author | oheimb |
Mon, 21 Sep 1998 23:12:31 +0200 | |
changeset 5521 | 7970832271cc |
parent 5459 | 1dbaf888f4e7 |
child 5535 | 678999604ee9 |
permissions | -rw-r--r-- |
5078 | 1 |
(* Title : Real.ML |
2 |
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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6 |
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(*** Proving that realrel is an equivalence relation ***) |
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More tidying and removal of "\!\!... from Goal commands
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5143
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changeset
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Goal "[| (x1::preal) + y2 = x2 + y1; x2 + y3 = x3 + y2 |] \ |
5078 | 10 |
\ ==> 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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20 |
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Goalw [realrel_def] |
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"(((x1,y1),(x2,y2)): realrel) = (x1 + y2 = x2 + y1)"; |
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5459 | 23 |
by (Blast_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"; |
5459 | 28 |
by (Blast_tac 1); |
5078 | 29 |
qed "realrelI"; |
30 |
||
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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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5459 | 34 |
by (Blast_tac 1); |
5078 | 35 |
qed "realrelE_lemma"; |
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||
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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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47 |
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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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51 |
||
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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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5459 | 91 |
by (Blast_tac 1); |
5078 | 92 |
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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105 |
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(**** real_minus: additive inverse on real ****) |
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107 |
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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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113 |
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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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116 |
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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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154 |
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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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168 |
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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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175 |
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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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181 |
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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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188 |
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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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198 |
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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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203 |
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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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207 |
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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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218 |
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Goal "? y. (x::real) + y = 0r"; |
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5459 | 220 |
by (blast_tac (claset() addIs [real_add_minus]) 1); |
5078 | 221 |
qed "real_minus_ex"; |
222 |
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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
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
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Goal "x + y = 0r ==> x = %~y"; |
5078 | 240 |
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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5459 | 248 |
by (Blast_tac 1); |
5078 | 249 |
qed "real_as_add_inverse_ex"; |
250 |
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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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257 |
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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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266 |
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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"; |
|
270 |
||
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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, |
|
280 |
preal_add_mult_distrib2 RS sym]) 1); |
|
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by (asm_full_simp_tac (simpset() addsimps [preal_add_commute]) 1); |
|
282 |
qed "real_mult_congruent2_lemma"; |
|
283 |
||
284 |
Goal |
|
285 |
"congruent2 realrel (%p1 p2. \ |
|
286 |
\ split (%x1 y1. split (%x2 y2. realrel^^{(x1*x2 + y1*y2, x1*y2+x2*y1)}) p2) p1)"; |
|
287 |
by (rtac (equiv_realrel RS congruent2_commuteI) 1); |
|
288 |
by Safe_tac; |
|
289 |
by (rewtac split_def); |
|
290 |
by (asm_simp_tac (simpset() addsimps [preal_mult_commute,preal_add_commute]) 1); |
|
291 |
by (auto_tac (claset(),simpset() addsimps [real_mult_congruent2_lemma])); |
|
292 |
qed "real_mult_congruent2"; |
|
293 |
||
294 |
(*Resolve th against the corresponding facts for real_mult*) |
|
295 |
val real_mult_ize = RSLIST [equiv_realrel, real_mult_congruent2]; |
|
296 |
||
297 |
Goalw [real_mult_def] |
|
298 |
"Abs_real((realrel^^{(x1,y1)})) * Abs_real((realrel^^{(x2,y2)})) = \ |
|
299 |
\ Abs_real(realrel ^^ {(x1*x2+y1*y2,x1*y2+x2*y1)})"; |
|
300 |
by (simp_tac (simpset() addsimps [real_mult_ize UN_equiv_class2]) 1); |
|
301 |
qed "real_mult"; |
|
302 |
||
303 |
Goal "(z::real) * w = w * z"; |
|
304 |
by (res_inst_tac [("z","z")] eq_Abs_real 1); |
|
305 |
by (res_inst_tac [("z","w")] eq_Abs_real 1); |
|
306 |
by (asm_simp_tac (simpset() addsimps ([real_mult] @ preal_add_ac @ preal_mult_ac)) 1); |
|
307 |
qed "real_mult_commute"; |
|
308 |
||
309 |
Goal "((z1::real) * z2) * z3 = z1 * (z2 * z3)"; |
|
310 |
by (res_inst_tac [("z","z1")] eq_Abs_real 1); |
|
311 |
by (res_inst_tac [("z","z2")] eq_Abs_real 1); |
|
312 |
by (res_inst_tac [("z","z3")] eq_Abs_real 1); |
|
313 |
by (asm_simp_tac (simpset() addsimps ([preal_add_mult_distrib2,real_mult] @ |
|
314 |
preal_add_ac @ preal_mult_ac)) 1); |
|
315 |
qed "real_mult_assoc"; |
|
316 |
||
317 |
qed_goal "real_mult_left_commute" thy |
|
318 |
"(z1::real) * (z2 * z3) = z2 * (z1 * z3)" |
|
319 |
(fn _ => [rtac (real_mult_commute RS trans) 1, rtac (real_mult_assoc RS trans) 1, |
|
320 |
rtac (real_mult_commute RS arg_cong) 1]); |
|
321 |
||
322 |
(* real multiplication is an AC operator *) |
|
323 |
val real_mult_ac = [real_mult_assoc, real_mult_commute, real_mult_left_commute]; |
|
324 |
||
325 |
Goalw [real_one_def,pnat_one_def] "1r * z = z"; |
|
326 |
by (res_inst_tac [("z","z")] eq_Abs_real 1); |
|
327 |
by (asm_full_simp_tac (simpset() addsimps [real_mult, |
|
328 |
preal_add_mult_distrib2,preal_mult_1_right] |
|
329 |
@ preal_mult_ac @ preal_add_ac) 1); |
|
330 |
qed "real_mult_1"; |
|
331 |
||
332 |
Goal "z * 1r = z"; |
|
333 |
by (simp_tac (simpset() addsimps [real_mult_commute, |
|
334 |
real_mult_1]) 1); |
|
335 |
qed "real_mult_1_right"; |
|
336 |
||
337 |
Goalw [real_zero_def,pnat_one_def] "0r * z = 0r"; |
|
338 |
by (res_inst_tac [("z","z")] eq_Abs_real 1); |
|
339 |
by (asm_full_simp_tac (simpset() addsimps [real_mult, |
|
340 |
preal_add_mult_distrib2,preal_mult_1_right] |
|
341 |
@ preal_mult_ac @ preal_add_ac) 1); |
|
342 |
qed "real_mult_0"; |
|
343 |
||
344 |
Goal "z * 0r = 0r"; |
|
345 |
by (simp_tac (simpset() addsimps [real_mult_commute, |
|
346 |
real_mult_0]) 1); |
|
347 |
qed "real_mult_0_right"; |
|
348 |
||
349 |
Addsimps [real_mult_0_right,real_mult_0]; |
|
350 |
||
351 |
Goal "%~(x * y) = %~x * y"; |
|
352 |
by (res_inst_tac [("z","x")] eq_Abs_real 1); |
|
353 |
by (res_inst_tac [("z","y")] eq_Abs_real 1); |
|
354 |
by (auto_tac (claset(),simpset() addsimps [real_minus,real_mult] |
|
355 |
@ preal_mult_ac @ preal_add_ac)); |
|
356 |
qed "real_minus_mult_eq1"; |
|
357 |
||
358 |
Goal "%~(x * y) = x * %~y"; |
|
359 |
by (res_inst_tac [("z","x")] eq_Abs_real 1); |
|
360 |
by (res_inst_tac [("z","y")] eq_Abs_real 1); |
|
361 |
by (auto_tac (claset(),simpset() addsimps [real_minus,real_mult] |
|
362 |
@ preal_mult_ac @ preal_add_ac)); |
|
363 |
qed "real_minus_mult_eq2"; |
|
364 |
||
365 |
Goal "%~x*%~y = x*y"; |
|
366 |
by (full_simp_tac (simpset() addsimps [real_minus_mult_eq2 RS sym, |
|
367 |
real_minus_mult_eq1 RS sym]) 1); |
|
368 |
qed "real_minus_mult_cancel"; |
|
369 |
||
370 |
Addsimps [real_minus_mult_cancel]; |
|
371 |
||
372 |
Goal "%~x*y = x*%~y"; |
|
373 |
by (full_simp_tac (simpset() addsimps [real_minus_mult_eq2 RS sym, |
|
374 |
real_minus_mult_eq1 RS sym]) 1); |
|
375 |
qed "real_minus_mult_commute"; |
|
376 |
||
377 |
(*----------------------------------------------------------------------------- |
|
378 |
||
379 |
-----------------------------------------------------------------------------*) |
|
380 |
||
381 |
(** Lemmas **) |
|
382 |
||
383 |
qed_goal "real_add_assoc_cong" thy |
|
384 |
"!!z. (z::real) + v = z' + v' ==> z + (v + w) = z' + (v' + w)" |
|
385 |
(fn _ => [(asm_simp_tac (simpset() addsimps [real_add_assoc RS sym]) 1)]); |
|
386 |
||
387 |
qed_goal "real_add_assoc_swap" thy "(z::real) + (v + w) = v + (z + w)" |
|
388 |
(fn _ => [(REPEAT (ares_tac [real_add_commute RS real_add_assoc_cong] 1))]); |
|
389 |
||
390 |
Goal "((z1::real) + z2) * w = (z1 * w) + (z2 * w)"; |
|
391 |
by (res_inst_tac [("z","z1")] eq_Abs_real 1); |
|
392 |
by (res_inst_tac [("z","z2")] eq_Abs_real 1); |
|
393 |
by (res_inst_tac [("z","w")] eq_Abs_real 1); |
|
394 |
by (asm_simp_tac |
|
395 |
(simpset() addsimps ([preal_add_mult_distrib2, real_add, real_mult] @ |
|
396 |
preal_add_ac @ preal_mult_ac)) 1); |
|
397 |
qed "real_add_mult_distrib"; |
|
398 |
||
399 |
val real_mult_commute'= read_instantiate [("z","w")] real_mult_commute; |
|
400 |
||
401 |
Goal "(w::real) * (z1 + z2) = (w * z1) + (w * z2)"; |
|
402 |
by (simp_tac (simpset() addsimps [real_mult_commute',real_add_mult_distrib]) 1); |
|
403 |
qed "real_add_mult_distrib2"; |
|
404 |
||
405 |
val real_mult_simps = [real_mult_1, real_mult_1_right]; |
|
406 |
Addsimps real_mult_simps; |
|
407 |
||
408 |
(*** one and zero are distinct ***) |
|
409 |
Goalw [real_zero_def,real_one_def] "0r ~= 1r"; |
|
410 |
by (auto_tac (claset(),simpset() addsimps |
|
411 |
[preal_self_less_add_left RS preal_not_refl2])); |
|
412 |
qed "real_zero_not_eq_one"; |
|
413 |
||
414 |
(*** existence of inverse ***) |
|
415 |
(** lemma -- alternative definition for 0r **) |
|
416 |
Goalw [real_zero_def] "0r = Abs_real (realrel ^^ {(x, x)})"; |
|
417 |
by (auto_tac (claset(),simpset() addsimps [preal_add_commute])); |
|
418 |
qed "real_zero_iff"; |
|
419 |
||
420 |
Goalw [real_zero_def,real_one_def] |
|
421 |
"!!(x::real). x ~= 0r ==> ? y. x*y = 1r"; |
|
422 |
by (res_inst_tac [("z","x")] eq_Abs_real 1); |
|
423 |
by (cut_inst_tac [("r1.0","xa"),("r2.0","y")] preal_linear 1); |
|
424 |
by (auto_tac (claset() addSDs [preal_less_add_left_Ex], |
|
425 |
simpset() addsimps [real_zero_iff RS sym])); |
|
426 |
by (res_inst_tac [("x","Abs_real (realrel ^^ {(@#$#1p,pinv(D)+@#$#1p)})")] exI 1); |
|
427 |
by (res_inst_tac [("x","Abs_real (realrel ^^ {(pinv(D)+@#$#1p,@#$#1p)})")] exI 2); |
|
428 |
by (auto_tac (claset(),simpset() addsimps [real_mult, |
|
429 |
pnat_one_def,preal_mult_1_right,preal_add_mult_distrib2, |
|
430 |
preal_add_mult_distrib,preal_mult_1,preal_mult_inv_right] |
|
431 |
@ preal_add_ac @ preal_mult_ac)); |
|
432 |
qed "real_mult_inv_right_ex"; |
|
433 |
||
434 |
Goal "!!(x::real). x ~= 0r ==> ? y. y*x = 1r"; |
|
435 |
by (asm_simp_tac (simpset() addsimps [real_mult_commute, |
|
436 |
real_mult_inv_right_ex]) 1); |
|
437 |
qed "real_mult_inv_left_ex"; |
|
438 |
||
439 |
Goalw [rinv_def] "!!(x::real). x ~= 0r ==> rinv(x)*x = 1r"; |
|
440 |
by (forward_tac [real_mult_inv_left_ex] 1); |
|
441 |
by (Step_tac 1); |
|
442 |
by (rtac selectI2 1); |
|
443 |
by Auto_tac; |
|
444 |
qed "real_mult_inv_left"; |
|
445 |
||
446 |
Goal "!!(x::real). x ~= 0r ==> x*rinv(x) = 1r"; |
|
447 |
by (auto_tac (claset() addIs [real_mult_commute RS subst], |
|
448 |
simpset() addsimps [real_mult_inv_left])); |
|
449 |
qed "real_mult_inv_right"; |
|
450 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
451 |
Goal "(c::real) ~= 0r ==> (c*a=c*b) = (a=b)"; |
5078 | 452 |
by Auto_tac; |
453 |
by (dres_inst_tac [("f","%x. x*rinv c")] arg_cong 1); |
|
454 |
by (asm_full_simp_tac (simpset() addsimps [real_mult_inv_right] @ real_mult_ac) 1); |
|
455 |
qed "real_mult_left_cancel"; |
|
456 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
457 |
Goal "(c::real) ~= 0r ==> (a*c=b*c) = (a=b)"; |
5078 | 458 |
by (Step_tac 1); |
459 |
by (dres_inst_tac [("f","%x. x*rinv c")] arg_cong 1); |
|
460 |
by (asm_full_simp_tac (simpset() addsimps [real_mult_inv_right] @ real_mult_ac) 1); |
|
461 |
qed "real_mult_right_cancel"; |
|
462 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
463 |
Goalw [rinv_def] "x ~= 0r ==> rinv(x) ~= 0r"; |
5078 | 464 |
by (forward_tac [real_mult_inv_left_ex] 1); |
465 |
by (etac exE 1); |
|
466 |
by (rtac selectI2 1); |
|
467 |
by (auto_tac (claset(),simpset() addsimps [real_mult_0, |
|
468 |
real_zero_not_eq_one])); |
|
469 |
qed "rinv_not_zero"; |
|
470 |
||
471 |
Addsimps [real_mult_inv_left,real_mult_inv_right]; |
|
472 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
473 |
Goal "x ~= 0r ==> rinv(rinv x) = x"; |
5078 | 474 |
by (res_inst_tac [("c1","rinv x")] (real_mult_right_cancel RS iffD1) 1); |
475 |
by (etac rinv_not_zero 1); |
|
476 |
by (auto_tac (claset() addDs [rinv_not_zero],simpset())); |
|
477 |
qed "real_rinv_rinv"; |
|
478 |
||
479 |
Goalw [rinv_def] "rinv(1r) = 1r"; |
|
480 |
by (cut_facts_tac [real_zero_not_eq_one RS |
|
481 |
not_sym RS real_mult_inv_left_ex] 1); |
|
482 |
by (etac exE 1); |
|
483 |
by (rtac selectI2 1); |
|
484 |
by (auto_tac (claset(),simpset() addsimps |
|
485 |
[real_zero_not_eq_one RS not_sym])); |
|
486 |
qed "real_rinv_1"; |
|
487 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
488 |
Goal "x ~= 0r ==> rinv(%~x) = %~rinv(x)"; |
5078 | 489 |
by (res_inst_tac [("c1","%~x")] (real_mult_right_cancel RS iffD1) 1); |
490 |
by Auto_tac; |
|
491 |
qed "real_minus_rinv"; |
|
492 |
||
493 |
(*** theorems for ordering ***) |
|
494 |
(* prove introduction and elimination rules for real_less *) |
|
495 |
||
496 |
Goalw [real_less_def] |
|
497 |
"P < (Q::real) = (EX x1 y1 x2 y2. x1 + y2 < x2 + y1 & \ |
|
498 |
\ (x1,y1::preal):Rep_real(P) & \ |
|
499 |
\ (x2,y2):Rep_real(Q))"; |
|
5459 | 500 |
by (Blast_tac 1); |
5078 | 501 |
qed "real_less_iff"; |
502 |
||
503 |
Goalw [real_less_def] |
|
5148
74919e8f221c
More tidying and removal of "\!\!... from Goal commands
paulson
parents:
5143
diff
changeset
|
504 |
"[| x1 + y2 < x2 + y1; (x1,y1::preal):Rep_real(P); \ |
5078 | 505 |
\ (x2,y2):Rep_real(Q) |] ==> P < (Q::real)"; |
5459 | 506 |
by (Blast_tac 1); |
5078 | 507 |
qed "real_lessI"; |
508 |
||
509 |
Goalw [real_less_def] |
|
5148
74919e8f221c
More tidying and removal of "\!\!... from Goal commands
paulson
parents:
5143
diff
changeset
|
510 |
"!!P. [| R1 < (R2::real); \ |
5078 | 511 |
\ !!x1 x2 y1 y2. x1 + y2 < x2 + y1 ==> P; \ |
512 |
\ !!x1 y1. (x1,y1::preal):Rep_real(R1) ==> P; \ |
|
513 |
\ !!x2 y2. (x2,y2::preal):Rep_real(R2) ==> P |] \ |
|
514 |
\ ==> P"; |
|
515 |
by Auto_tac; |
|
516 |
qed "real_lessE"; |
|
517 |
||
518 |
Goalw [real_less_def] |
|
5148
74919e8f221c
More tidying and removal of "\!\!... from Goal commands
paulson
parents:
5143
diff
changeset
|
519 |
"R1 < (R2::real) ==> (EX x1 y1 x2 y2. x1 + y2 < x2 + y1 & \ |
5078 | 520 |
\ (x1,y1::preal):Rep_real(R1) & \ |
521 |
\ (x2,y2):Rep_real(R2))"; |
|
5459 | 522 |
by (Blast_tac 1); |
5078 | 523 |
qed "real_lessD"; |
524 |
||
525 |
(* real_less is a strong order i.e nonreflexive and transitive *) |
|
526 |
(*** lemmas ***) |
|
527 |
Goal "!!(x::preal). [| x = y; x1 = y1 |] ==> x + y1 = x1 + y"; |
|
528 |
by (asm_simp_tac (simpset() addsimps [preal_add_commute]) 1); |
|
529 |
qed "preal_lemma_eq_rev_sum"; |
|
530 |
||
531 |
Goal "!!(b::preal). x + (b + y) = x1 + (b + y1) ==> x + y = x1 + y1"; |
|
532 |
by (asm_full_simp_tac (simpset() addsimps preal_add_ac) 1); |
|
533 |
qed "preal_add_left_commute_cancel"; |
|
534 |
||
535 |
Goal |
|
536 |
"!!(x::preal). [| x + y2a = x2a + y; \ |
|
537 |
\ x + y2b = x2b + y |] \ |
|
538 |
\ ==> x2a + y2b = x2b + y2a"; |
|
539 |
by (dtac preal_lemma_eq_rev_sum 1); |
|
540 |
by (assume_tac 1); |
|
541 |
by (thin_tac "x + y2b = x2b + y" 1); |
|
542 |
by (asm_full_simp_tac (simpset() addsimps preal_add_ac) 1); |
|
543 |
by (dtac preal_add_left_commute_cancel 1); |
|
544 |
by (asm_full_simp_tac (simpset() addsimps preal_add_ac) 1); |
|
545 |
qed "preal_lemma_for_not_refl"; |
|
546 |
||
547 |
Goal "~ (R::real) < R"; |
|
548 |
by (res_inst_tac [("z","R")] eq_Abs_real 1); |
|
549 |
by (auto_tac (claset(),simpset() addsimps [real_less_def])); |
|
550 |
by (dtac preal_lemma_for_not_refl 1); |
|
551 |
by (assume_tac 1 THEN rotate_tac 2 1); |
|
552 |
by (auto_tac (claset(),simpset() addsimps [preal_less_not_refl])); |
|
553 |
qed "real_less_not_refl"; |
|
554 |
||
555 |
(*** y < y ==> P ***) |
|
556 |
bind_thm("real_less_irrefl",real_less_not_refl RS notE); |
|
557 |
||
558 |
Goal "!!(x::real). x < y ==> x ~= y"; |
|
559 |
by (auto_tac (claset(),simpset() addsimps [real_less_not_refl])); |
|
560 |
qed "real_not_refl2"; |
|
561 |
||
562 |
(* lemma re-arranging and eliminating terms *) |
|
563 |
Goal "!! (a::preal). [| a + b = c + d; \ |
|
564 |
\ x2b + d + (c + y2e) < a + y2b + (x2e + b) |] \ |
|
565 |
\ ==> x2b + y2e < x2e + y2b"; |
|
566 |
by (asm_full_simp_tac (simpset() addsimps preal_add_ac) 1); |
|
567 |
by (res_inst_tac [("C","c+d")] preal_add_left_less_cancel 1); |
|
568 |
by (asm_full_simp_tac (simpset() addsimps [preal_add_assoc RS sym]) 1); |
|
569 |
qed "preal_lemma_trans"; |
|
570 |
||
571 |
(** heavy re-writing involved*) |
|
572 |
Goal "!!(R1::real). [| R1 < R2; R2 < R3 |] ==> R1 < R3"; |
|
573 |
by (res_inst_tac [("z","R1")] eq_Abs_real 1); |
|
574 |
by (res_inst_tac [("z","R2")] eq_Abs_real 1); |
|
575 |
by (res_inst_tac [("z","R3")] eq_Abs_real 1); |
|
576 |
by (auto_tac (claset(),simpset() addsimps [real_less_def])); |
|
577 |
by (REPEAT(rtac exI 1)); |
|
578 |
by (EVERY[rtac conjI 1, rtac conjI 2]); |
|
579 |
by (REPEAT(Blast_tac 2)); |
|
580 |
by (dtac preal_lemma_for_not_refl 1 THEN assume_tac 1); |
|
581 |
by (blast_tac (claset() addDs [preal_add_less_mono] |
|
582 |
addIs [preal_lemma_trans]) 1); |
|
583 |
qed "real_less_trans"; |
|
584 |
||
585 |
Goal "!! (R1::real). [| R1 < R2; R2 < R1 |] ==> P"; |
|
586 |
by (dtac real_less_trans 1 THEN assume_tac 1); |
|
587 |
by (asm_full_simp_tac (simpset() addsimps [real_less_not_refl]) 1); |
|
588 |
qed "real_less_asym"; |
|
589 |
||
590 |
(****)(****)(****)(****)(****)(****)(****)(****)(****)(****) |
|
591 |
(****** Map and more real_less ******) |
|
592 |
(*** mapping from preal into real ***) |
|
593 |
Goalw [real_preal_def] |
|
594 |
"%#((z1::preal) + z2) = %#z1 + %#z2"; |
|
595 |
by (asm_simp_tac (simpset() addsimps [real_add, |
|
596 |
preal_add_mult_distrib,preal_mult_1] addsimps preal_add_ac) 1); |
|
597 |
qed "real_preal_add"; |
|
598 |
||
599 |
Goalw [real_preal_def] |
|
600 |
"%#((z1::preal) * z2) = %#z1* %#z2"; |
|
601 |
by (full_simp_tac (simpset() addsimps [real_mult, |
|
602 |
preal_add_mult_distrib2,preal_mult_1, |
|
603 |
preal_mult_1_right,pnat_one_def] |
|
604 |
@ preal_add_ac @ preal_mult_ac) 1); |
|
605 |
qed "real_preal_mult"; |
|
606 |
||
607 |
Goalw [real_preal_def] |
|
608 |
"!!(x::preal). y < x ==> ? m. Abs_real (realrel ^^ {(x,y)}) = %#m"; |
|
609 |
by (auto_tac (claset() addSDs [preal_less_add_left_Ex], |
|
610 |
simpset() addsimps preal_add_ac)); |
|
611 |
qed "real_preal_ExI"; |
|
612 |
||
613 |
Goalw [real_preal_def] |
|
614 |
"!!(x::preal). ? m. Abs_real (realrel ^^ {(x,y)}) = %#m ==> y < x"; |
|
615 |
by (auto_tac (claset(),simpset() addsimps |
|
616 |
[preal_add_commute,preal_add_assoc])); |
|
617 |
by (asm_full_simp_tac (simpset() addsimps |
|
618 |
[preal_add_assoc RS sym,preal_self_less_add_left]) 1); |
|
619 |
qed "real_preal_ExD"; |
|
620 |
||
621 |
Goal "(? m. Abs_real (realrel ^^ {(x,y)}) = %#m) = (y < x)"; |
|
5459 | 622 |
by (blast_tac (claset() addSIs [real_preal_ExI,real_preal_ExD]) 1); |
5078 | 623 |
qed "real_preal_iff"; |
624 |
||
625 |
(*** Gleason prop 9-4.4 p 127 ***) |
|
626 |
Goalw [real_preal_def,real_zero_def] |
|
627 |
"? m. (x::real) = %#m | x = 0r | x = %~(%#m)"; |
|
628 |
by (res_inst_tac [("z","x")] eq_Abs_real 1); |
|
629 |
by (auto_tac (claset(),simpset() addsimps [real_minus] @ preal_add_ac)); |
|
630 |
by (cut_inst_tac [("r1.0","x"),("r2.0","y")] preal_linear 1); |
|
631 |
by (auto_tac (claset() addSDs [preal_less_add_left_Ex], |
|
632 |
simpset() addsimps [preal_add_assoc RS sym])); |
|
633 |
by (auto_tac (claset(),simpset() addsimps [preal_add_commute])); |
|
634 |
qed "real_preal_trichotomy"; |
|
635 |
||
5148
74919e8f221c
More tidying and removal of "\!\!... from Goal commands
paulson
parents:
5143
diff
changeset
|
636 |
Goal "!!P. [| !!m. x = %#m ==> P; \ |
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
637 |
\ x = 0r ==> P; \ |
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
638 |
\ !!m. x = %~(%#m) ==> P |] ==> P"; |
5078 | 639 |
by (cut_inst_tac [("x","x")] real_preal_trichotomy 1); |
640 |
by Auto_tac; |
|
641 |
qed "real_preal_trichotomyE"; |
|
642 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
643 |
Goalw [real_preal_def] "%#m1 < %#m2 ==> m1 < m2"; |
5078 | 644 |
by (auto_tac (claset(),simpset() addsimps [real_less_def] @ preal_add_ac)); |
645 |
by (auto_tac (claset(),simpset() addsimps [preal_add_assoc RS sym])); |
|
646 |
by (auto_tac (claset(),simpset() addsimps preal_add_ac)); |
|
647 |
qed "real_preal_lessD"; |
|
648 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
649 |
Goal "m1 < m2 ==> %#m1 < %#m2"; |
5078 | 650 |
by (dtac preal_less_add_left_Ex 1); |
651 |
by (auto_tac (claset(),simpset() addsimps [real_preal_add, |
|
652 |
real_preal_def,real_less_def])); |
|
653 |
by (REPEAT(rtac exI 1)); |
|
654 |
by (EVERY[rtac conjI 1, rtac conjI 2]); |
|
5459 | 655 |
by (REPEAT(Blast_tac 2)); |
5078 | 656 |
by (simp_tac (simpset() addsimps [preal_self_less_add_left] |
657 |
delsimps [preal_add_less_iff2]) 1); |
|
658 |
qed "real_preal_lessI"; |
|
659 |
||
660 |
Goal "(%#m1 < %#m2) = (m1 < m2)"; |
|
5459 | 661 |
by (blast_tac (claset() addIs [real_preal_lessI,real_preal_lessD]) 1); |
5078 | 662 |
qed "real_preal_less_iff1"; |
663 |
||
664 |
Addsimps [real_preal_less_iff1]; |
|
665 |
||
666 |
Goal "%~ %#m < %#m"; |
|
667 |
by (auto_tac (claset(),simpset() addsimps |
|
668 |
[real_preal_def,real_less_def,real_minus])); |
|
669 |
by (REPEAT(rtac exI 1)); |
|
670 |
by (EVERY[rtac conjI 1, rtac conjI 2]); |
|
5459 | 671 |
by (REPEAT(Blast_tac 2)); |
5078 | 672 |
by (full_simp_tac (simpset() addsimps preal_add_ac) 1); |
673 |
by (full_simp_tac (simpset() addsimps [preal_self_less_add_right, |
|
674 |
preal_add_assoc RS sym]) 1); |
|
675 |
qed "real_preal_minus_less_self"; |
|
676 |
||
677 |
Goalw [real_zero_def] "%~ %#m < 0r"; |
|
678 |
by (auto_tac (claset(),simpset() addsimps |
|
679 |
[real_preal_def,real_less_def,real_minus])); |
|
680 |
by (REPEAT(rtac exI 1)); |
|
681 |
by (EVERY[rtac conjI 1, rtac conjI 2]); |
|
5459 | 682 |
by (REPEAT(Blast_tac 2)); |
5078 | 683 |
by (full_simp_tac (simpset() addsimps |
684 |
[preal_self_less_add_right] @ preal_add_ac) 1); |
|
685 |
qed "real_preal_minus_less_zero"; |
|
686 |
||
687 |
Goal "~ 0r < %~ %#m"; |
|
688 |
by (cut_facts_tac [real_preal_minus_less_zero] 1); |
|
5521 | 689 |
by (fast_tac (claset() addDs [real_less_trans] |
5459 | 690 |
addEs [real_less_irrefl]) 1); |
5078 | 691 |
qed "real_preal_not_minus_gt_zero"; |
692 |
||
693 |
Goalw [real_zero_def] " 0r < %#m"; |
|
694 |
by (auto_tac (claset(),simpset() addsimps |
|
695 |
[real_preal_def,real_less_def,real_minus])); |
|
696 |
by (REPEAT(rtac exI 1)); |
|
697 |
by (EVERY[rtac conjI 1, rtac conjI 2]); |
|
5459 | 698 |
by (REPEAT(Blast_tac 2)); |
5078 | 699 |
by (full_simp_tac (simpset() addsimps |
700 |
[preal_self_less_add_right] @ preal_add_ac) 1); |
|
701 |
qed "real_preal_zero_less"; |
|
702 |
||
703 |
Goal "~ %#m < 0r"; |
|
704 |
by (cut_facts_tac [real_preal_zero_less] 1); |
|
5459 | 705 |
by (blast_tac (claset() addDs [real_less_trans] |
5078 | 706 |
addEs [real_less_irrefl]) 1); |
707 |
qed "real_preal_not_less_zero"; |
|
708 |
||
709 |
Goal "0r < %~ %~ %#m"; |
|
710 |
by (simp_tac (simpset() addsimps |
|
711 |
[real_preal_zero_less]) 1); |
|
712 |
qed "real_minus_minus_zero_less"; |
|
713 |
||
714 |
(* another lemma *) |
|
715 |
Goalw [real_zero_def] " 0r < %#m + %#m1"; |
|
716 |
by (auto_tac (claset(),simpset() addsimps |
|
717 |
[real_preal_def,real_less_def,real_add])); |
|
718 |
by (REPEAT(rtac exI 1)); |
|
719 |
by (EVERY[rtac conjI 1, rtac conjI 2]); |
|
5459 | 720 |
by (REPEAT(Blast_tac 2)); |
5078 | 721 |
by (full_simp_tac (simpset() addsimps preal_add_ac) 1); |
722 |
by (full_simp_tac (simpset() addsimps [preal_self_less_add_right, |
|
723 |
preal_add_assoc RS sym]) 1); |
|
724 |
qed "real_preal_sum_zero_less"; |
|
725 |
||
726 |
Goal "%~ %#m < %#m1"; |
|
727 |
by (auto_tac (claset(),simpset() addsimps |
|
728 |
[real_preal_def,real_less_def,real_minus])); |
|
729 |
by (REPEAT(rtac exI 1)); |
|
730 |
by (EVERY[rtac conjI 1, rtac conjI 2]); |
|
5459 | 731 |
by (REPEAT(Blast_tac 2)); |
5078 | 732 |
by (full_simp_tac (simpset() addsimps preal_add_ac) 1); |
733 |
by (full_simp_tac (simpset() addsimps [preal_self_less_add_right, |
|
734 |
preal_add_assoc RS sym]) 1); |
|
735 |
qed "real_preal_minus_less_all"; |
|
736 |
||
737 |
Goal "~ %#m < %~ %#m1"; |
|
738 |
by (cut_facts_tac [real_preal_minus_less_all] 1); |
|
5459 | 739 |
by (blast_tac (claset() addDs [real_less_trans] |
5078 | 740 |
addEs [real_less_irrefl]) 1); |
741 |
qed "real_preal_not_minus_gt_all"; |
|
742 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
743 |
Goal "%~ %#m1 < %~ %#m2 ==> %#m2 < %#m1"; |
5078 | 744 |
by (auto_tac (claset(),simpset() addsimps |
745 |
[real_preal_def,real_less_def,real_minus])); |
|
746 |
by (REPEAT(rtac exI 1)); |
|
747 |
by (EVERY[rtac conjI 1, rtac conjI 2]); |
|
5459 | 748 |
by (REPEAT(Blast_tac 2)); |
5078 | 749 |
by (auto_tac (claset(),simpset() addsimps preal_add_ac)); |
750 |
by (asm_full_simp_tac (simpset() addsimps [preal_add_assoc RS sym]) 1); |
|
751 |
by (auto_tac (claset(),simpset() addsimps preal_add_ac)); |
|
752 |
qed "real_preal_minus_less_rev1"; |
|
753 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
754 |
Goal "%#m1 < %#m2 ==> %~ %#m2 < %~ %#m1"; |
5078 | 755 |
by (auto_tac (claset(),simpset() addsimps |
756 |
[real_preal_def,real_less_def,real_minus])); |
|
757 |
by (REPEAT(rtac exI 1)); |
|
758 |
by (EVERY[rtac conjI 1, rtac conjI 2]); |
|
5459 | 759 |
by (REPEAT(Blast_tac 2)); |
5078 | 760 |
by (auto_tac (claset(),simpset() addsimps preal_add_ac)); |
761 |
by (asm_full_simp_tac (simpset() addsimps [preal_add_assoc RS sym]) 1); |
|
762 |
by (auto_tac (claset(),simpset() addsimps preal_add_ac)); |
|
763 |
qed "real_preal_minus_less_rev2"; |
|
764 |
||
765 |
Goal "(%~ %#m1 < %~ %#m2) = (%#m2 < %#m1)"; |
|
766 |
by (blast_tac (claset() addSIs [real_preal_minus_less_rev1, |
|
767 |
real_preal_minus_less_rev2]) 1); |
|
768 |
qed "real_preal_minus_less_rev_iff"; |
|
769 |
||
770 |
Addsimps [real_preal_minus_less_rev_iff]; |
|
771 |
||
772 |
(*** linearity ***) |
|
773 |
Goal "(R1::real) < R2 | R1 = R2 | R2 < R1"; |
|
774 |
by (res_inst_tac [("x","R1")] real_preal_trichotomyE 1); |
|
775 |
by (ALLGOALS(res_inst_tac [("x","R2")] real_preal_trichotomyE)); |
|
776 |
by (auto_tac (claset() addSDs [preal_le_anti_sym], |
|
777 |
simpset() addsimps [preal_less_le_iff,real_preal_minus_less_zero, |
|
778 |
real_preal_zero_less,real_preal_minus_less_all])); |
|
779 |
qed "real_linear"; |
|
780 |
||
5148
74919e8f221c
More tidying and removal of "\!\!... from Goal commands
paulson
parents:
5143
diff
changeset
|
781 |
Goal "!!(R1::real). [| R1 < R2 ==> P; R1 = R2 ==> P; \ |
74919e8f221c
More tidying and removal of "\!\!... from Goal commands
paulson
parents:
5143
diff
changeset
|
782 |
\ R2 < R1 ==> P |] ==> P"; |
5078 | 783 |
by (cut_inst_tac [("R1.0","R1"),("R2.0","R2")] real_linear 1); |
784 |
by Auto_tac; |
|
785 |
qed "real_linear_less2"; |
|
786 |
||
787 |
(*** Properties of <= ***) |
|
788 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
789 |
Goalw [real_le_def] "~(w < z) ==> z <= (w::real)"; |
5078 | 790 |
by (assume_tac 1); |
791 |
qed "real_leI"; |
|
792 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
793 |
Goalw [real_le_def] "z<=w ==> ~(w<(z::real))"; |
5078 | 794 |
by (assume_tac 1); |
795 |
qed "real_leD"; |
|
796 |
||
797 |
val real_leE = make_elim real_leD; |
|
798 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
799 |
Goal "(~(w < z)) = (z <= (w::real))"; |
5459 | 800 |
by (blast_tac (claset() addSIs [real_leI,real_leD]) 1); |
5078 | 801 |
qed "real_less_le_iff"; |
802 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
803 |
Goalw [real_le_def] "~ z <= w ==> w<(z::real)"; |
5459 | 804 |
by (Blast_tac 1); |
5078 | 805 |
qed "not_real_leE"; |
806 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
807 |
Goalw [real_le_def] "z < w ==> z <= (w::real)"; |
5459 | 808 |
by (blast_tac (claset() addEs [real_less_asym]) 1); |
5078 | 809 |
qed "real_less_imp_le"; |
810 |
||
811 |
Goalw [real_le_def] "!!(x::real). x <= y ==> x < y | x = y"; |
|
812 |
by (cut_facts_tac [real_linear] 1); |
|
5459 | 813 |
by (blast_tac (claset() addEs [real_less_irrefl,real_less_asym]) 1); |
5078 | 814 |
qed "real_le_imp_less_or_eq"; |
815 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
816 |
Goalw [real_le_def] "z<w | z=w ==> z <=(w::real)"; |
5078 | 817 |
by (cut_facts_tac [real_linear] 1); |
5521 | 818 |
by (fast_tac (claset() addEs [real_less_irrefl,real_less_asym]) 1); |
5078 | 819 |
qed "real_less_or_eq_imp_le"; |
820 |
||
821 |
Goal "(x <= (y::real)) = (x < y | x=y)"; |
|
822 |
by (REPEAT(ares_tac [iffI, real_less_or_eq_imp_le, real_le_imp_less_or_eq] 1)); |
|
823 |
qed "real_le_eq_less_or_eq"; |
|
824 |
||
825 |
Goal "w <= (w::real)"; |
|
826 |
by (simp_tac (simpset() addsimps [real_le_eq_less_or_eq]) 1); |
|
827 |
qed "real_le_refl"; |
|
828 |
||
829 |
val prems = goal Real.thy "!!i. [| i <= j; j < k |] ==> i < (k::real)"; |
|
830 |
by (dtac real_le_imp_less_or_eq 1); |
|
5459 | 831 |
by (blast_tac (claset() addIs [real_less_trans]) 1); |
5078 | 832 |
qed "real_le_less_trans"; |
833 |
||
834 |
Goal "!! (i::real). [| i < j; j <= k |] ==> i < k"; |
|
835 |
by (dtac real_le_imp_less_or_eq 1); |
|
5459 | 836 |
by (blast_tac (claset() addIs [real_less_trans]) 1); |
5078 | 837 |
qed "real_less_le_trans"; |
838 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
839 |
Goal "[| i <= j; j <= k |] ==> i <= (k::real)"; |
5078 | 840 |
by (EVERY1 [dtac real_le_imp_less_or_eq, dtac real_le_imp_less_or_eq, |
5459 | 841 |
rtac real_less_or_eq_imp_le, blast_tac (claset() addIs [real_less_trans])]); |
5078 | 842 |
qed "real_le_trans"; |
843 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
844 |
Goal "[| z <= w; w <= z |] ==> z = (w::real)"; |
5078 | 845 |
by (EVERY1 [dtac real_le_imp_less_or_eq, dtac real_le_imp_less_or_eq, |
5521 | 846 |
fast_tac (claset() addEs [real_less_irrefl,real_less_asym])]); |
5078 | 847 |
qed "real_le_anti_sym"; |
848 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
849 |
Goal "[| ~ y < x; y ~= x |] ==> x < (y::real)"; |
5078 | 850 |
by (rtac not_real_leE 1); |
5459 | 851 |
by (blast_tac (claset() addDs [real_le_imp_less_or_eq]) 1); |
5078 | 852 |
qed "not_less_not_eq_real_less"; |
853 |
||
854 |
Goal "(0r < %~R) = (R < 0r)"; |
|
855 |
by (res_inst_tac [("x","R")] real_preal_trichotomyE 1); |
|
856 |
by (auto_tac (claset(),simpset() addsimps [real_preal_not_minus_gt_zero, |
|
857 |
real_preal_not_less_zero,real_preal_zero_less, |
|
858 |
real_preal_minus_less_zero])); |
|
859 |
qed "real_minus_zero_less_iff"; |
|
860 |
||
861 |
Addsimps [real_minus_zero_less_iff]; |
|
862 |
||
863 |
Goal "(%~R < 0r) = (0r < R)"; |
|
864 |
by (res_inst_tac [("x","R")] real_preal_trichotomyE 1); |
|
865 |
by (auto_tac (claset(),simpset() addsimps [real_preal_not_minus_gt_zero, |
|
866 |
real_preal_not_less_zero,real_preal_zero_less, |
|
867 |
real_preal_minus_less_zero])); |
|
868 |
qed "real_minus_zero_less_iff2"; |
|
869 |
||
870 |
(** lemma **) |
|
871 |
Goal "(0r < x) = (? y. x = %#y)"; |
|
872 |
by (auto_tac (claset(),simpset() addsimps [real_preal_zero_less])); |
|
873 |
by (cut_inst_tac [("x","x")] real_preal_trichotomy 1); |
|
874 |
by (blast_tac (claset() addSEs [real_less_irrefl, |
|
875 |
real_preal_not_minus_gt_zero RS notE]) 1); |
|
876 |
qed "real_gt_zero_preal_Ex"; |
|
877 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
878 |
Goal "%#z < x ==> ? y. x = %#y"; |
5078 | 879 |
by (blast_tac (claset() addSDs [real_preal_zero_less RS real_less_trans] |
880 |
addIs [real_gt_zero_preal_Ex RS iffD1]) 1); |
|
881 |
qed "real_gt_preal_preal_Ex"; |
|
882 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
883 |
Goal "%#z <= x ==> ? y. x = %#y"; |
5078 | 884 |
by (blast_tac (claset() addDs [real_le_imp_less_or_eq, |
885 |
real_gt_preal_preal_Ex]) 1); |
|
886 |
qed "real_ge_preal_preal_Ex"; |
|
887 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
888 |
Goal "y <= 0r ==> !x. y < %#x"; |
5078 | 889 |
by (auto_tac (claset() addEs [real_le_imp_less_or_eq RS disjE] |
890 |
addIs [real_preal_zero_less RSN(2,real_less_trans)], |
|
891 |
simpset() addsimps [real_preal_zero_less])); |
|
892 |
qed "real_less_all_preal"; |
|
893 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
894 |
Goal "~ 0r < y ==> !x. y < %#x"; |
5078 | 895 |
by (blast_tac (claset() addSIs [real_less_all_preal,real_leI]) 1); |
896 |
qed "real_less_all_real2"; |
|
897 |
||
898 |
(**** Derive alternative definition for real_less ****) |
|
899 |
(** lemma **) |
|
900 |
Goal "!!(R::real). ? A. S + A = R"; |
|
901 |
by (res_inst_tac [("x","%~S + R")] exI 1); |
|
902 |
by (simp_tac (simpset() addsimps [real_add_minus, |
|
903 |
real_add_zero_right] @ real_add_ac) 1); |
|
904 |
qed "real_lemma_add_left_ex"; |
|
905 |
||
906 |
Goal "!!(R::real). R < S ==> ? T. R + T = S"; |
|
907 |
by (res_inst_tac [("x","R")] real_preal_trichotomyE 1); |
|
908 |
by (ALLGOALS(res_inst_tac [("x","S")] real_preal_trichotomyE)); |
|
909 |
by (auto_tac (claset() addSDs [preal_le_anti_sym] addSDs [preal_less_add_left_Ex], |
|
910 |
simpset() addsimps [preal_less_le_iff,real_preal_add,real_minus_add_eq, |
|
911 |
real_preal_minus_less_zero,real_less_not_refl,real_minus_ex,real_add_assoc, |
|
912 |
real_preal_zero_less,real_preal_minus_less_all,real_add_minus_left, |
|
913 |
real_preal_not_less_zero,real_add_zero_left,real_lemma_add_left_ex])); |
|
914 |
qed "real_less_add_left_Ex"; |
|
915 |
||
916 |
Goal "!!(R::real). R < S ==> ? T. 0r < T & R + T = S"; |
|
917 |
by (res_inst_tac [("x","R")] real_preal_trichotomyE 1); |
|
918 |
by (ALLGOALS(res_inst_tac [("x","S")] real_preal_trichotomyE)); |
|
919 |
by (auto_tac (claset() addSDs [preal_less_add_left_Ex], |
|
920 |
simpset() addsimps [real_preal_not_minus_gt_all, |
|
921 |
real_preal_add, real_preal_not_less_zero,real_less_not_refl, |
|
922 |
real_preal_not_minus_gt_zero,real_add_zero_left,real_minus_add_eq])); |
|
923 |
by (res_inst_tac [("x","%#D")] exI 1); |
|
924 |
by (res_inst_tac [("x","%#m+%#ma")] exI 2); |
|
925 |
by (res_inst_tac [("x","%#m")] exI 3); |
|
926 |
by (res_inst_tac [("x","%#D")] exI 4); |
|
927 |
by (auto_tac (claset(),simpset() addsimps [real_preal_zero_less, |
|
928 |
real_preal_sum_zero_less,real_add_minus_left,real_add_assoc, |
|
929 |
real_add_minus,real_add_zero_right])); |
|
930 |
by (simp_tac (simpset() addsimps [real_add_assoc RS sym, |
|
931 |
real_add_minus_left,real_add_zero_left]) 1); |
|
932 |
qed "real_less_add_positive_left_Ex"; |
|
933 |
||
934 |
(* lemmas *) |
|
935 |
(** change naff name(s)! **) |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
936 |
Goal "(W < S) ==> (0r < S + %~W)"; |
5078 | 937 |
by (dtac real_less_add_positive_left_Ex 1); |
938 |
by (auto_tac (claset(),simpset() addsimps [real_add_minus, |
|
939 |
real_add_zero_right] @ real_add_ac)); |
|
940 |
qed "real_less_sum_gt_zero"; |
|
941 |
||
942 |
Goal "!!S. T = S + W ==> S = T + %~W"; |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
943 |
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
|
944 |
@ real_add_ac) 1); |
5078 | 945 |
qed "real_lemma_change_eq_subj"; |
946 |
||
947 |
(* FIXME: long! *) |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
948 |
Goal "(0r < S + %~W) ==> (W < S)"; |
5078 | 949 |
by (rtac ccontr 1); |
950 |
by (dtac (real_leI RS real_le_imp_less_or_eq) 1); |
|
951 |
by (auto_tac (claset(), |
|
952 |
simpset() addsimps [real_less_not_refl,real_add_minus])); |
|
953 |
by (EVERY1[dtac real_less_add_positive_left_Ex, etac exE, etac conjE]); |
|
954 |
by (asm_full_simp_tac (simpset() addsimps [real_add_zero_left]) 1); |
|
955 |
by (dtac real_lemma_change_eq_subj 1); |
|
956 |
by (auto_tac (claset(),simpset() addsimps [real_minus_minus])); |
|
957 |
by (dtac real_less_sum_gt_zero 1); |
|
958 |
by (asm_full_simp_tac (simpset() addsimps [real_minus_add_eq] @ real_add_ac) 1); |
|
959 |
by (EVERY1[rotate_tac 1, dtac (real_add_left_commute RS ssubst)]); |
|
960 |
by (auto_tac (claset() addEs [real_less_asym], |
|
961 |
simpset() addsimps [real_add_minus,real_add_zero_right])); |
|
962 |
qed "real_sum_gt_zero_less"; |
|
963 |
||
964 |
Goal "(0r < S + %~W) = (W < S)"; |
|
5459 | 965 |
by (blast_tac (claset() addIs [real_less_sum_gt_zero, |
5078 | 966 |
real_sum_gt_zero_less]) 1); |
967 |
qed "real_less_sum_gt_0_iff"; |
|
968 |
||
969 |
Goal "((x::real) < y) = (%~y < %~x)"; |
|
970 |
by (rtac (real_less_sum_gt_0_iff RS subst) 1); |
|
971 |
by (res_inst_tac [("W1","x")] (real_less_sum_gt_0_iff RS subst) 1); |
|
972 |
by (simp_tac (simpset() addsimps [real_add_commute]) 1); |
|
973 |
qed "real_less_swap_iff"; |
|
974 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
975 |
Goal "[| R + L = S; 0r < L |] ==> R < S"; |
5078 | 976 |
by (rtac (real_less_sum_gt_0_iff RS iffD1) 1); |
977 |
by (auto_tac (claset(),simpset() addsimps [ |
|
978 |
real_add_minus,real_add_zero_right] @ real_add_ac)); |
|
979 |
qed "real_lemma_add_positive_imp_less"; |
|
980 |
||
981 |
Goal "!!(R::real). ? T. 0r < T & R + T = S ==> R < S"; |
|
982 |
by (blast_tac (claset() addIs [real_lemma_add_positive_imp_less]) 1); |
|
983 |
qed "real_ex_add_positive_left_less"; |
|
984 |
||
985 |
(*** alternative definition for real_less ***) |
|
986 |
Goal "!!(R::real). (? T. 0r < T & R + T = S) = (R < S)"; |
|
5459 | 987 |
by (blast_tac (claset() addSIs [real_less_add_positive_left_Ex, |
5078 | 988 |
real_ex_add_positive_left_less]) 1); |
989 |
qed "real_less_iffdef"; |
|
990 |
||
991 |
Goal "(0r < x) = (%~x < x)"; |
|
5459 | 992 |
by Safe_tac; |
5078 | 993 |
by (rtac ccontr 2 THEN forward_tac |
994 |
[real_leI RS real_le_imp_less_or_eq] 2); |
|
995 |
by (Step_tac 2); |
|
996 |
by (dtac (real_minus_zero_less_iff RS iffD2) 2); |
|
5459 | 997 |
by (blast_tac (claset() addIs [real_less_trans]) 2); |
5078 | 998 |
by (auto_tac (claset(),simpset() addsimps |
999 |
[real_gt_zero_preal_Ex,real_preal_minus_less_self])); |
|
1000 |
qed "real_gt_zero_iff"; |
|
1001 |
||
1002 |
Goal "(x < 0r) = (x < %~x)"; |
|
1003 |
by (rtac (real_minus_zero_less_iff RS subst) 1); |
|
1004 |
by (stac real_gt_zero_iff 1); |
|
1005 |
by (Full_simp_tac 1); |
|
1006 |
qed "real_lt_zero_iff"; |
|
1007 |
||
1008 |
Goalw [real_le_def] "(0r <= x) = (%~x <= x)"; |
|
1009 |
by (auto_tac (claset(),simpset() addsimps [real_lt_zero_iff RS sym])); |
|
1010 |
qed "real_ge_zero_iff"; |
|
1011 |
||
1012 |
Goalw [real_le_def] "(x <= 0r) = (x <= %~x)"; |
|
1013 |
by (auto_tac (claset(),simpset() addsimps [real_gt_zero_iff RS sym])); |
|
1014 |
qed "real_le_zero_iff"; |
|
1015 |
||
1016 |
Goal "(%#m1 <= %#m2) = (m1 <= m2)"; |
|
1017 |
by (auto_tac (claset() addSIs [preal_leI], |
|
1018 |
simpset() addsimps [real_less_le_iff RS sym])); |
|
1019 |
by (dtac preal_le_less_trans 1 THEN assume_tac 1); |
|
1020 |
by (etac preal_less_irrefl 1); |
|
1021 |
qed "real_preal_le_iff"; |
|
1022 |
||
1023 |
Goal "!!(x::real). [| 0r < x; 0r < y |] ==> 0r < x * y"; |
|
1024 |
by (auto_tac (claset(),simpset() addsimps [real_gt_zero_preal_Ex])); |
|
1025 |
by (res_inst_tac [("x","y*ya")] exI 1); |
|
1026 |
by (full_simp_tac (simpset() addsimps [real_preal_mult]) 1); |
|
1027 |
qed "real_mult_order"; |
|
1028 |
||
1029 |
Goal "!!(x::real). [| x < 0r; y < 0r |] ==> 0r < x * y"; |
|
1030 |
by (REPEAT(dtac (real_minus_zero_less_iff RS iffD2) 1)); |
|
1031 |
by (dtac real_mult_order 1 THEN assume_tac 1); |
|
1032 |
by (Asm_full_simp_tac 1); |
|
1033 |
qed "real_mult_less_zero1"; |
|
1034 |
||
1035 |
Goal "!!(x::real). [| 0r <= x; 0r <= y |] ==> 0r <= x * y"; |
|
1036 |
by (REPEAT(dtac real_le_imp_less_or_eq 1)); |
|
1037 |
by (auto_tac (claset() addIs [real_mult_order, |
|
1038 |
real_less_imp_le],simpset() addsimps [real_le_refl])); |
|
1039 |
qed "real_le_mult_order"; |
|
1040 |
||
1041 |
Goal "!!(x::real). [| x <= 0r; y <= 0r |] ==> 0r <= x * y"; |
|
1042 |
by (rtac real_less_or_eq_imp_le 1); |
|
1043 |
by (dtac real_le_imp_less_or_eq 1 THEN etac disjE 1); |
|
1044 |
by Auto_tac; |
|
1045 |
by (dtac real_le_imp_less_or_eq 1); |
|
1046 |
by (auto_tac (claset() addDs [real_mult_less_zero1],simpset())); |
|
1047 |
qed "real_mult_le_zero1"; |
|
1048 |
||
1049 |
Goal "!!(x::real). [| 0r <= x; y < 0r |] ==> x * y <= 0r"; |
|
1050 |
by (rtac real_less_or_eq_imp_le 1); |
|
1051 |
by (dtac real_le_imp_less_or_eq 1 THEN etac disjE 1); |
|
1052 |
by Auto_tac; |
|
1053 |
by (dtac (real_minus_zero_less_iff RS iffD2) 1); |
|
1054 |
by (rtac (real_minus_zero_less_iff RS subst) 1); |
|
1055 |
by (blast_tac (claset() addDs [real_mult_order] |
|
1056 |
addIs [real_minus_mult_eq2 RS ssubst]) 1); |
|
1057 |
qed "real_mult_le_zero"; |
|
1058 |
||
1059 |
Goal "!!(x::real). [| 0r < x; y < 0r |] ==> x*y < 0r"; |
|
1060 |
by (dtac (real_minus_zero_less_iff RS iffD2) 1); |
|
1061 |
by (dtac real_mult_order 1 THEN assume_tac 1); |
|
1062 |
by (rtac (real_minus_zero_less_iff RS iffD1) 1); |
|
1063 |
by (asm_full_simp_tac (simpset() addsimps [real_minus_mult_eq2]) 1); |
|
1064 |
qed "real_mult_less_zero"; |
|
1065 |
||
1066 |
Goalw [real_one_def] "0r < 1r"; |
|
1067 |
by (auto_tac (claset() addIs [real_gt_zero_preal_Ex RS iffD2], |
|
1068 |
simpset() addsimps [real_preal_def])); |
|
1069 |
qed "real_zero_less_one"; |
|
1070 |
||
1071 |
(*** Completeness of reals ***) |
|
1072 |
(** use supremum property of preal and theorems about real_preal **) |
|
1073 |
(*** a few lemmas ***) |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
1074 |
Goal "! x:P. 0r < x ==> ((? x:P. y < x) = (? X. %#X : P & y < %#X))"; |
5078 | 1075 |
by (blast_tac (claset() addSDs [bspec,real_gt_zero_preal_Ex RS iffD1]) 1); |
1076 |
qed "real_sup_lemma1"; |
|
1077 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
1078 |
Goal "[| ! x:P. 0r < x; ? x. x: P; ? y. !x: P. x < y |] \ |
5078 | 1079 |
\ ==> (? X. X: {w. %#w : P}) & (? Y. !X: {w. %#w : P}. X < Y)"; |
1080 |
by (rtac conjI 1); |
|
1081 |
by (blast_tac (claset() addDs [bspec,real_gt_zero_preal_Ex RS iffD1]) 1); |
|
1082 |
by Auto_tac; |
|
1083 |
by (dtac bspec 1 THEN assume_tac 1); |
|
1084 |
by (forward_tac [bspec] 1 THEN assume_tac 1); |
|
1085 |
by (dtac real_less_trans 1 THEN assume_tac 1); |
|
1086 |
by (dtac (real_gt_zero_preal_Ex RS iffD1) 1 THEN etac exE 1); |
|
1087 |
by (res_inst_tac [("x","ya")] exI 1); |
|
1088 |
by Auto_tac; |
|
1089 |
by (dres_inst_tac [("x","%#X")] bspec 1 THEN assume_tac 1); |
|
1090 |
by (etac real_preal_lessD 1); |
|
1091 |
qed "real_sup_lemma2"; |
|
1092 |
||
1093 |
(*------------------------------------------------------------- |
|
1094 |
Completeness of Positive Reals |
|
1095 |
-------------------------------------------------------------*) |
|
1096 |
||
1097 |
(* Supremum property for the set of positive reals *) |
|
1098 |
(* FIXME: long proof - can be improved - need only have one case split *) |
|
1099 |
(* will do for now *) |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
1100 |
Goal "[| ! x:P. 0r < x; ? x. x: P; ? y. !x: P. x < y |] \ |
5078 | 1101 |
\ ==> (? S. !y. (? x: P. y < x) = (y < S))"; |
1102 |
by (res_inst_tac [("x","%#psup({w. %#w : P})")] exI 1); |
|
1103 |
by Auto_tac; |
|
1104 |
by (forward_tac [real_sup_lemma2] 1 THEN Auto_tac); |
|
1105 |
by (case_tac "0r < ya" 1); |
|
1106 |
by (dtac (real_gt_zero_preal_Ex RS iffD1) 1); |
|
1107 |
by (dtac real_less_all_real2 2); |
|
1108 |
by Auto_tac; |
|
1109 |
by (rtac (preal_complete RS spec RS iffD1) 1); |
|
1110 |
by Auto_tac; |
|
1111 |
by (forward_tac [real_gt_preal_preal_Ex] 1); |
|
1112 |
by Auto_tac; |
|
1113 |
(* second part *) |
|
1114 |
by (rtac (real_sup_lemma1 RS iffD2) 1 THEN assume_tac 1); |
|
1115 |
by (case_tac "0r < ya" 1); |
|
1116 |
by (auto_tac (claset() addSDs [real_less_all_real2, |
|
1117 |
real_gt_zero_preal_Ex RS iffD1],simpset())); |
|
1118 |
by (forward_tac [real_sup_lemma2] 2 THEN Auto_tac); |
|
1119 |
by (forward_tac [real_sup_lemma2] 1 THEN Auto_tac); |
|
1120 |
by (rtac (preal_complete RS spec RS iffD2 RS bexE) 1); |
|
5459 | 1121 |
by (Blast_tac 3); |
1122 |
by (Blast_tac 1); |
|
1123 |
by (Blast_tac 1); |
|
5078 | 1124 |
by (Blast_tac 1); |
1125 |
qed "posreal_complete"; |
|
1126 |
||
1127 |
(*------------------------------------------------------------------ |
|
1128 |
||
1129 |
------------------------------------------------------------------*) |
|
1130 |
||
1131 |
Goal "!!(A::real). A < B ==> A + C < B + C"; |
|
1132 |
by (dtac (real_less_iffdef RS iffD2) 1); |
|
1133 |
by (rtac (real_less_iffdef RS iffD1) 1); |
|
1134 |
by (REPEAT(Step_tac 1)); |
|
1135 |
by (full_simp_tac (simpset() addsimps real_add_ac) 1); |
|
1136 |
qed "real_add_less_mono1"; |
|
1137 |
||
1138 |
Goal "!!(A::real). A < B ==> C + A < C + B"; |
|
1139 |
by (auto_tac (claset() addIs [real_add_less_mono1], |
|
1140 |
simpset() addsimps [real_add_commute])); |
|
1141 |
qed "real_add_less_mono2"; |
|
1142 |
||
1143 |
Goal "!!(A::real). A + C < B + C ==> A < B"; |
|
1144 |
by (dres_inst_tac [("C","%~C")] real_add_less_mono1 1); |
|
1145 |
by (asm_full_simp_tac (simpset() addsimps [real_add_assoc, |
|
1146 |
real_add_minus,real_add_zero_right]) 1); |
|
1147 |
qed "real_less_add_right_cancel"; |
|
1148 |
||
1149 |
Goal "!!(A::real). C + A < C + B ==> A < B"; |
|
1150 |
by (dres_inst_tac [("C","%~C")] real_add_less_mono2 1); |
|
1151 |
by (asm_full_simp_tac (simpset() addsimps [real_add_assoc RS sym, |
|
1152 |
real_add_minus_left,real_add_zero_left]) 1); |
|
1153 |
qed "real_less_add_left_cancel"; |
|
1154 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
1155 |
Goal "[| 0r < x; 0r < y |] ==> 0r < x + y"; |
5078 | 1156 |
by (REPEAT(dtac (real_gt_zero_preal_Ex RS iffD1) 1)); |
1157 |
by (rtac (real_gt_zero_preal_Ex RS iffD2) 1); |
|
1158 |
by (Step_tac 1); |
|
1159 |
by (res_inst_tac [("x","y + ya")] exI 1); |
|
1160 |
by (full_simp_tac (simpset() addsimps [real_preal_add]) 1); |
|
1161 |
qed "real_add_order"; |
|
1162 |
||
1163 |
Goal "!!(x::real). [| 0r <= x; 0r <= y |] ==> 0r <= x + y"; |
|
1164 |
by (REPEAT(dtac real_le_imp_less_or_eq 1)); |
|
1165 |
by (auto_tac (claset() addIs [real_add_order, |
|
1166 |
real_less_imp_le],simpset() addsimps [real_add_zero_left, |
|
1167 |
real_add_zero_right,real_le_refl])); |
|
1168 |
qed "real_le_add_order"; |
|
1169 |
||
1170 |
Goal |
|
5148
74919e8f221c
More tidying and removal of "\!\!... from Goal commands
paulson
parents:
5143
diff
changeset
|
1171 |
"[| R1 < S1; R2 < S2 |] ==> R1 + R2 < S1 + (S2::real)"; |
5078 | 1172 |
by (dtac (real_less_iffdef RS iffD2) 1); |
1173 |
by (dtac (real_less_iffdef RS iffD2) 1); |
|
1174 |
by (rtac (real_less_iffdef RS iffD1) 1); |
|
1175 |
by Auto_tac; |
|
1176 |
by (res_inst_tac [("x","T + Ta")] exI 1); |
|
1177 |
by (auto_tac (claset(),simpset() addsimps [real_add_order] @ real_add_ac)); |
|
1178 |
qed "real_add_less_mono"; |
|
1179 |
||
1180 |
Goal "!!(x::real). [| 0r <= x; 0r <= y |] ==> 0r <= x + y"; |
|
1181 |
by (REPEAT(dtac real_le_imp_less_or_eq 1)); |
|
1182 |
by (auto_tac (claset() addIs [real_add_order, |
|
1183 |
real_less_imp_le],simpset() addsimps [real_add_zero_left, |
|
1184 |
real_add_zero_right,real_le_refl])); |
|
1185 |
qed "real_le_add_order"; |
|
1186 |
||
1187 |
Goal "!!(q1::real). q1 <= q2 ==> x + q1 <= x + q2"; |
|
1188 |
by (dtac real_le_imp_less_or_eq 1); |
|
1189 |
by (Step_tac 1); |
|
1190 |
by (auto_tac (claset() addSIs [real_le_refl, |
|
1191 |
real_less_imp_le,real_add_less_mono1], |
|
1192 |
simpset() addsimps [real_add_commute])); |
|
1193 |
qed "real_add_left_le_mono1"; |
|
1194 |
||
1195 |
Goal "!!(q1::real). q1 <= q2 ==> q1 + x <= q2 + x"; |
|
1196 |
by (auto_tac (claset() addDs [real_add_left_le_mono1], |
|
1197 |
simpset() addsimps [real_add_commute])); |
|
1198 |
qed "real_add_le_mono1"; |
|
1199 |
||
1200 |
Goal "!!k l::real. [|i<=j; k<=l |] ==> i + k <= j + l"; |
|
1201 |
by (etac (real_add_le_mono1 RS real_le_trans) 1); |
|
1202 |
by (simp_tac (simpset() addsimps [real_add_commute]) 1); |
|
1203 |
(*j moves to the end because it is free while k, l are bound*) |
|
1204 |
by (etac real_add_le_mono1 1); |
|
1205 |
qed "real_add_le_mono"; |
|
1206 |
||
1207 |
Goal "EX (x::real). x < y"; |
|
1208 |
by (rtac (real_add_zero_right RS subst) 1); |
|
1209 |
by (res_inst_tac [("x","y + %~1r")] exI 1); |
|
1210 |
by (auto_tac (claset() addSIs [real_add_less_mono2], |
|
1211 |
simpset() addsimps [real_minus_zero_less_iff2, |
|
1212 |
real_zero_less_one])); |
|
1213 |
qed "real_less_Ex"; |
|
1214 |
(*--------------------------------------------------------------------------------- |
|
1215 |
An embedding of the naturals in the reals |
|
1216 |
---------------------------------------------------------------------------------*) |
|
1217 |
||
1218 |
Goalw [real_nat_def] "%%#0 = 1r"; |
|
1219 |
by (full_simp_tac (simpset() addsimps [pnat_one_iff RS sym,real_preal_def]) 1); |
|
1220 |
by (fold_tac [real_one_def]); |
|
1221 |
by (rtac refl 1); |
|
1222 |
qed "real_nat_one"; |
|
1223 |
||
1224 |
Goalw [real_nat_def] "%%#1 = 1r + 1r"; |
|
1225 |
by (full_simp_tac (simpset() addsimps [real_preal_def,real_one_def, |
|
1226 |
pnat_two_eq,real_add,prat_pnat_add RS sym,preal_prat_add RS sym |
|
1227 |
] @ pnat_add_ac) 1); |
|
1228 |
qed "real_nat_two"; |
|
1229 |
||
1230 |
Goalw [real_nat_def] |
|
1231 |
"%%#n1 + %%#n2 = %%#(n1 + n2) + 1r"; |
|
1232 |
by (full_simp_tac (simpset() addsimps [real_nat_one RS sym, |
|
1233 |
real_nat_def,real_preal_add RS sym,preal_prat_add RS sym, |
|
1234 |
prat_pnat_add RS sym,pnat_nat_add]) 1); |
|
1235 |
qed "real_nat_add"; |
|
1236 |
||
1237 |
Goal "%%#(n + 1) = %%#n + 1r"; |
|
1238 |
by (res_inst_tac [("x1","1r")] (real_add_right_cancel RS iffD1) 1); |
|
1239 |
by (rtac (real_nat_add RS subst) 1); |
|
1240 |
by (full_simp_tac (simpset() addsimps [real_nat_two,real_add_assoc]) 1); |
|
1241 |
qed "real_nat_add_one"; |
|
1242 |
||
1243 |
Goal "Suc n = n + 1"; |
|
1244 |
by Auto_tac; |
|
1245 |
qed "lemma"; |
|
1246 |
||
1247 |
Goal "%%#Suc n = %%#n + 1r"; |
|
1248 |
by (stac lemma 1); |
|
1249 |
by (rtac real_nat_add_one 1); |
|
1250 |
qed "real_nat_Suc"; |
|
1251 |
||
1252 |
Goal "inj(real_nat)"; |
|
1253 |
by (rtac injI 1); |
|
1254 |
by (rewtac real_nat_def); |
|
1255 |
by (dtac (inj_real_preal RS injD) 1); |
|
1256 |
by (dtac (inj_preal_prat RS injD) 1); |
|
1257 |
by (dtac (inj_prat_pnat RS injD) 1); |
|
1258 |
by (etac (inj_pnat_nat RS injD) 1); |
|
1259 |
qed "inj_real_nat"; |
|
1260 |
||
1261 |
Goalw [real_nat_def] "0r < %%#n"; |
|
1262 |
by (rtac (real_gt_zero_preal_Ex RS iffD2) 1); |
|
1263 |
by (Blast_tac 1); |
|
1264 |
qed "real_nat_less_zero"; |
|
1265 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
1266 |
Goal "1r <= %%#n"; |
5078 | 1267 |
by (simp_tac (simpset() addsimps [real_nat_one RS sym]) 1); |
5184 | 1268 |
by (induct_tac "n" 1); |
5078 | 1269 |
by (auto_tac (claset(),simpset () |
1270 |
addsimps [real_nat_Suc,real_le_refl,real_nat_one])); |
|
1271 |
by (res_inst_tac [("t","1r")] (real_add_zero_left RS subst) 1); |
|
1272 |
by (rtac real_add_le_mono 1); |
|
1273 |
by (auto_tac (claset(),simpset () |
|
1274 |
addsimps [real_le_refl,real_nat_less_zero, |
|
1275 |
real_less_imp_le,real_add_zero_left])); |
|
1276 |
qed "real_nat_less_one"; |
|
1277 |
||
1278 |
Goal "rinv(%%#n) ~= 0r"; |
|
1279 |
by (rtac ((real_nat_less_zero RS |
|
1280 |
real_not_refl2 RS not_sym) RS rinv_not_zero) 1); |
|
1281 |
qed "real_nat_rinv_not_zero"; |
|
1282 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
1283 |
Goal "rinv(%%#x) = rinv(%%#y) ==> x = y"; |
5078 | 1284 |
by (rtac (inj_real_nat RS injD) 1); |
1285 |
by (res_inst_tac [("n2","x")] |
|
1286 |
(real_nat_rinv_not_zero RS real_mult_left_cancel RS iffD1) 1); |
|
1287 |
by (full_simp_tac (simpset() addsimps [(real_nat_less_zero RS |
|
1288 |
real_not_refl2 RS not_sym) RS real_mult_inv_left]) 1); |
|
1289 |
by (asm_full_simp_tac (simpset() addsimps [(real_nat_less_zero RS |
|
1290 |
real_not_refl2 RS not_sym)]) 1); |
|
1291 |
qed "real_nat_rinv_inj"; |
|
1292 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
1293 |
Goal "0r < x ==> 0r < rinv x"; |
5078 | 1294 |
by (EVERY1[rtac ccontr, dtac real_leI]); |
1295 |
by (forward_tac [real_minus_zero_less_iff2 RS iffD2] 1); |
|
1296 |
by (forward_tac [real_not_refl2 RS not_sym] 1); |
|
1297 |
by (dtac (real_not_refl2 RS not_sym RS rinv_not_zero) 1); |
|
1298 |
by (EVERY1[dtac real_le_imp_less_or_eq, Step_tac]); |
|
1299 |
by (dtac real_mult_less_zero1 1 THEN assume_tac 1); |
|
1300 |
by (auto_tac (claset() addIs [real_zero_less_one RS real_less_asym], |
|
1301 |
simpset() addsimps [real_minus_mult_eq1 RS sym])); |
|
1302 |
qed "real_rinv_gt_zero"; |
|
1303 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
1304 |
Goal "x < 0r ==> rinv x < 0r"; |
5078 | 1305 |
by (forward_tac [real_not_refl2] 1); |
1306 |
by (dtac (real_minus_zero_less_iff RS iffD2) 1); |
|
1307 |
by (rtac (real_minus_zero_less_iff RS iffD1) 1); |
|
1308 |
by (dtac (real_minus_rinv RS sym) 1); |
|
1309 |
by (auto_tac (claset() addIs [real_rinv_gt_zero], |
|
1310 |
simpset())); |
|
1311 |
qed "real_rinv_less_zero"; |
|
1312 |
||
1313 |
Goal "x+x=x*(1r+1r)"; |
|
1314 |
by (simp_tac (simpset() addsimps [real_add_mult_distrib2]) 1); |
|
1315 |
qed "real_add_self"; |
|
1316 |
||
1317 |
Goal "x < x + 1r"; |
|
1318 |
by (rtac (real_less_sum_gt_0_iff RS iffD1) 1); |
|
1319 |
by (full_simp_tac (simpset() addsimps [real_zero_less_one, |
|
1320 |
real_add_assoc,real_add_minus,real_add_zero_right, |
|
1321 |
real_add_left_commute]) 1); |
|
1322 |
qed "real_self_less_add_one"; |
|
1323 |
||
1324 |
Goal "1r < 1r + 1r"; |
|
1325 |
by (rtac real_self_less_add_one 1); |
|
1326 |
qed "real_one_less_two"; |
|
1327 |
||
1328 |
Goal "0r < 1r + 1r"; |
|
1329 |
by (rtac ([real_zero_less_one, |
|
1330 |
real_one_less_two] MRS real_less_trans) 1); |
|
1331 |
qed "real_zero_less_two"; |
|
1332 |
||
1333 |
Goal "1r + 1r ~= 0r"; |
|
1334 |
by (rtac (real_zero_less_two RS real_not_refl2 RS not_sym) 1); |
|
1335 |
qed "real_two_not_zero"; |
|
1336 |
||
1337 |
Addsimps [real_two_not_zero]; |
|
1338 |
||
1339 |
Goal "x*rinv(1r + 1r) + x*rinv(1r + 1r) = x"; |
|
1340 |
by (stac real_add_self 1); |
|
1341 |
by (full_simp_tac (simpset() addsimps [real_mult_assoc]) 1); |
|
1342 |
qed "real_sum_of_halves"; |
|
1343 |
||
1344 |
Goal "!!(x::real). [| 0r<z; x<y |] ==> x*z<y*z"; |
|
1345 |
by (rotate_tac 1 1); |
|
1346 |
by (dtac real_less_sum_gt_zero 1); |
|
1347 |
by (rtac real_sum_gt_zero_less 1); |
|
1348 |
by (dtac real_mult_order 1 THEN assume_tac 1); |
|
1349 |
by (asm_full_simp_tac (simpset() addsimps [real_add_mult_distrib2, |
|
1350 |
real_minus_mult_eq2 RS sym, real_mult_commute ]) 1); |
|
1351 |
qed "real_mult_less_mono1"; |
|
1352 |
||
1353 |
Goal "!!(y::real). [| 0r<z; x<y |] ==> z*x<z*y"; |
|
1354 |
by (asm_simp_tac (simpset() addsimps [real_mult_commute,real_mult_less_mono1]) 1); |
|
1355 |
qed "real_mult_less_mono2"; |
|
1356 |
||
1357 |
Goal "!!(x::real). [| 0r<z; x*z<y*z |] ==> x<y"; |
|
1358 |
by (forw_inst_tac [("x","x*z")] (real_rinv_gt_zero |
|
1359 |
RS real_mult_less_mono1) 1); |
|
1360 |
by (auto_tac (claset(),simpset() addsimps |
|
1361 |
[real_mult_assoc,real_not_refl2 RS not_sym])); |
|
1362 |
qed "real_mult_less_cancel1"; |
|
1363 |
||
1364 |
Goal "!!(x::real). [| 0r<z; z*x<z*y |] ==> x<y"; |
|
1365 |
by (etac real_mult_less_cancel1 1); |
|
1366 |
by (asm_full_simp_tac (simpset() addsimps [real_mult_commute]) 1); |
|
1367 |
qed "real_mult_less_cancel2"; |
|
1368 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
1369 |
Goal "0r < z ==> (x*z < y*z) = (x < y)"; |
5078 | 1370 |
by (blast_tac (claset() addIs [real_mult_less_mono1, |
1371 |
real_mult_less_cancel1]) 1); |
|
1372 |
qed "real_mult_less_iff1"; |
|
1373 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
1374 |
Goal "0r < z ==> (z*x < z*y) = (x < y)"; |
5078 | 1375 |
by (blast_tac (claset() addIs [real_mult_less_mono2, |
1376 |
real_mult_less_cancel2]) 1); |
|
1377 |
qed "real_mult_less_iff2"; |
|
1378 |
||
1379 |
Addsimps [real_mult_less_iff1,real_mult_less_iff2]; |
|
1380 |
||
1381 |
Goal "!!(x::real). [| 0r<=z; x<y |] ==> x*z<=y*z"; |
|
1382 |
by (EVERY1 [rtac real_less_or_eq_imp_le, dtac real_le_imp_less_or_eq]); |
|
1383 |
by (auto_tac (claset() addIs [real_mult_less_mono1],simpset())); |
|
1384 |
qed "real_mult_le_less_mono1"; |
|
1385 |
||
1386 |
Goal "!!(x::real). [| 0r<=z; x<y |] ==> z*x<=z*y"; |
|
1387 |
by (asm_simp_tac (simpset() addsimps [real_mult_commute,real_mult_le_less_mono1]) 1); |
|
1388 |
qed "real_mult_le_less_mono2"; |
|
1389 |
||
1390 |
Goal "!!x y (z::real). [| 0r<=z; x<=y |] ==> z*x<=z*y"; |
|
1391 |
by (dres_inst_tac [("x","x")] real_le_imp_less_or_eq 1); |
|
1392 |
by (auto_tac (claset() addIs [real_mult_le_less_mono2,real_le_refl],simpset())); |
|
1393 |
qed "real_mult_le_le_mono1"; |
|
1394 |
||
1395 |
Goal "!!(x::real). x < y ==> x < (x + y)*rinv(1r + 1r)"; |
|
1396 |
by (dres_inst_tac [("C","x")] real_add_less_mono2 1); |
|
1397 |
by (dtac (real_add_self RS subst) 1); |
|
1398 |
by (dtac (real_zero_less_two RS real_rinv_gt_zero RS |
|
1399 |
real_mult_less_mono1) 1); |
|
1400 |
by (asm_full_simp_tac (simpset() addsimps [real_mult_assoc]) 1); |
|
1401 |
qed "real_less_half_sum"; |
|
1402 |
||
1403 |
Goal "!!(x::real). x < y ==> (x + y)*rinv(1r + 1r) < y"; |
|
1404 |
by (dres_inst_tac [("C","y")] real_add_less_mono1 1); |
|
1405 |
by (dtac (real_add_self RS subst) 1); |
|
1406 |
by (dtac (real_zero_less_two RS real_rinv_gt_zero RS |
|
1407 |
real_mult_less_mono1) 1); |
|
1408 |
by (asm_full_simp_tac (simpset() addsimps [real_mult_assoc]) 1); |
|
1409 |
qed "real_gt_half_sum"; |
|
1410 |
||
1411 |
Goal "!!(x::real). x < y ==> EX r. x < r & r < y"; |
|
1412 |
by (blast_tac (claset() addSIs [real_less_half_sum,real_gt_half_sum]) 1); |
|
1413 |
qed "real_dense"; |
|
1414 |
||
1415 |
Goal "(EX n. rinv(%%#n) < r) = (EX n. 1r < r * %%#n)"; |
|
1416 |
by (Step_tac 1); |
|
1417 |
by (dres_inst_tac [("n1","n")] (real_nat_less_zero |
|
1418 |
RS real_mult_less_mono1) 1); |
|
1419 |
by (dres_inst_tac [("n2","n")] (real_nat_less_zero RS |
|
1420 |
real_rinv_gt_zero RS real_mult_less_mono1) 2); |
|
1421 |
by (auto_tac (claset(),simpset() addsimps [(real_nat_less_zero RS |
|
1422 |
real_not_refl2 RS not_sym),real_mult_assoc])); |
|
1423 |
qed "real_nat_rinv_Ex_iff"; |
|
1424 |
||
1425 |
Goalw [real_nat_def] "(%%#n < %%#m) = (n < m)"; |
|
1426 |
by Auto_tac; |
|
1427 |
qed "real_nat_less_iff"; |
|
1428 |
||
1429 |
Addsimps [real_nat_less_iff]; |
|
1430 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
1431 |
Goal "0r < u ==> (u < rinv (%%#n)) = (%%#n < rinv(u))"; |
5078 | 1432 |
by (Step_tac 1); |
1433 |
by (res_inst_tac [("n2","n")] (real_nat_less_zero RS |
|
1434 |
real_rinv_gt_zero RS real_mult_less_cancel1) 1); |
|
1435 |
by (res_inst_tac [("x1","u")] ( real_rinv_gt_zero |
|
1436 |
RS real_mult_less_cancel1) 2); |
|
1437 |
by (auto_tac (claset(),simpset() addsimps [real_nat_less_zero, |
|
1438 |
real_not_refl2 RS not_sym])); |
|
1439 |
by (res_inst_tac [("z","u")] real_mult_less_cancel2 1); |
|
1440 |
by (res_inst_tac [("n1","n")] (real_nat_less_zero RS |
|
1441 |
real_mult_less_cancel2) 3); |
|
1442 |
by (auto_tac (claset(),simpset() addsimps [real_nat_less_zero, |
|
1443 |
real_not_refl2 RS not_sym,real_mult_assoc RS sym])); |
|
1444 |
qed "real_nat_less_rinv_iff"; |
|
1445 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
1446 |
Goal "0r < u ==> (u = rinv(%%#n)) = (%%#n = rinv u)"; |
5078 | 1447 |
by (auto_tac (claset(),simpset() addsimps [real_rinv_rinv, |
1448 |
real_nat_less_zero,real_not_refl2 RS not_sym])); |
|
1449 |
qed "real_nat_rinv_eq_iff"; |
|
1450 |
||
1451 |
(* |
|
1452 |
(*------------------------------------------------------------------ |
|
1453 |
lemmas about upper bounds and least upper bound |
|
1454 |
------------------------------------------------------------------*) |
|
5148
74919e8f221c
More tidying and removal of "\!\!... from Goal commands
paulson
parents:
5143
diff
changeset
|
1455 |
Goalw [real_ub_def] "[| real_ub u S; x : S |] ==> x <= u"; |
5078 | 1456 |
by Auto_tac; |
1457 |
qed "real_ubD"; |
|
1458 |
||
1459 |
*) |