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(* Title : NthRoot.ML
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Author : Jacques D. Fleuriot
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Copyright : 1998 University of Cambridge
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Description : Existence of nth root. Adapted from
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http://www.math.unl.edu/~webnotes
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*)
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(*----------------------------------------------------------------------
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Existence of nth root
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Various lemmas needed for this result. We follow the proof
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given by John Lindsay Orr (jorr@math.unl.edu) in his Analysis
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Webnotes available on the www at http://www.math.unl.edu/~webnotes
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Lemmas about sequences of reals are used to reach the result.
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---------------------------------------------------------------------*)
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Goal "[| (0::real) < a; 0 < n |] \
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\ ==> EX s. s : {x. x ^ n <= a & 0 < x}";
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by (case_tac "1 <= a" 1);
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by (res_inst_tac [("x","1")] exI 1);
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by (dtac not_real_leE 2);
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by (dtac (less_not_refl2 RS not0_implies_Suc) 2);
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by (auto_tac (claset() addSIs [realpow_Suc_le_self],
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simpset() addsimps [real_zero_less_one]));
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qed "lemma_nth_realpow_non_empty";
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Goal "[| (0::real) < a; 0 < n |] \
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\ ==> EX u. isUb (UNIV::real set) {x. x ^ n <= a & 0 < x} u";
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by (case_tac "1 <= a" 1);
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by (res_inst_tac [("x","a")] exI 1);
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by (dtac not_real_leE 2);
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by (res_inst_tac [("x","1")] exI 2);
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by (ALLGOALS(rtac (setleI RS isUbI)));
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by (Auto_tac);
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by (ALLGOALS(rtac ccontr));
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by (ALLGOALS(dtac not_real_leE));
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by (dtac realpow_ge_self2 1 THEN assume_tac 1);
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by (dres_inst_tac [("n","n")] (conjI
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RSN (2,conjI RS realpow_less)) 1);
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by (REPEAT(assume_tac 1));
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by (dtac real_le_trans 1 THEN assume_tac 1);
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by (dres_inst_tac [("y","y ^ n")] order_less_le_trans 1);
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by (assume_tac 1 THEN etac real_less_irrefl 1);
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by (dres_inst_tac [("n","n")] ((real_zero_less_one) RS (conjI
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RSN (2,conjI RS realpow_less))) 1);
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by (Auto_tac);
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qed "lemma_nth_realpow_isUb_ex";
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Goal "[| (0::real) < a; 0 < n |] \
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\ ==> EX u. isLub (UNIV::real set) {x. x ^ n <= a & 0 < x} u";
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by (blast_tac (claset() addIs [lemma_nth_realpow_isUb_ex,
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lemma_nth_realpow_non_empty,reals_complete]) 1);
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qed "nth_realpow_isLub_ex";
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(*---------------------------------------------
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First Half -- lemmas first
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--------------------------------------------*)
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Goal "isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u \
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\ ==> u + inverse(real_of_posnat k) ~: {x. x ^ n <= a & 0 < x}";
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by (Step_tac 1 THEN dtac isLubD2 1 THEN Blast_tac 1);
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by (asm_full_simp_tac (simpset() addsimps [real_le_def]) 1);
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val lemma_nth_realpow_seq = result();
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Goal "[| isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u; \
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\ 0 < a; 0 < n |] ==> 0 < u";
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by (dtac lemma_nth_realpow_non_empty 1 THEN Auto_tac);
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by (dres_inst_tac [("y","s")] (isLub_isUb RS isUbD) 1);
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by (auto_tac (claset() addIs [order_less_le_trans],simpset()));
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val lemma_nth_realpow_isLub_gt_zero = result();
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Goal "[| isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u; \
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\ 0 < a; 0 < n |] ==> ALL k. a <= (u + inverse(real_of_posnat k)) ^ n";
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by (Step_tac 1);
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by (forward_tac [lemma_nth_realpow_seq] 1 THEN Step_tac 1);
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by (auto_tac (claset() addEs [real_less_asym],
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simpset() addsimps [real_le_def]));
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by (fold_tac [real_le_def]);
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by (rtac real_less_trans 1);
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by (auto_tac (claset() addIs [real_inv_real_of_posnat_gt_zero,
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lemma_nth_realpow_isLub_gt_zero],simpset()));
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val lemma_nth_realpow_isLub_ge = result();
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(*-----------------------
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First result we want
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----------------------*)
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Goal "[| (0::real) < a; 0 < n; \
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\ isLub (UNIV::real set) \
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\ {x. x ^ n <= a & 0 < x} u |] ==> a <= u ^ n";
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by (forward_tac [lemma_nth_realpow_isLub_ge] 1 THEN Step_tac 1);
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by (rtac (LIMSEQ_inverse_real_of_posnat_add RS LIMSEQ_pow
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RS LIMSEQ_le_const) 1);
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by (auto_tac (claset(),simpset() addsimps [real_of_nat_def,
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real_of_posnat_Suc]));
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qed "realpow_nth_ge";
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(*---------------------------------------------
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Second Half
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--------------------------------------------*)
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Goal "[| isLub (UNIV::real set) S u; x < u |] \
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\ ==> ~ isUb (UNIV::real set) S x";
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by (Step_tac 1);
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by (dtac isLub_le_isUb 1);
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by (assume_tac 1);
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by (dtac order_less_le_trans 1);
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by (auto_tac (claset(),simpset()
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addsimps [real_less_not_refl]));
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qed "less_isLub_not_isUb";
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Goal "~ isUb (UNIV::real set) S u \
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\ ==> EX x: S. u < x";
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by (rtac ccontr 1 THEN etac swap 1);
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by (rtac (setleI RS isUbI) 1);
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by (auto_tac (claset(),simpset() addsimps [real_le_def]));
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qed "not_isUb_less_ex";
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Goal "[| isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u; \
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\ 0 < a; 0 < n |] ==> ALL k. (u*(1 + -inverse(real_of_posnat k))) ^ n <= a";
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by (Step_tac 1);
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by (forward_tac [less_isLub_not_isUb RS not_isUb_less_ex] 1);
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by (res_inst_tac [("n","k")] real_mult_less_self 1);
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by (blast_tac (claset() addIs [lemma_nth_realpow_isLub_gt_zero]) 1);
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by (Step_tac 1);
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by (dres_inst_tac [("n","k")] (lemma_nth_realpow_isLub_gt_zero RS
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real_mult_add_one_minus_ge_zero) 1);
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by (dtac (conjI RS realpow_le2) 3);
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by (rtac (CLAIM "x < y ==> (x::real) <= y") 3);
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by (auto_tac (claset() addIs [real_le_trans],simpset()));
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val lemma_nth_realpow_isLub_le = result();
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(*-----------------------
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Second result we want
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----------------------*)
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Goal "[| (0::real) < a; 0 < n; \
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\ isLub (UNIV::real set) \
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\ {x. x ^ n <= a & 0 < x} u |] ==> u ^ n <= a";
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by (forward_tac [lemma_nth_realpow_isLub_le] 1 THEN Step_tac 1);
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by (rtac (LIMSEQ_inverse_real_of_posnat_add_minus_mult RS LIMSEQ_pow
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RS LIMSEQ_le_const2) 1);
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by (auto_tac (claset(),simpset() addsimps [real_of_nat_def,real_of_posnat_Suc]));
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qed "realpow_nth_le";
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(*----------- The theorem at last! -----------*)
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Goal "[| (0::real) < a; 0 < n |] ==> EX r. r ^ n = a";
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by (forward_tac [nth_realpow_isLub_ex] 1 THEN Auto_tac);
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by (auto_tac (claset() addIs [realpow_nth_le,
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realpow_nth_ge,real_le_anti_sym],simpset()));
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qed "realpow_nth";
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(* positive only *)
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Goal "[| (0::real) < a; 0 < n |] ==> EX r. 0 < r & r ^ n = a";
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by (forward_tac [nth_realpow_isLub_ex] 1 THEN Auto_tac);
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by (auto_tac (claset() addIs [realpow_nth_le,
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realpow_nth_ge,real_le_anti_sym,
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lemma_nth_realpow_isLub_gt_zero],simpset()));
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qed "realpow_pos_nth";
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Goal "(0::real) < a ==> EX r. 0 < r & r ^ Suc n = a";
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by (blast_tac (claset() addIs [realpow_pos_nth]) 1);
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qed "realpow_pos_nth2";
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(* uniqueness of nth positive root *)
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Goal "[| (0::real) < a; 0 < n |] ==> EX! r. 0 < r & r ^ n = a";
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by (auto_tac (claset() addSIs [realpow_pos_nth],simpset()));
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by (cut_inst_tac [("R1.0","r"),("R2.0","y")] real_linear 1);
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by (Auto_tac);
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by (dres_inst_tac [("x","r")] (conjI RS realpow_less) 1);
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by (dres_inst_tac [("x","y")] (conjI RS realpow_less) 3);
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by (auto_tac (claset(),simpset() addsimps [real_less_not_refl]));
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qed "realpow_pos_nth_unique";
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