--- a/src/HOL/Hyperreal/IntFloor.ML Tue Mar 02 01:46:26 2004 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,302 +0,0 @@
-(* Title: IntFloor.ML
- Author: Jacques D. Fleuriot
- Copyright: 2001,2002 University of Edinburgh
- Description: Floor and ceiling operations over reals
-*)
-
-Goal "((number_of n) < real (m::int)) = (number_of n < m)";
-by Auto_tac;
-by (rtac (real_of_int_less_iff RS iffD1) 1);
-by (dtac (real_of_int_less_iff RS iffD2) 2);
-by Auto_tac;
-qed "number_of_less_real_of_int_iff";
-Addsimps [number_of_less_real_of_int_iff];
-
-Goal "(real (m::int) < (number_of n)) = (m < number_of n)";
-by Auto_tac;
-by (rtac (real_of_int_less_iff RS iffD1) 1);
-by (dtac (real_of_int_less_iff RS iffD2) 2);
-by Auto_tac;
-qed "number_of_less_real_of_int_iff2";
-Addsimps [number_of_less_real_of_int_iff2];
-
-Goal "((number_of n) <= real (m::int)) = (number_of n <= m)";
-by (auto_tac (claset(), simpset() addsimps [linorder_not_less RS sym]));
-qed "number_of_le_real_of_int_iff";
-Addsimps [number_of_le_real_of_int_iff];
-
-Goal "(real (m::int) <= (number_of n)) = (m <= number_of n)";
-by (auto_tac (claset(), simpset() addsimps [linorder_not_less RS sym]));
-qed "number_of_le_real_of_int_iff2";
-Addsimps [number_of_le_real_of_int_iff2];
-
-Goalw [floor_def] "floor 0 = 0";
-by (rtac Least_equality 1);
-by Auto_tac;
-qed "floor_zero";
-Addsimps [floor_zero];
-
-Goal "floor (real (0::nat)) = 0";
-by Auto_tac;
-qed "floor_real_of_nat_zero";
-Addsimps [floor_real_of_nat_zero];
-
-Goalw [floor_def] "floor (real (n::nat)) = int n";
-by (rtac Least_equality 1);
-by (dtac (real_of_int_real_of_nat RS ssubst) 2);
-by (dtac (real_of_int_less_iff RS iffD1) 2);
-by (auto_tac (claset(),simpset() addsimps [real_of_int_real_of_nat]));
-qed "floor_real_of_nat";
-Addsimps [floor_real_of_nat];
-
-Goalw [floor_def] "floor (- real (n::nat)) = - int n";
-by (rtac Least_equality 1);
-by (dtac (real_of_int_real_of_nat RS ssubst) 2);
-by (dtac (real_of_int_minus RS subst) 2);
-by (dtac (real_of_int_less_iff RS iffD1) 2);
-by (auto_tac (claset(),simpset() addsimps [real_of_int_real_of_nat]));
-qed "floor_minus_real_of_nat";
-Addsimps [floor_minus_real_of_nat];
-
-Goalw [floor_def] "floor (real (n::int)) = n";
-by (rtac Least_equality 1);
-by (dtac (real_of_int_real_of_nat RS ssubst) 2);
-by (dtac (real_of_int_less_iff RS iffD1) 2);
-by Auto_tac;
-qed "floor_real_of_int";
-Addsimps [floor_real_of_int];
-
-Goalw [floor_def] "floor (- real (n::int)) = - n";
-by (rtac Least_equality 1);
-by (dtac (real_of_int_minus RS subst) 2);
-by (dtac (real_of_int_real_of_nat RS ssubst) 2);
-by (dtac (real_of_int_less_iff RS iffD1) 2);
-by Auto_tac;
-qed "floor_minus_real_of_int";
-Addsimps [floor_minus_real_of_int];
-
-Goal "0 <= r ==> EX (n::nat). real (n - 1) <= r & r < real (n)";
-by (cut_inst_tac [("x","r")] reals_Archimedean2 1);
-by (Step_tac 1);
-by (forw_inst_tac [("P","%k. r < real k"),("k","n"),("m","%x. x")]
- (thm "ex_has_least_nat") 1);
-by Auto_tac;
-by (res_inst_tac [("x","x")] exI 1);
-by (dres_inst_tac [("x","x - 1")] spec 1);
-by (auto_tac (claset() addDs [ARITH_PROVE "x <= x - Suc 0 ==> x = (0::nat)"],
- simpset()));
-qed "reals_Archimedean6";
-
-Goal "0 <= r ==> EX n. real (n) <= r & r < real (Suc n)";
-by (dtac reals_Archimedean6 1);
-by Auto_tac;
-qed "reals_Archimedean6a";
-
-Goal "0 <= r ==> EX n. real n <= r & r < real ((n::int) + 1)";
-by (dtac reals_Archimedean6a 1);
-by Auto_tac;
-by (res_inst_tac [("x","int n")] exI 1);
-by (auto_tac (claset(),simpset() addsimps [real_of_int_real_of_nat,
- real_of_nat_Suc]));
-qed "reals_Archimedean_6b_int";
-
-Goal "r < 0 ==> EX n. real n <= r & r < real ((n::int) + 1)";
-by (dtac (CLAIM "r < (0::real) ==> 0 <= -r") 1);
-by (dtac reals_Archimedean_6b_int 1);
-by Auto_tac;
-by (dtac real_le_imp_less_or_eq 1 THEN Auto_tac);
-by (res_inst_tac [("x","- n - 1")] exI 1);
-by (res_inst_tac [("x","- n")] exI 2);
-by Auto_tac;
-qed "reals_Archimedean_6c_int";
-
-Goal " EX (n::int). real n <= r & r < real ((n::int) + 1)";
-by (cut_inst_tac [("r","r")] (CLAIM "0 <= r | r < (0::real)") 1);
-by (blast_tac (claset() addIs [reals_Archimedean_6b_int,
- reals_Archimedean_6c_int]) 1);
-qed "real_lb_ub_int";
-
-Goal "[| real n <= r; r < real y + 1 |] ==> n <= (y::int)";
-by (dres_inst_tac [("x","real n"),("z","real y + 1")] order_le_less_trans 1);
-by (rotate_tac 1 2);
-by (dtac ((CLAIM "real (1::int) = 1") RS ssubst) 2);
-by (rotate_tac 1 2);
-by (dres_inst_tac [("x1","y")] (real_of_int_add RS subst) 2);
-by (dtac (real_of_int_less_iff RS iffD1) 2);
-by Auto_tac;
-val lemma_floor = result();
-
-Goalw [floor_def,Least_def] "real (floor r) <= r";
-by (cut_inst_tac [("r","r")] real_lb_ub_int 1 THEN Step_tac 1);
-by (rtac theI2 1);
-by Auto_tac;
-qed "real_of_int_floor_le";
-Addsimps [real_of_int_floor_le];
-
-Goalw [floor_def,Least_def]
- "x < y ==> floor x <= floor y";
-by (cut_inst_tac [("r","x")] real_lb_ub_int 1 THEN Step_tac 1);
-by (cut_inst_tac [("r","y")] real_lb_ub_int 1 THEN Step_tac 1);
-by (rtac theI2 1);
-by (rtac theI2 3);
-by Auto_tac;
-by (auto_tac (claset() addIs [lemma_floor],simpset()));
-by (ALLGOALS(force_tac (claset() addDs [lemma_floor],simpset())));
-qed "floor_le";
-
-Goal "x <= y ==> floor x <= floor y";
-by (auto_tac (claset() addDs [real_le_imp_less_or_eq],simpset()
- addsimps [floor_le]));
-qed "floor_le2";
-
-Goal "real na < real (x::int) + 1 ==> na <= x";
-by (auto_tac (claset() addIs [lemma_floor],simpset()));
-val lemma_floor2 = result();
-
-Goalw [floor_def,Least_def]
- "(real (floor x) = x) = (EX (n::int). x = real n)";
-by (cut_inst_tac [("r","x")] real_lb_ub_int 1 THEN etac exE 1);
-by (rtac theI2 1);
-by (auto_tac (claset() addIs [lemma_floor],simpset()));
-qed "real_of_int_floor_cancel";
-Addsimps [real_of_int_floor_cancel];
-
-Goalw [floor_def]
- "[| real n < x; x < real n + 1 |] ==> floor x = n";
-by (rtac Least_equality 1);
-by (auto_tac (claset() addIs [lemma_floor],simpset()));
-qed "floor_eq";
-
-Goalw [floor_def]
- "[| real n <= x; x < real n + 1 |] ==> floor x = n";
-by (rtac Least_equality 1);
-by (auto_tac (claset() addIs [lemma_floor],simpset()));
-qed "floor_eq2";
-
-Goal "[| real n < x; x < real (Suc n) |] ==> nat(floor x) = n";
-by (rtac (inj_int RS injD) 1);
-by (rtac (CLAIM "0 <= x ==> int (nat x) = x" RS ssubst) 1);
-by (rtac floor_eq 2);
-by (auto_tac (claset(),simpset() addsimps [real_of_nat_Suc,
- real_of_int_real_of_nat]));
-by (rtac (floor_le RSN (2,zle_trans)) 1 THEN Auto_tac);
-qed "floor_eq3";
-
-Goal "[| real n <= x; x < real (Suc n) |] ==> nat(floor x) = n";
-by (dtac order_le_imp_less_or_eq 1);
-by (auto_tac (claset() addIs [floor_eq3],simpset()));
-qed "floor_eq4";
-
-Goal "floor(number_of n :: real) = (number_of n :: int)";
-by (stac (real_number_of RS sym) 1);
-by (rtac floor_eq2 1);
-by (rtac (CLAIM "x < x + (1::real)") 2);
-by (rtac real_le_refl 1);
-qed "floor_number_of_eq";
-Addsimps [floor_number_of_eq];
-
-Goalw [floor_def,Least_def] "r - 1 <= real(floor r)";
-by (cut_inst_tac [("r","r")] real_lb_ub_int 1 THEN Step_tac 1);
-by (rtac theI2 1);
-by (auto_tac (claset() addIs [lemma_floor],simpset()));
-qed "real_of_int_floor_ge_diff_one";
-Addsimps [real_of_int_floor_ge_diff_one];
-
-Goal "r <= real(floor r) + 1";
-by (cut_inst_tac [("r","r")] real_of_int_floor_ge_diff_one 1);
-by (auto_tac (claset(),simpset() delsimps [real_of_int_floor_ge_diff_one]));
-qed "real_of_int_floor_add_one_ge";
-Addsimps [real_of_int_floor_add_one_ge];
-
-
-(*--------------------------------------------------------------------------------*)
-(* ceiling function for positive reals *)
-(*--------------------------------------------------------------------------------*)
-
-Goalw [ceiling_def] "ceiling 0 = 0";
-by Auto_tac;
-qed "ceiling_zero";
-Addsimps [ceiling_zero];
-
-Goalw [ceiling_def] "ceiling (real (n::nat)) = int n";
-by Auto_tac;
-qed "ceiling_real_of_nat";
-Addsimps [ceiling_real_of_nat];
-
-Goal "ceiling (real (0::nat)) = 0";
-by Auto_tac;
-qed "ceiling_real_of_nat_zero";
-Addsimps [ceiling_real_of_nat_zero];
-
-Goalw [ceiling_def] "ceiling (real (floor r)) = floor r";
-by Auto_tac;
-qed "ceiling_floor";
-Addsimps [ceiling_floor];
-
-Goalw [ceiling_def] "floor (real (ceiling r)) = ceiling r";
-by Auto_tac;
-qed "floor_ceiling";
-Addsimps [floor_ceiling];
-
-Goalw [ceiling_def] "r <= real (ceiling r)";
-by Auto_tac;
-by (rtac (CLAIM "x <= -y ==> (y::real) <= - x") 1);
-by Auto_tac;
-qed "real_of_int_ceiling_ge";
-Addsimps [real_of_int_ceiling_ge];
-
-Goalw [ceiling_def] "x < y ==> ceiling x <= ceiling y";
-by (auto_tac (claset() addIs [floor_le],simpset()));
-qed "ceiling_le";
-
-Goalw [ceiling_def] "x <= y ==> ceiling x <= ceiling y";
-by (auto_tac (claset() addIs [floor_le2],simpset()));
-qed "ceiling_le2";
-
-Goalw [ceiling_def] "(real (ceiling x) = x) = (EX (n::int). x = real n)";
-by Auto_tac;
-by (dtac (CLAIM "- x = y ==> (x::real) = -y") 1);
-by Auto_tac;
-by (res_inst_tac [("x","-n")] exI 1);
-by Auto_tac;
-qed "real_of_int_ceiling_cancel";
-Addsimps [real_of_int_ceiling_cancel];
-
-Goalw [ceiling_def]
- "[| real n < x; x < real n + 1 |] ==> ceiling x = n + 1";
-by (rtac (CLAIM "x = - y ==> - (x::int) = y") 1);
-by (auto_tac (claset() addIs [floor_eq],simpset()));
-qed "ceiling_eq";
-
-Goalw [ceiling_def]
- "[| real n < x; x <= real n + 1 |] ==> ceiling x = n + 1";
-by (rtac (CLAIM "x = - y ==> - (x::int) = y") 1);
-by (auto_tac (claset() addIs [floor_eq2],simpset()));
-qed "ceiling_eq2";
-
-Goalw [ceiling_def] "[| real n - 1 < x; x <= real n |] ==> ceiling x = n";
-by (rtac (CLAIM "x = -(y::int) ==> - x = y") 1);
-by (auto_tac (claset() addIs [floor_eq2],simpset()));
-qed "ceiling_eq3";
-
-Goalw [ceiling_def]
- "ceiling (number_of n :: real) = (number_of n :: int)";
-by (stac (real_number_of RS sym) 1);
-by (rtac (CLAIM "x = - y ==> - (x::int) = y") 1);
-by (rtac (floor_minus_real_of_int RS ssubst) 1);
-by Auto_tac;
-qed "ceiling_number_of_eq";
-Addsimps [ceiling_number_of_eq];
-
-Goalw [ceiling_def] "real (ceiling r) - 1 <= r";
-by (rtac (CLAIM "-x <= -y ==> (y::real) <= x") 1);
-by (auto_tac (claset(),simpset() addsimps [real_diff_def]));
-qed "real_of_int_ceiling_diff_one_le";
-Addsimps [real_of_int_ceiling_diff_one_le];
-
-Goal "real (ceiling r) <= r + 1";
-by (cut_inst_tac [("r","r")] real_of_int_ceiling_diff_one_le 1);
-by (auto_tac (claset(),simpset() delsimps [real_of_int_ceiling_diff_one_le]));
-qed "real_of_int_ceiling_le_add_one";
-Addsimps [real_of_int_ceiling_le_add_one];
-
--- a/src/HOL/Hyperreal/IntFloor.thy Tue Mar 02 01:46:26 2004 +0100
+++ b/src/HOL/Hyperreal/IntFloor.thy Tue Mar 02 11:05:55 2004 +0100
@@ -1,17 +1,319 @@
(* Title: IntFloor.thy
Author: Jacques D. Fleuriot
Copyright: 2001,2002 University of Edinburgh
- Description: Floor and ceiling operations over reals
+Converted to Isar and polished by lcp
*)
-IntFloor = Integration +
+header{*Floor and Ceiling Functions from the Reals to the Integers*}
+
+theory IntFloor = Integration:
constdefs
-
- floor :: real => int
+
+ floor :: "real => int"
"floor r == (LEAST n. r < real (n + (1::int)))"
- ceiling :: real => int
+ ceiling :: "real => int"
"ceiling r == - floor (- r)"
-
+
+syntax (xsymbols)
+ floor :: "real => int" ("\<lfloor>_\<rfloor>")
+ ceiling :: "real => int" ("\<lceil>_\<rceil>")
+
+
+
+lemma number_of_less_real_of_int_iff [simp]:
+ "((number_of n) < real (m::int)) = (number_of n < m)"
+apply auto
+apply (rule real_of_int_less_iff [THEN iffD1])
+apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)
+done
+
+lemma number_of_less_real_of_int_iff2 [simp]:
+ "(real (m::int) < (number_of n)) = (m < number_of n)"
+apply auto
+apply (rule real_of_int_less_iff [THEN iffD1])
+apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)
+done
+
+lemma number_of_le_real_of_int_iff [simp]:
+ "((number_of n) \<le> real (m::int)) = (number_of n \<le> m)"
+by (simp add: linorder_not_less [symmetric])
+
+lemma number_of_le_real_of_int_iff2 [simp]:
+ "(real (m::int) \<le> (number_of n)) = (m \<le> number_of n)"
+by (simp add: linorder_not_less [symmetric])
+
+lemma floor_zero [simp]: "floor 0 = 0"
+apply (simp add: floor_def)
+apply (rule Least_equality, auto)
+done
+
+lemma floor_real_of_nat_zero [simp]: "floor (real (0::nat)) = 0"
+by auto
+
+lemma floor_real_of_nat [simp]: "floor (real (n::nat)) = int n"
+apply (simp only: floor_def)
+apply (rule Least_equality)
+apply (drule_tac [2] real_of_int_real_of_nat [THEN ssubst])
+apply (drule_tac [2] real_of_int_less_iff [THEN iffD1])
+apply (simp_all add: real_of_int_real_of_nat)
+done
+
+lemma floor_minus_real_of_nat [simp]: "floor (- real (n::nat)) = - int n"
+apply (simp only: floor_def)
+apply (rule Least_equality)
+apply (drule_tac [2] real_of_int_real_of_nat [THEN ssubst])
+apply (drule_tac [2] real_of_int_minus [THEN subst])
+apply (drule_tac [2] real_of_int_less_iff [THEN iffD1])
+apply (simp_all add: real_of_int_real_of_nat)
+done
+
+lemma floor_real_of_int [simp]: "floor (real (n::int)) = n"
+apply (simp only: floor_def)
+apply (rule Least_equality)
+apply (drule_tac [2] real_of_int_real_of_nat [THEN ssubst])
+apply (drule_tac [2] real_of_int_less_iff [THEN iffD1], auto)
+done
+
+lemma floor_minus_real_of_int [simp]: "floor (- real (n::int)) = - n"
+apply (simp only: floor_def)
+apply (rule Least_equality)
+apply (drule_tac [2] real_of_int_minus [THEN subst])
+apply (drule_tac [2] real_of_int_real_of_nat [THEN ssubst])
+apply (drule_tac [2] real_of_int_less_iff [THEN iffD1], auto)
+done
+
+lemma reals_Archimedean6:
+ "0 \<le> r ==> \<exists>(n::nat). real (n - 1) \<le> r & r < real (n)"
+apply (insert reals_Archimedean2 [of r], safe)
+apply (frule_tac P = "%k. r < real k" and k = n and m = "%x. x"
+ in ex_has_least_nat, auto)
+apply (rule_tac x = x in exI)
+apply (case_tac x, simp)
+apply (rename_tac x')
+apply (drule_tac x = x' in spec, simp)
+done
+
+lemma reals_Archimedean6a: "0 \<le> r ==> \<exists>n. real (n) \<le> r & r < real (Suc n)"
+by (drule reals_Archimedean6, auto)
+
+lemma reals_Archimedean_6b_int:
+ "0 \<le> r ==> \<exists>n. real n \<le> r & r < real ((n::int) + 1)"
+apply (drule reals_Archimedean6a, auto)
+apply (rule_tac x = "int n" in exI)
+apply (simp add: real_of_int_real_of_nat real_of_nat_Suc)
+done
+
+lemma reals_Archimedean_6c_int:
+ "r < 0 ==> \<exists>n. real n \<le> r & r < real ((n::int) + 1)"
+apply (rule reals_Archimedean_6b_int [of "-r", THEN exE], simp, auto)
+apply (rename_tac n)
+apply (drule real_le_imp_less_or_eq, auto)
+apply (rule_tac x = "- n - 1" in exI)
+apply (rule_tac [2] x = "- n" in exI, auto)
+done
+
+lemma real_lb_ub_int: " \<exists>(n::int). real n \<le> r & r < real ((n::int) + 1)"
+apply (case_tac "r < 0")
+apply (blast intro: reals_Archimedean_6c_int)
+apply (simp only: linorder_not_less)
+apply (blast intro: reals_Archimedean_6b_int reals_Archimedean_6c_int)
+done
+
+lemma lemma_floor:
+ assumes a1: "real m \<le> r" and a2: "r < real n + 1"
+ shows "m \<le> (n::int)"
+proof -
+ have "real m < real n + 1" by (rule order_le_less_trans)
+ also have "... = real(n+1)" by simp
+ finally have "m < n+1" by (simp only: real_of_int_less_iff)
+ thus ?thesis by arith
+qed
+
+lemma real_of_int_floor_le [simp]: "real (floor r) \<le> r"
+apply (simp add: floor_def Least_def)
+apply (insert real_lb_ub_int [of r], safe)
+apply (rule theI2, auto)
+done
+
+lemma floor_le: "x < y ==> floor x \<le> floor y"
+apply (simp add: floor_def Least_def)
+apply (insert real_lb_ub_int [of x])
+apply (insert real_lb_ub_int [of y], safe)
+apply (rule theI2)
+apply (rule_tac [3] theI2, auto)
+done
+
+lemma floor_le2: "x \<le> y ==> floor x \<le> floor y"
+by (auto dest: real_le_imp_less_or_eq simp add: floor_le)
+
+lemma lemma_floor2: "real na < real (x::int) + 1 ==> na \<le> x"
+by (auto intro: lemma_floor)
+
+lemma real_of_int_floor_cancel [simp]:
+ "(real (floor x) = x) = (\<exists>n::int. x = real n)"
+apply (simp add: floor_def Least_def)
+apply (insert real_lb_ub_int [of x], erule exE)
+apply (rule theI2)
+apply (auto intro: lemma_floor)
+done
+
+lemma floor_eq: "[| real n < x; x < real n + 1 |] ==> floor x = n"
+apply (simp add: floor_def)
+apply (rule Least_equality)
+apply (auto intro: lemma_floor)
+done
+
+lemma floor_eq2: "[| real n \<le> x; x < real n + 1 |] ==> floor x = n"
+apply (simp add: floor_def)
+apply (rule Least_equality)
+apply (auto intro: lemma_floor)
+done
+
+lemma floor_eq3: "[| real n < x; x < real (Suc n) |] ==> nat(floor x) = n"
+apply (rule inj_int [THEN injD])
+apply (simp add: real_of_nat_Suc)
+apply (simp add: real_of_nat_Suc floor_eq floor_eq [where n = "of_nat n"])
+done
+
+lemma floor_eq4: "[| real n \<le> x; x < real (Suc n) |] ==> nat(floor x) = n"
+apply (drule order_le_imp_less_or_eq)
+apply (auto intro: floor_eq3)
+done
+
+lemma floor_number_of_eq [simp]:
+ "floor(number_of n :: real) = (number_of n :: int)"
+apply (subst real_number_of [symmetric])
+apply (rule floor_real_of_int)
+done
+
+lemma real_of_int_floor_ge_diff_one [simp]: "r - 1 \<le> real(floor r)"
+apply (simp add: floor_def Least_def)
+apply (insert real_lb_ub_int [of r], safe)
+apply (rule theI2)
+apply (auto intro: lemma_floor)
+done
+
+lemma real_of_int_floor_add_one_ge [simp]: "r \<le> real(floor r) + 1"
+apply (insert real_of_int_floor_ge_diff_one [of r])
+apply (auto simp del: real_of_int_floor_ge_diff_one)
+done
+
+
+subsection{*Ceiling Function for Positive Reals*}
+
+lemma ceiling_zero [simp]: "ceiling 0 = 0"
+by (simp add: ceiling_def)
+
+lemma ceiling_real_of_nat [simp]: "ceiling (real (n::nat)) = int n"
+by (simp add: ceiling_def)
+
+lemma ceiling_real_of_nat_zero [simp]: "ceiling (real (0::nat)) = 0"
+by auto
+
+lemma ceiling_floor [simp]: "ceiling (real (floor r)) = floor r"
+by (simp add: ceiling_def)
+
+lemma floor_ceiling [simp]: "floor (real (ceiling r)) = ceiling r"
+by (simp add: ceiling_def)
+
+lemma real_of_int_ceiling_ge [simp]: "r \<le> real (ceiling r)"
+apply (simp add: ceiling_def)
+apply (subst le_minus_iff, simp)
+done
+
+lemma ceiling_le: "x < y ==> ceiling x \<le> ceiling y"
+by (simp add: floor_le ceiling_def)
+
+lemma ceiling_le2: "x \<le> y ==> ceiling x \<le> ceiling y"
+by (simp add: floor_le2 ceiling_def)
+
+lemma real_of_int_ceiling_cancel [simp]:
+ "(real (ceiling x) = x) = (\<exists>n::int. x = real n)"
+apply (auto simp add: ceiling_def)
+apply (drule arg_cong [where f = uminus], auto)
+apply (rule_tac x = "-n" in exI, auto)
+done
+
+lemma ceiling_eq: "[| real n < x; x < real n + 1 |] ==> ceiling x = n + 1"
+apply (simp add: ceiling_def)
+apply (rule minus_equation_iff [THEN iffD1])
+apply (simp add: floor_eq [where n = "-(n+1)"])
+done
+
+lemma ceiling_eq2: "[| real n < x; x \<le> real n + 1 |] ==> ceiling x = n + 1"
+by (simp add: ceiling_def floor_eq2 [where n = "-(n+1)"])
+
+lemma ceiling_eq3: "[| real n - 1 < x; x \<le> real n |] ==> ceiling x = n"
+by (simp add: ceiling_def floor_eq2 [where n = "-n"])
+
+lemma ceiling_real_of_int [simp]: "ceiling (real (n::int)) = n"
+by (simp add: ceiling_def)
+
+lemma ceiling_number_of_eq [simp]:
+ "ceiling (number_of n :: real) = (number_of n)"
+apply (subst real_number_of [symmetric])
+apply (rule ceiling_real_of_int)
+done
+
+lemma real_of_int_ceiling_diff_one_le [simp]: "real (ceiling r) - 1 \<le> r"
+apply (rule neg_le_iff_le [THEN iffD1])
+apply (simp add: ceiling_def diff_minus)
+done
+
+lemma real_of_int_ceiling_le_add_one [simp]: "real (ceiling r) \<le> r + 1"
+apply (insert real_of_int_ceiling_diff_one_le [of r])
+apply (simp del: real_of_int_ceiling_diff_one_le)
+done
+
+ML
+{*
+val number_of_less_real_of_int_iff = thm "number_of_less_real_of_int_iff";
+val number_of_less_real_of_int_iff2 = thm "number_of_less_real_of_int_iff2";
+val number_of_le_real_of_int_iff = thm "number_of_le_real_of_int_iff";
+val number_of_le_real_of_int_iff2 = thm "number_of_le_real_of_int_iff2";
+val floor_zero = thm "floor_zero";
+val floor_real_of_nat_zero = thm "floor_real_of_nat_zero";
+val floor_real_of_nat = thm "floor_real_of_nat";
+val floor_minus_real_of_nat = thm "floor_minus_real_of_nat";
+val floor_real_of_int = thm "floor_real_of_int";
+val floor_minus_real_of_int = thm "floor_minus_real_of_int";
+val reals_Archimedean6 = thm "reals_Archimedean6";
+val reals_Archimedean6a = thm "reals_Archimedean6a";
+val reals_Archimedean_6b_int = thm "reals_Archimedean_6b_int";
+val reals_Archimedean_6c_int = thm "reals_Archimedean_6c_int";
+val real_lb_ub_int = thm "real_lb_ub_int";
+val lemma_floor = thm "lemma_floor";
+val real_of_int_floor_le = thm "real_of_int_floor_le";
+val floor_le = thm "floor_le";
+val floor_le2 = thm "floor_le2";
+val lemma_floor2 = thm "lemma_floor2";
+val real_of_int_floor_cancel = thm "real_of_int_floor_cancel";
+val floor_eq = thm "floor_eq";
+val floor_eq2 = thm "floor_eq2";
+val floor_eq3 = thm "floor_eq3";
+val floor_eq4 = thm "floor_eq4";
+val floor_number_of_eq = thm "floor_number_of_eq";
+val real_of_int_floor_ge_diff_one = thm "real_of_int_floor_ge_diff_one";
+val real_of_int_floor_add_one_ge = thm "real_of_int_floor_add_one_ge";
+val ceiling_zero = thm "ceiling_zero";
+val ceiling_real_of_nat = thm "ceiling_real_of_nat";
+val ceiling_real_of_nat_zero = thm "ceiling_real_of_nat_zero";
+val ceiling_floor = thm "ceiling_floor";
+val floor_ceiling = thm "floor_ceiling";
+val real_of_int_ceiling_ge = thm "real_of_int_ceiling_ge";
+val ceiling_le = thm "ceiling_le";
+val ceiling_le2 = thm "ceiling_le2";
+val real_of_int_ceiling_cancel = thm "real_of_int_ceiling_cancel";
+val ceiling_eq = thm "ceiling_eq";
+val ceiling_eq2 = thm "ceiling_eq2";
+val ceiling_eq3 = thm "ceiling_eq3";
+val ceiling_real_of_int = thm "ceiling_real_of_int";
+val ceiling_number_of_eq = thm "ceiling_number_of_eq";
+val real_of_int_ceiling_diff_one_le = thm "real_of_int_ceiling_diff_one_le";
+val real_of_int_ceiling_le_add_one = thm "real_of_int_ceiling_le_add_one";
+*}
+
+
end
--- a/src/HOL/IsaMakefile Tue Mar 02 01:46:26 2004 +0100
+++ b/src/HOL/IsaMakefile Tue Mar 02 11:05:55 2004 +0100
@@ -148,8 +148,7 @@
Hyperreal/Filter.ML Hyperreal/Filter.thy Hyperreal/HSeries.thy\
Hyperreal/HTranscendental.thy Hyperreal/HyperArith.thy\
Hyperreal/HyperDef.thy Hyperreal/HyperNat.thy\
- Hyperreal/HyperPow.thy Hyperreal/Hyperreal.thy\
- Hyperreal/IntFloor.thy Hyperreal/IntFloor.ML\
+ Hyperreal/HyperPow.thy Hyperreal/Hyperreal.thy Hyperreal/IntFloor.thy\
Hyperreal/Lim.ML Hyperreal/Lim.thy Hyperreal/Log.thy\
Hyperreal/MacLaurin.ML Hyperreal/MacLaurin.thy Hyperreal/NatStar.thy\
Hyperreal/NSA.thy Hyperreal/NthRoot.thy\