--- a/src/HOL/Real/RComplete.ML Tue Jan 27 09:44:14 2004 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,250 +0,0 @@
-(* Title : HOL/Real/RComplete.ML
- ID : $Id$
- Author : Jacques D. Fleuriot
- Copyright : 1998 University of Cambridge
-
-Completeness theorems for positive reals and reals.
-*)
-
-Goal "x/2 + x/2 = (x::real)";
-by (Simp_tac 1);
-qed "real_sum_of_halves";
-
-(*---------------------------------------------------------
- Completeness of reals: use supremum property of
- preal and theorems about real_preal. Theorems
- previously in Real.ML.
- ---------------------------------------------------------*)
- (*a few lemmas*)
-Goal "ALL x:P. 0 < x ==> \
-\ ((EX x:P. y < x) = (EX X. real_of_preal X : P & \
-\ y < real_of_preal X))";
-by (blast_tac (claset() addSDs [bspec,
- real_gt_zero_preal_Ex RS iffD1]) 1);
-qed "real_sup_lemma1";
-
-Goal "[| ALL x:P. 0 < x; a: P; ALL x: P. x < y |] \
-\ ==> (EX X. X: {w. real_of_preal w : P}) & \
-\ (EX Y. ALL X: {w. real_of_preal w : P}. X < Y)";
-by (rtac conjI 1);
-by (blast_tac (claset() addDs [bspec,
- real_gt_zero_preal_Ex RS iffD1]) 1);
-by Auto_tac;
-by (dtac bspec 1 THEN assume_tac 1);
-by (ftac bspec 1 THEN assume_tac 1);
-by (dtac order_less_trans 1 THEN assume_tac 1);
-by (dtac (real_gt_zero_preal_Ex RS iffD1) 1);
-by (Force_tac 1);
-qed "real_sup_lemma2";
-
-(*-------------------------------------------------------------
- Completeness of Positive Reals
- -------------------------------------------------------------*)
-
-(**
- Supremum property for the set of positive reals
- FIXME: long proof - should be improved
-**)
-
-(*Let P be a non-empty set of positive reals, with an upper bound y.
- Then P has a least upper bound (written S).*)
-Goal "[| ALL x:P. (0::real) < x; EX x. x: P; EX y. ALL x: P. x<y |] \
-\ ==> (EX S. ALL y. (EX x: P. y < x) = (y < S))";
-by (res_inst_tac
- [("x","real_of_preal (psup({w. real_of_preal w : P}))")] exI 1);
-by (Clarify_tac 1);
-by (case_tac "0 < ya" 1);
-by Auto_tac;
-by (ftac real_sup_lemma2 1 THEN REPEAT (assume_tac 1));
-by (dtac (real_gt_zero_preal_Ex RS iffD1) 1);
-by (dtac (real_less_all_real2) 3);
-by Auto_tac;
-by (rtac (preal_complete RS spec RS iffD1) 1);
-by Auto_tac;
-by (ftac real_gt_preal_preal_Ex 1);
-by (Force_tac 1);
-(* second part *)
-by (rtac (real_sup_lemma1 RS iffD2) 1 THEN assume_tac 1);
-by (auto_tac (claset() addSDs [real_less_all_real2,
- real_gt_zero_preal_Ex RS iffD1],
- simpset()));
-by (ftac real_sup_lemma2 2 THEN REPEAT (assume_tac 1));
-by (ftac real_sup_lemma2 1 THEN REPEAT (assume_tac 1));
-by (rtac (preal_complete RS spec RS iffD2 RS bexE) 1);
-by (Blast_tac 3);
-by (ALLGOALS(Blast_tac));
-qed "posreal_complete";
-
-(*--------------------------------------------------------
- Completeness properties using isUb, isLub etc.
- -------------------------------------------------------*)
-
-Goal "[| isLub R S x; isLub R S y |] ==> x = (y::real)";
-by (ftac isLub_isUb 1);
-by (forw_inst_tac [("x","y")] isLub_isUb 1);
-by (blast_tac (claset() addSIs [real_le_anti_sym]
- addSDs [isLub_le_isUb]) 1);
-qed "real_isLub_unique";
-
-Goalw [setle_def,setge_def] "[| (x::real) <=* S'; S <= S' |] ==> x <=* S";
-by (Blast_tac 1);
-qed "real_order_restrict";
-
-(*----------------------------------------------------------------
- Completeness theorem for the positive reals(again)
- ----------------------------------------------------------------*)
-
-Goal "[| ALL x: S. 0 < x; \
-\ EX x. x: S; \
-\ EX u. isUb (UNIV::real set) S u \
-\ |] ==> EX t. isLub (UNIV::real set) S t";
-by (res_inst_tac
- [("x","real_of_preal(psup({w. real_of_preal w : S}))")] exI 1);
-by (auto_tac (claset(), simpset() addsimps
- [isLub_def,leastP_def,isUb_def]));
-by (auto_tac (claset() addSIs [setleI,setgeI]
- addSDs [(real_gt_zero_preal_Ex) RS iffD1],
- simpset()));
-by (forw_inst_tac [("x","y")] bspec 1 THEN assume_tac 1);
-by (dtac ((real_gt_zero_preal_Ex) RS iffD1) 1);
-by (auto_tac (claset(), simpset() addsimps [real_of_preal_le_iff]));
-by (rtac preal_psup_leI2a 1);
-by (forw_inst_tac [("y","real_of_preal ya")] setleD 1 THEN assume_tac 1);
-by (ftac real_ge_preal_preal_Ex 1);
-by (Step_tac 1);
-by (res_inst_tac [("x","y")] exI 1);
-by (blast_tac (claset() addSDs [setleD] addIs [real_of_preal_le_iff RS iffD1]) 1);
-by (forw_inst_tac [("x","x")] bspec 1 THEN assume_tac 1);
-by (ftac isUbD2 1);
-by (dtac ((real_gt_zero_preal_Ex) RS iffD1) 1);
-by (auto_tac (claset() addSDs [isUbD, real_ge_preal_preal_Ex],
- simpset() addsimps [real_of_preal_le_iff]));
-by (blast_tac (claset() addSDs [setleD] addSIs [psup_le_ub1]
- addIs [real_of_preal_le_iff RS iffD1]) 1);
-qed "posreals_complete";
-
-
-(*-------------------------------
- Lemmas
- -------------------------------*)
-Goal "ALL y : {z. EX x: P. z = x + (-xa) + 1} Int {x. 0 < x}. 0 < y";
-by Auto_tac;
-qed "real_sup_lemma3";
-
-Goal "(xa <= S + X + (-Z)) = (xa + (-X) + Z <= (S::real))";
-by Auto_tac;
-qed "lemma_le_swap2";
-
-Goal "[| (x::real) + (-X) + 1 <= S; xa <= x |] ==> xa <= S + X + (- 1)";
-by (arith_tac 1);
-by Auto_tac;
-qed "lemma_real_complete2b";
-
-(*----------------------------------------------------------
- reals Completeness (again!)
- ----------------------------------------------------------*)
-Goal "[| EX X. X: S; EX Y. isUb (UNIV::real set) S Y |] \
-\ ==> EX t. isLub (UNIV :: real set) S t";
-by (Step_tac 1);
-by (subgoal_tac "EX u. u: {z. EX x: S. z = x + (-X) + 1} \
-\ Int {x. 0 < x}" 1);
-by (subgoal_tac "isUb (UNIV::real set) ({z. EX x: S. z = x + (-X) + 1} \
-\ Int {x. 0 < x}) (Y + (-X) + 1)" 1);
-by (cut_inst_tac [("P","S"),("xa","X")] real_sup_lemma3 1);
-by (EVERY1[forward_tac [exI RSN (3,posreals_complete)], Blast_tac, Blast_tac,
- Step_tac]);
-by (res_inst_tac [("x","t + X + (- 1)")] exI 1);
-by (rtac isLubI2 1);
-by (rtac setgeI 2 THEN Step_tac 2);
-by (subgoal_tac "isUb (UNIV:: real set) ({z. EX x: S. z = x + (-X) + 1} \
-\ Int {x. 0 < x}) (y + (-X) + 1)" 2);
-by (dres_inst_tac [("y","(y + (- X) + 1)")] isLub_le_isUb 2
- THEN assume_tac 2);
-by (full_simp_tac
- (simpset() addsimps [real_diff_def, diff_le_eq RS sym] @
- add_ac) 2);
-by (rtac (setleI RS isUbI) 1);
-by (Step_tac 1);
-by (res_inst_tac [("R1.0","x"),("R2.0","y")] real_linear_less2 1);
-by (stac lemma_le_swap2 1);
-by (ftac isLubD2 1 THEN assume_tac 2);
-by (Step_tac 1);
-by (Blast_tac 1);
-by (arith_tac 1);
-by (stac lemma_le_swap2 1);
-by (ftac isLubD2 1 THEN assume_tac 2);
-by (Blast_tac 1);
-by (rtac lemma_real_complete2b 1);
-by (etac order_less_imp_le 2);
-by (blast_tac (claset() addSIs [isLubD2]) 1 THEN Step_tac 1);
-by (full_simp_tac (simpset() addsimps [real_add_assoc]) 1);
-by (blast_tac (claset() addDs [isUbD] addSIs [setleI RS isUbI]
- addIs [add_right_mono]) 1);
-by (blast_tac (claset() addDs [isUbD] addSIs [setleI RS isUbI]
- addIs [add_right_mono]) 1);
-by (Auto_tac);
-qed "reals_complete";
-
-(*----------------------------------------------------------------
- Related: Archimedean property of reals
- ----------------------------------------------------------------*)
-
-Goal "0 < real (Suc n)";
-by (res_inst_tac [("y","real n")] order_le_less_trans 1);
-by (rtac (real_of_nat_ge_zero) 1);
-by (simp_tac (simpset() addsimps [real_of_nat_Suc]) 1);
-qed "real_of_nat_Suc_gt_zero";
-
-Goal "0 < x ==> EX n. inverse (real(Suc n)) < x";
-by (rtac ccontr 1);
-by (subgoal_tac "ALL n. x * real (Suc n) <= 1" 1);
-by (asm_full_simp_tac
- (simpset() addsimps [linorder_not_less, inverse_eq_divide]) 2);
-by (Clarify_tac 2);
-by (dres_inst_tac [("x","n")] spec 2);
-by (dres_inst_tac [("c","real (Suc n)")] (mult_right_mono) 2);
-by (rtac real_of_nat_ge_zero 2);
-by (asm_full_simp_tac (simpset()
- addsimps [real_of_nat_Suc_gt_zero RS real_not_refl2 RS not_sym,
- real_mult_commute]) 2);
-by (subgoal_tac "isUb (UNIV::real set) \
-\ {z. EX n. z = x*(real (Suc n))} 1" 1);
-by (subgoal_tac "EX X. X : {z. EX n. z = x*(real (Suc n))}" 1);
-by (dtac reals_complete 1);
-by (auto_tac (claset() addIs [isUbI,setleI],simpset()));
-by (subgoal_tac "ALL m. x*(real(Suc m)) <= t" 1);
-by (asm_full_simp_tac (simpset() addsimps
- [real_of_nat_Suc, right_distrib]) 1);
-by (blast_tac (claset() addIs [isLubD2]) 2);
-by (asm_full_simp_tac
- (simpset() addsimps [le_diff_eq RS sym, real_diff_def]) 1);
-by (subgoal_tac "isUb (UNIV::real set) \
-\ {z. EX n. z = x*(real (Suc n))} (t + (-x))" 1);
-by (blast_tac (claset() addSIs [isUbI,setleI]) 2);
-by (dres_inst_tac [("y","t+(-x)")] isLub_le_isUb 1);
-by (auto_tac (claset(),
- simpset() addsimps [real_of_nat_Suc,right_distrib]));
-qed "reals_Archimedean";
-
-(*There must be other proofs, e.g. Suc of the largest integer in the
- cut representing x*)
-Goal "EX n. (x::real) < real (n::nat)";
-by (res_inst_tac [("R1.0","x"),("R2.0","0")] real_linear_less2 1);
-by (res_inst_tac [("x","0")] exI 1);
-by (res_inst_tac [("x","1")] exI 2);
-by (auto_tac (claset() addEs [order_less_trans],
- simpset() addsimps [real_of_nat_one]));
-by (ftac (positive_imp_inverse_positive RS reals_Archimedean) 1);
-by (Step_tac 1 THEN res_inst_tac [("x","Suc n")] exI 1);
-by (forw_inst_tac [("b","inverse x")] mult_strict_right_mono 1);
-by Auto_tac;
-qed "reals_Archimedean2";
-
-Goal "0 < x ==> ALL y. EX (n::nat). y < real n * x";
-by (Step_tac 1);
-by (cut_inst_tac [("x","y*inverse(x)")] reals_Archimedean2 1);
-by (Step_tac 1);
-by (forw_inst_tac [("a","y * inverse x")] (mult_strict_right_mono) 1);
-by (auto_tac (claset(),simpset() addsimps [real_mult_assoc,real_of_nat_def]));
-qed "reals_Archimedean3";
-