (fast_tac (claset() addDs [real_le_imp_less_or_eq]) 1); |
|
854 |
qed "not_less_not_eq_real_less"; |
|
855 |
||
856 |
Goal "(0r < %~R) = (R < 0r)"; |
|
857 |
by (res_inst_tac [("x","R")] real_preal_trichotomyE 1); |
|
858 |
by (auto_tac (claset(),simpset() addsimps [real_preal_not_minus_gt_zero, |
|
859 |
real_preal_not_less_zero,real_preal_zero_less, |
|
860 |
real_preal_minus_less_zero])); |
|
861 |
qed "real_minus_zero_less_iff"; |
|
862 |
||
863 |
Addsimps [real_minus_zero_less_iff]; |
|
864 |
||
865 |
Goal "(%~R < 0r) = (0r < R)"; |
|
866 |
by (res_inst_tac [("x","R")] real_preal_trichotomyE 1); |
|
867 |
by (auto_tac (claset(),simpset() addsimps [real_preal_not_minus_gt_zero, |
|
868 |
real_preal_not_less_zero,real_preal_zero_less, |
|
869 |
real_preal_minus_less_zero])); |
|
870 |
qed "real_minus_zero_less_iff2"; |
|
871 |
||
872 |
(** lemma **) |
|
873 |
Goal "(0r < x) = (? y. x = %#y)"; |
|
874 |
by (auto_tac (claset(),simpset() addsimps [real_preal_zero_less])); |
|
875 |
by (cut_inst_tac [("x","x")] real_preal_trichotomy 1); |
|
876 |
by (blast_tac (claset() addSEs [real_less_irrefl, |
|
877 |
real_preal_not_minus_gt_zero RS notE]) 1); |
|
878 |
qed "real_gt_zero_preal_Ex"; |
|
879 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
880 |
Goal "%#z < x ==> ? y. x = %#y"; |
5078 | 881 |
by (blast_tac (claset() addSDs [real_preal_zero_less RS real_less_trans] |
882 |
addIs [real_gt_zero_preal_Ex RS iffD1]) 1); |
|
883 |
qed "real_gt_preal_preal_Ex"; |
|
884 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
885 |
Goal "%#z <= x ==> ? y. x = %#y"; |
5078 | 886 |
by (blast_tac (claset() addDs [real_le_imp_less_or_eq, |
887 |
real_gt_preal_preal_Ex]) 1); |
|
888 |
qed "real_ge_preal_preal_Ex"; |
|
889 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
890 |
Goal "y <= 0r ==> !x. y < %#x"; |
5078 | 891 |
by (auto_tac (claset() addEs [real_le_imp_less_or_eq RS disjE] |
892 |
addIs [real_preal_zero_less RSN(2,real_less_trans)], |
|
893 |
simpset() addsimps [real_preal_zero_less])); |
|
894 |
qed "real_less_all_preal"; |
|
895 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
896 |
Goal "~ 0r < y ==> !x. y < %#x"; |
5078 | 897 |
by (blast_tac (claset() addSIs [real_less_all_preal,real_leI]) 1); |
898 |
qed "real_less_all_real2"; |
|
899 |
||
900 |
(**** Derive alternative definition for real_less ****) |
|
901 |
(** lemma **) |
|
902 |
Goal "!!(R::real). ? A. S + A = R"; |
|
903 |
by (res_inst_tac [("x","%~S + R")] exI 1); |
|
904 |
by (simp_tac (simpset() addsimps [real_add_minus, |
|
905 |
real_add_zero_right] @ real_add_ac) 1); |
|
906 |
qed "real_lemma_add_left_ex"; |
|
907 |
||
908 |
Goal "!!(R::real). R < S ==> ? T. R + T = S"; |
|
909 |
by (res_inst_tac [("x","R")] real_preal_trichotomyE 1); |
|
910 |
by (ALLGOALS(res_inst_tac [("x","S")] real_preal_trichotomyE)); |
|
911 |
by (auto_tac (claset() addSDs [preal_le_anti_sym] addSDs [preal_less_add_left_Ex], |
|
912 |
simpset() addsimps [preal_less_le_iff,real_preal_add,real_minus_add_eq, |
|
913 |
real_preal_minus_less_zero,real_less_not_refl,real_minus_ex,real_add_assoc, |
|
914 |
real_preal_zero_less,real_preal_minus_less_all,real_add_minus_left, |
|
915 |
real_preal_not_less_zero,real_add_zero_left,real_lemma_add_left_ex])); |
|
916 |
qed "real_less_add_left_Ex"; |
|
917 |
||
918 |
Goal "!!(R::real). R < S ==> ? T. 0r < T & R + T = S"; |
|
919 |
by (res_inst_tac [("x","R")] real_preal_trichotomyE 1); |
|
920 |
by (ALLGOALS(res_inst_tac [("x","S")] real_preal_trichotomyE)); |
|
921 |
by (auto_tac (claset() addSDs [preal_less_add_left_Ex], |
|
922 |
simpset() addsimps [real_preal_not_minus_gt_all, |
|
923 |
real_preal_add, real_preal_not_less_zero,real_less_not_refl, |
|
924 |
real_preal_not_minus_gt_zero,real_add_zero_left,real_minus_add_eq])); |
|
925 |
by (res_inst_tac [("x","%#D")] exI 1); |
|
926 |
by (res_inst_tac [("x","%#m+%#ma")] exI 2); |
|
927 |
by (res_inst_tac [("x","%#m")] exI 3); |
|
928 |
by (res_inst_tac [("x","%#D")] exI 4); |
|
929 |
by (auto_tac (claset(),simpset() addsimps [real_preal_zero_less, |
|
930 |
real_preal_sum_zero_less,real_add_minus_left,real_add_assoc, |
|
931 |
real_add_minus,real_add_zero_right])); |
|
932 |
by (simp_tac (simpset() addsimps [real_add_assoc RS sym, |
|
933 |
real_add_minus_left,real_add_zero_left]) 1); |
|
934 |
qed "real_less_add_positive_left_Ex"; |
|
935 |
||
936 |
(* lemmas *) |
|
937 |
(** change naff name(s)! **) |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
938 |
Goal "(W < S) ==> (0r < S + %~W)"; |
5078 | 939 |
by (dtac real_less_add_positive_left_Ex 1); |
940 |
by (auto_tac (claset(),simpset() addsimps [real_add_minus, |
|
941 |
real_add_zero_right] @ real_add_ac)); |
|
942 |
qed "real_less_sum_gt_zero"; |
|
943 |
||
944 |
Goal "!!S. T = S + W ==> S = T + %~W"; |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
945 |
by (asm_simp_tac (simpset() addsimps [real_add_minus, real_add_zero_right] |
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
946 |
@ real_add_ac) 1); |
5078 | 947 |
qed "real_lemma_change_eq_subj"; |
948 |
||
949 |
(* FIXME: long! *) |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
950 |
Goal "(0r < S + %~W) ==> (W < S)"; |
5078 | 951 |
by (rtac ccontr 1); |
952 |
by (dtac (real_leI RS real_le_imp_less_or_eq) 1); |
|
953 |
by (auto_tac (claset(), |
|
954 |
simpset() addsimps [real_less_not_refl,real_add_minus])); |
|
955 |
by (EVERY1[dtac real_less_add_positive_left_Ex, etac exE, etac conjE]); |
|
956 |
by (asm_full_simp_tac (simpset() addsimps [real_add_zero_left]) 1); |
|
957 |
by (dtac real_lemma_change_eq_subj 1); |
|
958 |
by (auto_tac (claset(),simpset() addsimps [real_minus_minus])); |
|
959 |
by (dtac real_less_sum_gt_zero 1); |
|
960 |
by (asm_full_simp_tac (simpset() addsimps [real_minus_add_eq] @ real_add_ac) 1); |
|
961 |
by (EVERY1[rotate_tac 1, dtac (real_add_left_commute RS ssubst)]); |
|
962 |
by (auto_tac (claset() addEs [real_less_asym], |
|
963 |
simpset() addsimps [real_add_minus,real_add_zero_right])); |
|
964 |
qed "real_sum_gt_zero_less"; |
|
965 |
||
966 |
Goal "(0r < S + %~W) = (W < S)"; |
|
967 |
by (fast_tac (claset() addIs [real_less_sum_gt_zero, |
|
968 |
real_sum_gt_zero_less]) 1); |
|
969 |
qed "real_less_sum_gt_0_iff"; |
|
970 |
||
971 |
Goal "((x::real) < y) = (%~y < %~x)"; |
|
972 |
by (rtac (real_less_sum_gt_0_iff RS subst) 1); |
|
973 |
by (res_inst_tac [("W1","x")] (real_less_sum_gt_0_iff RS subst) 1); |
|
974 |
by (simp_tac (simpset() addsimps [real_add_commute]) 1); |
|
975 |
qed "real_less_swap_iff"; |
|
976 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
977 |
Goal "[| R + L = S; 0r < L |] ==> R < S"; |
5078 | 978 |
by (rtac (real_less_sum_gt_0_iff RS iffD1) 1); |
979 |
by (auto_tac (claset(),simpset() addsimps [ |
|
980 |
real_add_minus,real_add_zero_right] @ real_add_ac)); |
|
981 |
qed "real_lemma_add_positive_imp_less"; |
|
982 |
||
983 |
Goal "!!(R::real). ? T. 0r < T & R + T = S ==> R < S"; |
|
984 |
by (blast_tac (claset() addIs [real_lemma_add_positive_imp_less]) 1); |
|
985 |
qed "real_ex_add_positive_left_less"; |
|
986 |
||
987 |
(*** alternative definition for real_less ***) |
|
988 |
Goal "!!(R::real). (? T. 0r < T & R + T = S) = (R < S)"; |
|
989 |
by (fast_tac (claset() addSIs [real_less_add_positive_left_Ex, |
|
990 |
real_ex_add_positive_left_less]) 1); |
|
991 |
qed "real_less_iffdef"; |
|
992 |
||
993 |
Goal "(0r < x) = (%~x < x)"; |
|
994 |
by (Step_tac 1); |
|
995 |
by (rtac ccontr 2 THEN forward_tac |
|
996 |
[real_leI RS real_le_imp_less_or_eq] 2); |
|
997 |
by (Step_tac 2); |
|
998 |
by (dtac (real_minus_zero_less_iff RS iffD2) 2); |
|
999 |
by (fast_tac (claset() addDs [real_less_trans]) 2); |
|
1000 |
by (auto_tac (claset(),simpset() addsimps |
|
1001 |
[real_gt_zero_preal_Ex,real_preal_minus_less_self])); |
|
1002 |
qed "real_gt_zero_iff"; |
|
1003 |
||
1004 |
Goal "(x < 0r) = (x < %~x)"; |
|
1005 |
by (rtac (real_minus_zero_less_iff RS subst) 1); |
|
1006 |
by (stac real_gt_zero_iff 1); |
|
1007 |
by (Full_simp_tac 1); |
|
1008 |
qed "real_lt_zero_iff"; |
|
1009 |
||
1010 |
Goalw [real_le_def] "(0r <= x) = (%~x <= x)"; |
|
1011 |
by (auto_tac (claset(),simpset() addsimps [real_lt_zero_iff RS sym])); |
|
1012 |
qed "real_ge_zero_iff"; |
|
1013 |
||
1014 |
Goalw [real_le_def] "(x <= 0r) = (x <= %~x)"; |
|
1015 |
by (auto_tac (claset(),simpset() addsimps [real_gt_zero_iff RS sym])); |
|
1016 |
qed "real_le_zero_iff"; |
|
1017 |
||
1018 |
Goal "(%#m1 <= %#m2) = (m1 <= m2)"; |
|
1019 |
by (auto_tac (claset() addSIs [preal_leI], |
|
1020 |
simpset() addsimps [real_less_le_iff RS sym])); |
|
1021 |
by (dtac preal_le_less_trans 1 THEN assume_tac 1); |
|
1022 |
by (etac preal_less_irrefl 1); |
|
1023 |
qed "real_preal_le_iff"; |
|
1024 |
||
1025 |
Goal "!!(x::real). [| 0r < x; 0r < y |] ==> 0r < x * y"; |
|
1026 |
by (auto_tac (claset(),simpset() addsimps [real_gt_zero_preal_Ex])); |
|
1027 |
by (res_inst_tac [("x","y*ya")] exI 1); |
|
1028 |
by (full_simp_tac (simpset() addsimps [real_preal_mult]) 1); |
|
1029 |
qed "real_mult_order"; |
|
1030 |
||
1031 |
Goal "!!(x::real). [| x < 0r; y < 0r |] ==> 0r < x * y"; |
|
1032 |
by (REPEAT(dtac (real_minus_zero_less_iff RS iffD2) 1)); |
|
1033 |
by (dtac real_mult_order 1 THEN assume_tac 1); |
|
1034 |
by (Asm_full_simp_tac 1); |
|
1035 |
qed "real_mult_less_zero1"; |
|
1036 |
||
1037 |
Goal "!!(x::real). [| 0r <= x; 0r <= y |] ==> 0r <= x * y"; |
|
1038 |
by (REPEAT(dtac real_le_imp_less_or_eq 1)); |
|
1039 |
by (auto_tac (claset() addIs [real_mult_order, |
|
1040 |
real_less_imp_le],simpset() addsimps [real_le_refl])); |
|
1041 |
qed "real_le_mult_order"; |
|
1042 |
||
1043 |
Goal "!!(x::real). [| x <= 0r; y <= 0r |] ==> 0r <= x * y"; |
|
1044 |
by (rtac real_less_or_eq_imp_le 1); |
|
1045 |
by (dtac real_le_imp_less_or_eq 1 THEN etac disjE 1); |
|
1046 |
by Auto_tac; |
|
1047 |
by (dtac real_le_imp_less_or_eq 1); |
|
1048 |
by (auto_tac (claset() addDs [real_mult_less_zero1],simpset())); |
|
1049 |
qed "real_mult_le_zero1"; |
|
1050 |
||
1051 |
Goal "!!(x::real). [| 0r <= x; y < 0r |] ==> x * y <= 0r"; |
|
1052 |
by (rtac real_less_or_eq_imp_le 1); |
|
1053 |
by (dtac real_le_imp_less_or_eq 1 THEN etac disjE 1); |
|
1054 |
by Auto_tac; |
|
1055 |
by (dtac (real_minus_zero_less_iff RS iffD2) 1); |
|
1056 |
by (rtac (real_minus_zero_less_iff RS subst) 1); |
|
1057 |
by (blast_tac (claset() addDs [real_mult_order] |
|
1058 |
addIs [real_minus_mult_eq2 RS ssubst]) 1); |
|
1059 |
qed "real_mult_le_zero"; |
|
1060 |
||
1061 |
Goal "!!(x::real). [| 0r < x; y < 0r |] ==> x*y < 0r"; |
|
1062 |
by (dtac (real_minus_zero_less_iff RS iffD2) 1); |
|
1063 |
by (dtac real_mult_order 1 THEN assume_tac 1); |
|
1064 |
by (rtac (real_minus_zero_less_iff RS iffD1) 1); |
|
1065 |
by (asm_full_simp_tac (simpset() addsimps [real_minus_mult_eq2]) 1); |
|
1066 |
qed "real_mult_less_zero"; |
|
1067 |
||
1068 |
Goalw [real_one_def] "0r < 1r"; |
|
1069 |
by (auto_tac (claset() addIs [real_gt_zero_preal_Ex RS iffD2], |
|
1070 |
simpset() addsimps [real_preal_def])); |
|
1071 |
qed "real_zero_less_one"; |
|
1072 |
||
1073 |
(*** Completeness of reals ***) |
|
1074 |
(** use supremum property of preal and theorems about real_preal **) |
|
1075 |
(*** a few lemmas ***) |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
1076 |
Goal "! x:P. 0r < x ==> ((? x:P. y < x) = (? X. %#X : P & y < %#X))"; |
5078 | 1077 |
by (blast_tac (claset() addSDs [bspec,real_gt_zero_preal_Ex RS iffD1]) 1); |
1078 |
qed "real_sup_lemma1"; |
|
1079 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
1080 |
Goal "[| ! x:P. 0r < x; ? x. x: P; ? y. !x: P. x < y |] \ |
5078 | 1081 |
\ ==> (? X. X: {w. %#w : P}) & (? Y. !X: {w. %#w : P}. X < Y)"; |
1082 |
by (rtac conjI 1); |
|
1083 |
by (blast_tac (claset() addDs [bspec,real_gt_zero_preal_Ex RS iffD1]) 1); |
|
1084 |
by Auto_tac; |
|
1085 |
by (dtac bspec 1 THEN assume_tac 1); |
|
1086 |
by (forward_tac [bspec] 1 THEN assume_tac 1); |
|
1087 |
by (dtac real_less_trans 1 THEN assume_tac 1); |
|
1088 |
by (dtac (real_gt_zero_preal_Ex RS iffD1) 1 THEN etac exE 1); |
|
1089 |
by (res_inst_tac [("x","ya")] exI 1); |
|
1090 |
by Auto_tac; |
|
1091 |
by (dres_inst_tac [("x","%#X")] bspec 1 THEN assume_tac 1); |
|
1092 |
by (etac real_preal_lessD 1); |
|
1093 |
qed "real_sup_lemma2"; |
|
1094 |
||
1095 |
(*------------------------------------------------------------- |
|
1096 |
Completeness of Positive Reals |
|
1097 |
-------------------------------------------------------------*) |
|
1098 |
||
1099 |
(* Supremum property for the set of positive reals *) |
|
1100 |
(* FIXME: long proof - can be improved - need only have one case split *) |
|
1101 |
(* will do for now *) |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
1102 |
Goal "[| ! x:P. 0r < x; ? x. x: P; ? y. !x: P. x < y |] \ |
5078 | 1103 |
\ ==> (? S. !y. (? x: P. y < x) = (y < S))"; |
1104 |
by (res_inst_tac [("x","%#psup({w. %#w : P})")] exI 1); |
|
1105 |
by Auto_tac; |
|
1106 |
by (forward_tac [real_sup_lemma2] 1 THEN Auto_tac); |
|
1107 |
by (case_tac "0r < ya" 1); |
|
1108 |
by (dtac (real_gt_zero_preal_Ex RS iffD1) 1); |
|
1109 |
by (dtac real_less_all_real2 2); |
|
1110 |
by Auto_tac; |
|
1111 |
by (rtac (preal_complete RS spec RS iffD1) 1); |
|
1112 |
by Auto_tac; |
|
1113 |
by (forward_tac [real_gt_preal_preal_Ex] 1); |
|
1114 |
by Auto_tac; |
|
1115 |
(* second part *) |
|
1116 |
by (rtac (real_sup_lemma1 RS iffD2) 1 THEN assume_tac 1); |
|
1117 |
by (case_tac "0r < ya" 1); |
|
1118 |
by (auto_tac (claset() addSDs [real_less_all_real2, |
|
1119 |
real_gt_zero_preal_Ex RS iffD1],simpset())); |
|
1120 |
by (forward_tac [real_sup_lemma2] 2 THEN Auto_tac); |
|
1121 |
by (forward_tac [real_sup_lemma2] 1 THEN Auto_tac); |
|
1122 |
by (rtac (preal_complete RS spec RS iffD2 RS bexE) 1); |
|
1123 |
by (Fast_tac 3); |
|
1124 |
by (Fast_tac 1); |
|
1125 |
by (Fast_tac 1); |
|
1126 |
by (Blast_tac 1); |
|
1127 |
qed "posreal_complete"; |
|
1128 |
||
1129 |
(*------------------------------------------------------------------ |
|
1130 |
||
1131 |
------------------------------------------------------------------*) |
|
1132 |
||
1133 |
Goal "!!(A::real). A < B ==> A + C < B + C"; |
|
1134 |
by (dtac (real_less_iffdef RS iffD2) 1); |
|
1135 |
by (rtac (real_less_iffdef RS iffD1) 1); |
|
1136 |
by (REPEAT(Step_tac 1)); |
|
1137 |
by (full_simp_tac (simpset() addsimps real_add_ac) 1); |
|
1138 |
qed "real_add_less_mono1"; |
|
1139 |
||
1140 |
Goal "!!(A::real). A < B ==> C + A < C + B"; |
|
1141 |
by (auto_tac (claset() addIs [real_add_less_mono1], |
|
1142 |
simpset() addsimps [real_add_commute])); |
|
1143 |
qed "real_add_less_mono2"; |
|
1144 |
||
1145 |
Goal "!!(A::real). A + C < B + C ==> A < B"; |
|
1146 |
by (dres_inst_tac [("C","%~C")] real_add_less_mono1 1); |
|
1147 |
by (asm_full_simp_tac (simpset() addsimps [real_add_assoc, |
|
1148 |
real_add_minus,real_add_zero_right]) 1); |
|
1149 |
qed "real_less_add_right_cancel"; |
|
1150 |
||
1151 |
Goal "!!(A::real). C + A < C + B ==> A < B"; |
|
1152 |
by (dres_inst_tac [("C","%~C")] real_add_less_mono2 1); |
|
1153 |
by (asm_full_simp_tac (simpset() addsimps [real_add_assoc RS sym, |
|
1154 |
real_add_minus_left,real_add_zero_left]) 1); |
|
1155 |
qed "real_less_add_left_cancel"; |
|
1156 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
1157 |
Goal "[| 0r < x; 0r < y |] ==> 0r < x + y"; |
5078 | 1158 |
by (REPEAT(dtac (real_gt_zero_preal_Ex RS iffD1) 1)); |
1159 |
by (rtac (real_gt_zero_preal_Ex RS iffD2) 1); |
|
1160 |
by (Step_tac 1); |
|
1161 |
by (res_inst_tac [("x","y + ya")] exI 1); |
|
1162 |
by (full_simp_tac (simpset() addsimps [real_preal_add]) 1); |
|
1163 |
qed "real_add_order"; |
|
1164 |
||
1165 |
Goal "!!(x::real). [| 0r <= x; 0r <= y |] ==> 0r <= x + y"; |
|
1166 |
by (REPEAT(dtac real_le_imp_less_or_eq 1)); |
|
1167 |
by (auto_tac (claset() addIs [real_add_order, |
|
1168 |
real_less_imp_le],simpset() addsimps [real_add_zero_left, |
|
1169 |
real_add_zero_right,real_le_refl])); |
|
1170 |
qed "real_le_add_order"; |
|
1171 |
||
1172 |
Goal |
|
5148
74919e8f221c
More tidying and removal of "\!\!... from Goal commands
paulson
parents:
5143
diff
changeset
|
1173 |
"[| R1 < S1; R2 < S2 |] ==> R1 + R2 < S1 + (S2::real)"; |
5078 | 1174 |
by (dtac (real_less_iffdef RS iffD2) 1); |
1175 |
by (dtac (real_less_iffdef RS iffD2) 1); |
|
1176 |
by (rtac (real_less_iffdef RS iffD1) 1); |
|
1177 |
by Auto_tac; |
|
1178 |
by (res_inst_tac [("x","T + Ta")] exI 1); |
|
1179 |
by (auto_tac (claset(),simpset() addsimps [real_add_order] @ real_add_ac)); |
|
1180 |
qed "real_add_less_mono"; |
|
1181 |
||
1182 |
Goal "!!(x::real). [| 0r <= x; 0r <= y |] ==> 0r <= x + y"; |
|
1183 |
by (REPEAT(dtac real_le_imp_less_or_eq 1)); |
|
1184 |
by (auto_tac (claset() addIs [real_add_order, |
|
1185 |
real_less_imp_le],simpset() addsimps [real_add_zero_left, |
|
1186 |
real_add_zero_right,real_le_refl])); |
|
1187 |
qed "real_le_add_order"; |
|
1188 |
||
1189 |
Goal "!!(q1::real). q1 <= q2 ==> x + q1 <= x + q2"; |
|
1190 |
by (dtac real_le_imp_less_or_eq 1); |
|
1191 |
by (Step_tac 1); |
|
1192 |
by (auto_tac (claset() addSIs [real_le_refl, |
|
1193 |
real_less_imp_le,real_add_less_mono1], |
|
1194 |
simpset() addsimps [real_add_commute])); |
|
1195 |
qed "real_add_left_le_mono1"; |
|
1196 |
||
1197 |
Goal "!!(q1::real). q1 <= q2 ==> q1 + x <= q2 + x"; |
|
1198 |
by (auto_tac (claset() addDs [real_add_left_le_mono1], |
|
1199 |
simpset() addsimps [real_add_commute])); |
|
1200 |
qed "real_add_le_mono1"; |
|
1201 |
||
1202 |
Goal "!!k l::real. [|i<=j; k<=l |] ==> i + k <= j + l"; |
|
1203 |
by (etac (real_add_le_mono1 RS real_le_trans) 1); |
|
1204 |
by (simp_tac (simpset() addsimps [real_add_commute]) 1); |
|
1205 |
(*j moves to the end because it is free while k, l are bound*) |
|
1206 |
by (etac real_add_le_mono1 1); |
|
1207 |
qed "real_add_le_mono"; |
|
1208 |
||
1209 |
Goal "EX (x::real). x < y"; |
|
1210 |
by (rtac (real_add_zero_right RS subst) 1); |
|
1211 |
by (res_inst_tac [("x","y + %~1r")] exI 1); |
|
1212 |
by (auto_tac (claset() addSIs [real_add_less_mono2], |
|
1213 |
simpset() addsimps [real_minus_zero_less_iff2, |
|
1214 |
real_zero_less_one])); |
|
1215 |
qed "real_less_Ex"; |
|
1216 |
(*--------------------------------------------------------------------------------- |
|
1217 |
An embedding of the naturals in the reals |
|
1218 |
---------------------------------------------------------------------------------*) |
|
1219 |
||
1220 |
Goalw [real_nat_def] "%%#0 = 1r"; |
|
1221 |
by (full_simp_tac (simpset() addsimps [pnat_one_iff RS sym,real_preal_def]) 1); |
|
1222 |
by (fold_tac [real_one_def]); |
|
1223 |
by (rtac refl 1); |
|
1224 |
qed "real_nat_one"; |
|
1225 |
||
1226 |
Goalw [real_nat_def] "%%#1 = 1r + 1r"; |
|
1227 |
by (full_simp_tac (simpset() addsimps [real_preal_def,real_one_def, |
|
1228 |
pnat_two_eq,real_add,prat_pnat_add RS sym,preal_prat_add RS sym |
|
1229 |
] @ pnat_add_ac) 1); |
|
1230 |
qed "real_nat_two"; |
|
1231 |
||
1232 |
Goalw [real_nat_def] |
|
1233 |
"%%#n1 + %%#n2 = %%#(n1 + n2) + 1r"; |
|
1234 |
by (full_simp_tac (simpset() addsimps [real_nat_one RS sym, |
|
1235 |
real_nat_def,real_preal_add RS sym,preal_prat_add RS sym, |
|
1236 |
prat_pnat_add RS sym,pnat_nat_add]) 1); |
|
1237 |
qed "real_nat_add"; |
|
1238 |
||
1239 |
Goal "%%#(n + 1) = %%#n + 1r"; |
|
1240 |
by (res_inst_tac [("x1","1r")] (real_add_right_cancel RS iffD1) 1); |
|
1241 |
by (rtac (real_nat_add RS subst) 1); |
|
1242 |
by (full_simp_tac (simpset() addsimps [real_nat_two,real_add_assoc]) 1); |
|
1243 |
qed "real_nat_add_one"; |
|
1244 |
||
1245 |
Goal "Suc n = n + 1"; |
|
1246 |
by Auto_tac; |
|
1247 |
qed "lemma"; |
|
1248 |
||
1249 |
Goal "%%#Suc n = %%#n + 1r"; |
|
1250 |
by (stac lemma 1); |
|
1251 |
by (rtac real_nat_add_one 1); |
|
1252 |
qed "real_nat_Suc"; |
|
1253 |
||
1254 |
Goal "inj(real_nat)"; |
|
1255 |
by (rtac injI 1); |
|
1256 |
by (rewtac real_nat_def); |
|
1257 |
by (dtac (inj_real_preal RS injD) 1); |
|
1258 |
by (dtac (inj_preal_prat RS injD) 1); |
|
1259 |
by (dtac (inj_prat_pnat RS injD) 1); |
|
1260 |
by (etac (inj_pnat_nat RS injD) 1); |
|
1261 |
qed "inj_real_nat"; |
|
1262 |
||
1263 |
Goalw [real_nat_def] "0r < %%#n"; |
|
1264 |
by (rtac (real_gt_zero_preal_Ex RS iffD2) 1); |
|
1265 |
by (Blast_tac 1); |
|
1266 |
qed "real_nat_less_zero"; |
|
1267 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
1268 |
Goal "1r <= %%#n"; |
5078 | 1269 |
by (simp_tac (simpset() addsimps [real_nat_one RS sym]) 1); |
5184 | 1270 |
by (induct_tac "n" 1); |
5078 | 1271 |
by (auto_tac (claset(),simpset () |
1272 |
addsimps [real_nat_Suc,real_le_refl,real_nat_one])); |
|
1273 |
by (res_inst_tac [("t","1r")] (real_add_zero_left RS subst) 1); |
|
1274 |
by (rtac real_add_le_mono 1); |
|
1275 |
by (auto_tac (claset(),simpset () |
|
1276 |
addsimps [real_le_refl,real_nat_less_zero, |
|
1277 |
real_less_imp_le,real_add_zero_left])); |
|
1278 |
qed "real_nat_less_one"; |
|
1279 |
||
1280 |
Goal "rinv(%%#n) ~= 0r"; |
|
1281 |
by (rtac ((real_nat_less_zero RS |
|
1282 |
real_not_refl2 RS not_sym) RS rinv_not_zero) 1); |
|
1283 |
qed "real_nat_rinv_not_zero"; |
|
1284 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
1285 |
Goal "rinv(%%#x) = rinv(%%#y) ==> x = y"; |
5078 | 1286 |
by (rtac (inj_real_nat RS injD) 1); |
1287 |
by (res_inst_tac [("n2","x")] |
|
1288 |
(real_nat_rinv_not_zero RS real_mult_left_cancel RS iffD1) 1); |
|
1289 |
by (full_simp_tac (simpset() addsimps [(real_nat_less_zero RS |
|
1290 |
real_not_refl2 RS not_sym) RS real_mult_inv_left]) 1); |
|
1291 |
by (asm_full_simp_tac (simpset() addsimps [(real_nat_less_zero RS |
|
1292 |
real_not_refl2 RS not_sym)]) 1); |
|
1293 |
qed "real_nat_rinv_inj"; |
|
1294 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
1295 |
Goal "0r < x ==> 0r < rinv x"; |
5078 | 1296 |
by (EVERY1[rtac ccontr, dtac real_leI]); |
1297 |
by (forward_tac [real_minus_zero_less_iff2 RS iffD2] 1); |
|
1298 |
by (forward_tac [real_not_refl2 RS not_sym] 1); |
|
1299 |
by (dtac (real_not_refl2 RS not_sym RS rinv_not_zero) 1); |
|
1300 |
by (EVERY1[dtac real_le_imp_less_or_eq, Step_tac]); |
|
1301 |
by (dtac real_mult_less_zero1 1 THEN assume_tac 1); |
|
1302 |
by (auto_tac (claset() addIs [real_zero_less_one RS real_less_asym], |
|
1303 |
simpset() addsimps [real_minus_mult_eq1 RS sym])); |
|
1304 |
qed "real_rinv_gt_zero"; |
|
1305 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
1306 |
Goal "x < 0r ==> rinv x < 0r"; |
5078 | 1307 |
by (forward_tac [real_not_refl2] 1); |
1308 |
by (dtac (real_minus_zero_less_iff RS iffD2) 1); |
|
1309 |
by (rtac (real_minus_zero_less_iff RS iffD1) 1); |
|
1310 |
by (dtac (real_minus_rinv RS sym) 1); |
|
1311 |
by (auto_tac (claset() addIs [real_rinv_gt_zero], |
|
1312 |
simpset())); |
|
1313 |
qed "real_rinv_less_zero"; |
|
1314 |
||
1315 |
Goal "x+x=x*(1r+1r)"; |
|
1316 |
by (simp_tac (simpset() addsimps [real_add_mult_distrib2]) 1); |
|
1317 |
qed "real_add_self"; |
|
1318 |
||
1319 |
Goal "x < x + 1r"; |
|
1320 |
by (rtac (real_less_sum_gt_0_iff RS iffD1) 1); |
|
1321 |
by (full_simp_tac (simpset() addsimps [real_zero_less_one, |
|
1322 |
real_add_assoc,real_add_minus,real_add_zero_right, |
|
1323 |
real_add_left_commute]) 1); |
|
1324 |
qed "real_self_less_add_one"; |
|
1325 |
||
1326 |
Goal "1r < 1r + 1r"; |
|
1327 |
by (rtac real_self_less_add_one 1); |
|
1328 |
qed "real_one_less_two"; |
|
1329 |
||
1330 |
Goal "0r < 1r + 1r"; |
|
1331 |
by (rtac ([real_zero_less_one, |
|
1332 |
real_one_less_two] MRS real_less_trans) 1); |
|
1333 |
qed "real_zero_less_two"; |
|
1334 |
||
1335 |
Goal "1r + 1r ~= 0r"; |
|
1336 |
by (rtac (real_zero_less_two RS real_not_refl2 RS not_sym) 1); |
|
1337 |
qed "real_two_not_zero"; |
|
1338 |
||
1339 |
Addsimps [real_two_not_zero]; |
|
1340 |
||
1341 |
Goal "x*rinv(1r + 1r) + x*rinv(1r + 1r) = x"; |
|
1342 |
by (stac real_add_self 1); |
|
1343 |
by (full_simp_tac (simpset() addsimps [real_mult_assoc]) 1); |
|
1344 |
qed "real_sum_of_halves"; |
|
1345 |
||
1346 |
Goal "!!(x::real). [| 0r<z; x<y |] ==> x*z<y*z"; |
|
1347 |
by (rotate_tac 1 1); |
|
1348 |
by (dtac real_less_sum_gt_zero 1); |
|
1349 |
by (rtac real_sum_gt_zero_less 1); |
|
1350 |
by (dtac real_mult_order 1 THEN assume_tac 1); |
|
1351 |
by (asm_full_simp_tac (simpset() addsimps [real_add_mult_distrib2, |
|
1352 |
real_minus_mult_eq2 RS sym, real_mult_commute ]) 1); |
|
1353 |
qed "real_mult_less_mono1"; |
|
1354 |
||
1355 |
Goal "!!(y::real). [| 0r<z; x<y |] ==> z*x<z*y"; |
|
1356 |
by (asm_simp_tac (simpset() addsimps [real_mult_commute,real_mult_less_mono1]) 1); |
|
1357 |
qed "real_mult_less_mono2"; |
|
1358 |
||
1359 |
Goal "!!(x::real). [| 0r<z; x*z<y*z |] ==> x<y"; |
|
1360 |
by (forw_inst_tac [("x","x*z")] (real_rinv_gt_zero |
|
1361 |
RS real_mult_less_mono1) 1); |
|
1362 |
by (auto_tac (claset(),simpset() addsimps |
|
1363 |
[real_mult_assoc,real_not_refl2 RS not_sym])); |
|
1364 |
qed "real_mult_less_cancel1"; |
|
1365 |
||
1366 |
Goal "!!(x::real). [| 0r<z; z*x<z*y |] ==> x<y"; |
|
1367 |
by (etac real_mult_less_cancel1 1); |
|
1368 |
by (asm_full_simp_tac (simpset() addsimps [real_mult_commute]) 1); |
|
1369 |
qed "real_mult_less_cancel2"; |
|
1370 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
1371 |
Goal "0r < z ==> (x*z < y*z) = (x < y)"; |
5078 | 1372 |
by (blast_tac (claset() addIs [real_mult_less_mono1, |
1373 |
real_mult_less_cancel1]) 1); |
|
1374 |
qed "real_mult_less_iff1"; |
|
1375 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
1376 |
Goal "0r < z ==> (z*x < z*y) = (x < y)"; |
5078 | 1377 |
by (blast_tac (claset() addIs [real_mult_less_mono2, |
1378 |
real_mult_less_cancel2]) 1); |
|
1379 |
qed "real_mult_less_iff2"; |
|
1380 |
||
1381 |
Addsimps [real_mult_less_iff1,real_mult_less_iff2]; |
|
1382 |
||
1383 |
Goal "!!(x::real). [| 0r<=z; x<y |] ==> x*z<=y*z"; |
|
1384 |
by (EVERY1 [rtac real_less_or_eq_imp_le, dtac real_le_imp_less_or_eq]); |
|
1385 |
by (auto_tac (claset() addIs [real_mult_less_mono1],simpset())); |
|
1386 |
qed "real_mult_le_less_mono1"; |
|
1387 |
||
1388 |
Goal "!!(x::real). [| 0r<=z; x<y |] ==> z*x<=z*y"; |
|
1389 |
by (asm_simp_tac (simpset() addsimps [real_mult_commute,real_mult_le_less_mono1]) 1); |
|
1390 |
qed "real_mult_le_less_mono2"; |
|
1391 |
||
1392 |
Goal "!!x y (z::real). [| 0r<=z; x<=y |] ==> z*x<=z*y"; |
|
1393 |
by (dres_inst_tac [("x","x")] real_le_imp_less_or_eq 1); |
|
1394 |
by (auto_tac (claset() addIs [real_mult_le_less_mono2,real_le_refl],simpset())); |
|
1395 |
qed "real_mult_le_le_mono1"; |
|
1396 |
||
1397 |
Goal "!!(x::real). x < y ==> x < (x + y)*rinv(1r + 1r)"; |
|
1398 |
by (dres_inst_tac [("C","x")] real_add_less_mono2 1); |
|
1399 |
by (dtac (real_add_self RS subst) 1); |
|
1400 |
by (dtac (real_zero_less_two RS real_rinv_gt_zero RS |
|
1401 |
real_mult_less_mono1) 1); |
|
1402 |
by (asm_full_simp_tac (simpset() addsimps [real_mult_assoc]) 1); |
|
1403 |
qed "real_less_half_sum"; |
|
1404 |
||
1405 |
Goal "!!(x::real). x < y ==> (x + y)*rinv(1r + 1r) < y"; |
|
1406 |
by (dres_inst_tac [("C","y")] real_add_less_mono1 1); |
|
1407 |
by (dtac (real_add_self RS subst) 1); |
|
1408 |
by (dtac (real_zero_less_two RS real_rinv_gt_zero RS |
|
1409 |
real_mult_less_mono1) 1); |
|
1410 |
by (asm_full_simp_tac (simpset() addsimps [real_mult_assoc]) 1); |
|
1411 |
qed "real_gt_half_sum"; |
|
1412 |
||
1413 |
Goal "!!(x::real). x < y ==> EX r. x < r & r < y"; |
|
1414 |
by (blast_tac (claset() addSIs [real_less_half_sum,real_gt_half_sum]) 1); |
|
1415 |
qed "real_dense"; |
|
1416 |
||
1417 |
Goal "(EX n. rinv(%%#n) < r) = (EX n. 1r < r * %%#n)"; |
|
1418 |
by (Step_tac 1); |
|
1419 |
by (dres_inst_tac [("n1","n")] (real_nat_less_zero |
|
1420 |
RS real_mult_less_mono1) 1); |
|
1421 |
by (dres_inst_tac [("n2","n")] (real_nat_less_zero RS |
|
1422 |
real_rinv_gt_zero RS real_mult_less_mono1) 2); |
|
1423 |
by (auto_tac (claset(),simpset() addsimps [(real_nat_less_zero RS |
|
1424 |
real_not_refl2 RS not_sym),real_mult_assoc])); |
|
1425 |
qed "real_nat_rinv_Ex_iff"; |
|
1426 |
||
1427 |
Goalw [real_nat_def] "(%%#n < %%#m) = (n < m)"; |
|
1428 |
by Auto_tac; |
|
1429 |
qed "real_nat_less_iff"; |
|
1430 |
||
1431 |
Addsimps [real_nat_less_iff]; |
|
1432 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
1433 |
Goal "0r < u ==> (u < rinv (%%#n)) = (%%#n < rinv(u))"; |
5078 | 1434 |
by (Step_tac 1); |
1435 |
by (res_inst_tac [("n2","n")] (real_nat_less_zero RS |
|
1436 |
real_rinv_gt_zero RS real_mult_less_cancel1) 1); |
|
1437 |
by (res_inst_tac [("x1","u")] ( real_rinv_gt_zero |
|
1438 |
RS real_mult_less_cancel1) 2); |
|
1439 |
by (auto_tac (claset(),simpset() addsimps [real_nat_less_zero, |
|
1440 |
real_not_refl2 RS not_sym])); |
|
1441 |
by (res_inst_tac [("z","u")] real_mult_less_cancel2 1); |
|
1442 |
by (res_inst_tac [("n1","n")] (real_nat_less_zero RS |
|
1443 |
real_mult_less_cancel2) 3); |
|
1444 |
by (auto_tac (claset(),simpset() addsimps [real_nat_less_zero, |
|
1445 |
real_not_refl2 RS not_sym,real_mult_assoc RS sym])); |
|
1446 |
qed "real_nat_less_rinv_iff"; |
|
1447 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
1448 |
Goal "0r < u ==> (u = rinv(%%#n)) = (%%#n = rinv u)"; |
5078 | 1449 |
by (auto_tac (claset(),simpset() addsimps [real_rinv_rinv, |
1450 |
real_nat_less_zero,real_not_refl2 RS not_sym])); |
|
1451 |
qed "real_nat_rinv_eq_iff"; |
|
1452 |
||
1453 |
(* |
|
1454 |
(*------------------------------------------------------------------ |
|
1455 |
lemmas about upper bounds and least upper bound |
|
1456 |
------------------------------------------------------------------*) |
|
5148
74919e8f221c
More tidying and removal of "\!\!... from Goal commands
paulson
parents:
5143
diff
changeset
|
1457 |
Goalw [real_ub_def] "[| real_ub u S; x : S |] ==> x <= u"; |
5078 | 1458 |
by Auto_tac; |
1459 |
qed "real_ubD"; |
|
1460 |
||
1461 |
*) |