--- a/src/HOL/Nat.thy Mon Apr 16 16:11:03 2007 +0200
+++ b/src/HOL/Nat.thy Tue Apr 17 00:30:44 2007 +0200
@@ -48,13 +48,15 @@
consts
Suc :: "nat => nat"
- pred_nat :: "(nat * nat) set"
local
defs
Suc_def: "Suc == (%n. Abs_Nat (Suc_Rep (Rep_Nat n)))"
- pred_nat_def: "pred_nat == {(m, n). n = Suc m}"
+
+definition
+ pred_nat :: "(nat * nat) set" where
+ "pred_nat = {(m, n). n = Suc m}"
instance nat :: "{ord, zero, one}"
Zero_nat_def: "0 == Abs_Nat Zero_Rep"
@@ -64,8 +66,11 @@
text {* Induction *}
-lemmas Rep_Nat' = Rep_Nat [simplified mem_Collect_eq]
-lemmas Abs_Nat_inverse' = Abs_Nat_inverse [simplified mem_Collect_eq]
+lemma Rep_Nat': "Nat (Rep_Nat x)"
+ by (rule Rep_Nat [simplified mem_Collect_eq])
+
+lemma Abs_Nat_inverse': "Nat y \<Longrightarrow> Rep_Nat (Abs_Nat y) = y"
+ by (rule Abs_Nat_inverse [simplified mem_Collect_eq])
theorem nat_induct: "P 0 ==> (!!n. P n ==> P (Suc n)) ==> P n"
apply (unfold Zero_nat_def Suc_def)
@@ -78,7 +83,7 @@
lemma Suc_not_Zero [iff]: "Suc m \<noteq> 0"
by (simp add: Zero_nat_def Suc_def Abs_Nat_inject Rep_Nat' Suc_RepI Zero_RepI
- Suc_Rep_not_Zero_Rep)
+ Suc_Rep_not_Zero_Rep)
lemma Zero_not_Suc [iff]: "0 \<noteq> Suc m"
by (rule not_sym, rule Suc_not_Zero not_sym)
@@ -92,8 +97,8 @@
text {* Injectiveness of @{term Suc} *}
lemma inj_Suc[simp]: "inj_on Suc N"
- by (simp add: Suc_def inj_on_def Abs_Nat_inject Rep_Nat' Suc_RepI
- inj_Suc_Rep [THEN inj_eq] Rep_Nat_inject)
+ by (simp add: Suc_def inj_on_def Abs_Nat_inject Rep_Nat' Suc_RepI
+ inj_Suc_Rep [THEN inj_eq] Rep_Nat_inject)
lemma Suc_inject: "Suc x = Suc y ==> x = y"
by (rule inj_Suc [THEN injD])
@@ -259,9 +264,9 @@
text {* "Less than" is antisymmetric, sort of *}
lemma less_antisym: "\<lbrakk> \<not> n < m; n < Suc m \<rbrakk> \<Longrightarrow> m = n"
-apply(simp only:less_Suc_eq)
-apply blast
-done
+ apply(simp only:less_Suc_eq)
+ apply blast
+ done
lemma nat_neq_iff: "((m::nat) \<noteq> n) = (m < n | n < m)"
using less_linear by blast
@@ -318,13 +323,12 @@
text {* Can be used with @{text less_Suc_eq} to get @{term "n = m | n < m"} *}
lemma not_less_eq: "(~ m < n) = (n < Suc m)"
-by (rule_tac m = m and n = n in diff_induct, simp_all)
+ by (induct m n rule: diff_induct) simp_all
text {* Complete induction, aka course-of-values induction *}
lemma nat_less_induct:
assumes prem: "!!n. \<forall>m::nat. m < n --> P m ==> P n" shows "P n"
- apply (rule_tac a=n in wf_induct)
- apply (rule wf_pred_nat [THEN wf_trancl])
+ apply (induct n rule: wf_induct [OF wf_pred_nat [THEN wf_trancl]])
apply (rule prem)
apply (unfold less_def, assumption)
done
@@ -336,13 +340,13 @@
text {* Was @{text le_eq_less_Suc}, but this orientation is more useful *}
lemma less_Suc_eq_le: "(m < Suc n) = (m \<le> n)"
- by (unfold le_def, rule not_less_eq [symmetric])
+ unfolding le_def by (rule not_less_eq [symmetric])
lemma le_imp_less_Suc: "m \<le> n ==> m < Suc n"
by (rule less_Suc_eq_le [THEN iffD2])
lemma le0 [iff]: "(0::nat) \<le> n"
- by (unfold le_def, rule not_less0)
+ unfolding le_def by (rule not_less0)
lemma Suc_n_not_le_n: "~ Suc n \<le> n"
by (simp add: le_def)
@@ -387,23 +391,21 @@
text {* Equivalence of @{term "m \<le> n"} and @{term "m < n | m = n"} *}
lemma le_imp_less_or_eq: "m \<le> n ==> m < n | m = (n::nat)"
- apply (unfold le_def)
+ unfolding le_def
using less_linear
- apply (blast elim: less_irrefl less_asym)
- done
+ by (blast elim: less_irrefl less_asym)
lemma less_or_eq_imp_le: "m < n | m = n ==> m \<le> (n::nat)"
- apply (unfold le_def)
+ unfolding le_def
using less_linear
- apply (blast elim!: less_irrefl elim: less_asym)
- done
+ by (blast elim!: less_irrefl elim: less_asym)
lemma le_eq_less_or_eq: "(m \<le> (n::nat)) = (m < n | m=n)"
by (iprover intro: less_or_eq_imp_le le_imp_less_or_eq)
-text {* Useful with @{text Blast}. *}
+text {* Useful with @{text blast}. *}
lemma eq_imp_le: "(m::nat) = n ==> m \<le> n"
- by (rule less_or_eq_imp_le, rule disjI2)
+ by (rule less_or_eq_imp_le) (rule disjI2)
lemma le_refl: "n \<le> (n::nat)"
by (simp add: le_eq_less_or_eq)
@@ -433,9 +435,7 @@
text {* Axiom @{text linorder_linear} of class @{text linorder}: *}
lemma nat_le_linear: "(m::nat) \<le> n | n \<le> m"
apply (simp add: le_eq_less_or_eq)
- using less_linear
- apply blast
- done
+ using less_linear by blast
text {* Type {@typ nat} is a wellfounded linear order *}
@@ -444,7 +444,7 @@
(assumption |
rule le_refl le_trans le_anti_sym nat_less_le nat_le_linear wf_less)+
-lemmas linorder_neqE_nat = linorder_neqE[where 'a = nat]
+lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat]
lemma not_less_less_Suc_eq: "~ n < m ==> (n < Suc m) = (n = m)"
by (blast elim!: less_SucE)
@@ -461,8 +461,8 @@
text {*
- Re-orientation of the equations @{text "0 = x"} and @{text "1 = x"}.
- No longer added as simprules (they loop)
+ Re-orientation of the equations @{text "0 = x"} and @{text "1 = x"}.
+ No longer added as simprules (they loop)
but via @{text reorient_simproc} in Bin
*}
@@ -495,7 +495,7 @@
mult_0: "0 * n = 0"
mult_Suc: "Suc m * n = n + (m * n)"
-text {* These two rules ease the use of primitive recursion.
+text {* These two rules ease the use of primitive recursion.
NOTE USE OF @{text "=="} *}
lemma def_nat_rec_0: "(!!n. f n == nat_rec c h n) ==> f 0 = c"
by simp
@@ -504,13 +504,13 @@
by simp
lemma not0_implies_Suc: "n \<noteq> 0 ==> \<exists>m. n = Suc m"
- by (case_tac n) simp_all
+ by (cases n) simp_all
-lemma gr_implies_not0: "!!n::nat. m<n ==> n \<noteq> 0"
- by (case_tac n) simp_all
+lemma gr_implies_not0: fixes n :: nat shows "m<n ==> n \<noteq> 0"
+ by (cases n) simp_all
-lemma neq0_conv [iff]: "!!n::nat. (n \<noteq> 0) = (0 < n)"
- by (case_tac n) simp_all
+lemma neq0_conv [iff]: fixes n :: nat shows "(n \<noteq> 0) = (0 < n)"
+ by (cases n) simp_all
text {* This theorem is useful with @{text blast} *}
lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n"
@@ -521,7 +521,8 @@
lemma not_gr0 [iff]: "!!n::nat. (~ (0 < n)) = (n = 0)"
apply (rule iffI)
- apply (rule ccontr, simp_all)
+ apply (rule ccontr)
+ apply simp_all
done
lemma Suc_le_D: "(Suc n \<le> m') ==> (? m. m' = Suc m)"
@@ -529,7 +530,7 @@
text {* Useful in certain inductive arguments *}
lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0 | (\<exists>j. m = Suc j & j < n))"
- by (case_tac m) simp_all
+ by (cases m) simp_all
lemma nat_induct2: "[|P 0; P (Suc 0); !!k. P k ==> P (Suc (Suc k))|] ==> P n"
apply (rule nat_less_induct)
@@ -571,7 +572,7 @@
lemma min_Suc1:
"min (Suc n) m = (case m of 0 => 0 | Suc m' => Suc(min n m'))"
- by (simp split: nat.split)
+ by (simp split: nat.split)
lemma min_Suc2:
"min m (Suc n) = (case m of 0 => 0 | Suc m' => Suc(min m' n))"
@@ -588,7 +589,7 @@
lemma max_Suc1:
"max (Suc n) m = (case m of 0 => Suc n | Suc m' => Suc(max n m'))"
- by (simp split: nat.split)
+ by (simp split: nat.split)
lemma max_Suc2:
"max m (Suc n) = (case m of 0 => Suc n | Suc m' => Suc(max m' n))"
@@ -657,11 +658,11 @@
text {* Reasoning about @{text "m + 0 = 0"}, etc. *}
-lemma add_is_0 [iff]: "!!m::nat. (m + n = 0) = (m = 0 & n = 0)"
- by (case_tac m) simp_all
+lemma add_is_0 [iff]: fixes m :: nat shows "(m + n = 0) = (m = 0 & n = 0)"
+ by (cases m) simp_all
lemma add_is_1: "(m+n= Suc 0) = (m= Suc 0 & n=0 | m=0 & n= Suc 0)"
- by (case_tac m) simp_all
+ by (cases m) simp_all
lemma one_is_add: "(Suc 0 = m + n) = (m = Suc 0 & n = 0 | m = 0 & n = Suc 0)"
by (rule trans, rule eq_commute, rule add_is_1)
@@ -674,13 +675,12 @@
apply (simp add: nat_add_assoc del: add_0_right)
done
-
lemma inj_on_add_nat[simp]: "inj_on (%n::nat. n+k) N"
-apply(induct k)
- apply simp
-apply(drule comp_inj_on[OF _ inj_Suc])
-apply (simp add:o_def)
-done
+ apply (induct k)
+ apply simp
+ apply(drule comp_inj_on[OF _ inj_Suc])
+ apply (simp add:o_def)
+ done
subsection {* Multiplication *}
@@ -726,7 +726,8 @@
lemma mult_is_0 [simp]: "((m::nat) * n = 0) = (m=0 | n=0)"
apply (induct m)
- apply (induct_tac [2] n, simp_all)
+ apply (induct_tac [2] n)
+ apply simp_all
done
@@ -746,14 +747,14 @@
lemma less_imp_Suc_add: "m < n ==> (\<exists>k. n = Suc (m + k))"
apply (induct n)
apply (simp_all add: order_le_less)
- apply (blast elim!: less_SucE
+ apply (blast elim!: less_SucE
intro!: add_0_right [symmetric] add_Suc_right [symmetric])
done
text {* strict, in 1st argument; proof is by induction on @{text "k > 0"} *}
lemma mult_less_mono2: "(i::nat) < j ==> 0 < k ==> k * i < k * j"
apply (erule_tac m1 = 0 in less_imp_Suc_add [THEN exE], simp)
- apply (induct_tac x)
+ apply (induct_tac x)
apply (simp_all add: add_less_mono)
done
@@ -777,22 +778,22 @@
subsection {* Additional theorems about "less than" *}
text{*An induction rule for estabilishing binary relations*}
-lemma less_Suc_induct:
+lemma less_Suc_induct:
assumes less: "i < j"
and step: "!!i. P i (Suc i)"
and trans: "!!i j k. P i j ==> P j k ==> P i k"
shows "P i j"
proof -
- from less obtain k where j: "j = Suc(i+k)" by (auto dest: less_imp_Suc_add)
- have "P i (Suc(i+k))"
+ from less obtain k where j: "j = Suc(i+k)" by (auto dest: less_imp_Suc_add)
+ have "P i (Suc (i + k))"
proof (induct k)
- case 0
- show ?case by (simp add: step)
+ case 0
+ show ?case by (simp add: step)
next
case (Suc k)
- thus ?case by (auto intro: prems)
+ thus ?case by (auto intro: assms)
qed
- thus "P i j" by (simp add: j)
+ thus "P i j" by (simp add: j)
qed
@@ -800,7 +801,9 @@
monotonicity to @{text "\<le>"} monotonicity *}
lemma less_mono_imp_le_mono:
assumes lt_mono: "!!i j::nat. i < j ==> f i < f j"
- and le: "i \<le> j" shows "f i \<le> ((f j)::nat)" using le
+ and le: "i \<le> j"
+ shows "f i \<le> ((f j)::nat)"
+ using le
apply (simp add: order_le_less)
apply (blast intro!: lt_mono)
done
@@ -814,7 +817,7 @@
by (rule add_mono)
lemma le_add2: "n \<le> ((m + n)::nat)"
- by (insert add_right_mono [of 0 m n], simp)
+ by (insert add_right_mono [of 0 m n], simp)
lemma le_add1: "n \<le> ((n + m)::nat)"
by (simp add: add_commute, rule le_add2)
@@ -841,7 +844,7 @@
by (rule less_le_trans, assumption, rule le_add2)
lemma add_lessD1: "i + j < (k::nat) ==> i < k"
- apply (rule le_less_trans [of _ "i+j"])
+ apply (rule le_less_trans [of _ "i+j"])
apply (simp_all add: le_add1)
done
@@ -854,7 +857,7 @@
by (simp add: add_commute not_add_less1)
lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)"
- apply (rule order_trans [of _ "m+k"])
+ apply (rule order_trans [of _ "m+k"])
apply (simp_all add: le_add1)
done
@@ -913,9 +916,7 @@
by (simp add: diff_diff_left)
lemma diff_Suc_less [simp]: "0<n ==> n - Suc i < n"
- apply (case_tac "n", safe)
- apply (simp add: le_simps)
- done
+ by (cases n) (auto simp add: le_simps)
text {* This and the next few suggested by Florian Kammueller *}
lemma diff_commute: "(i::nat) - j - k = i - k - j"
@@ -934,9 +935,7 @@
by (simp add: diff_add_assoc)
lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j - i = k) = (j = k + i)"
- apply safe
- apply (simp_all add: diff_add_inverse2)
- done
+ by (auto simp add: diff_add_inverse2)
lemma diff_is_0_eq [simp]: "((m::nat) - n = 0) = (m \<le> n)"
by (induct m n rule: diff_induct) simp_all
@@ -947,10 +946,13 @@
lemma zero_less_diff [simp]: "(0 < n - (m::nat)) = (m < n)"
by (induct m n rule: diff_induct) simp_all
-lemma less_imp_add_positive: "i < j ==> \<exists>k::nat. 0 < k & i + k = j"
- apply (rule_tac x = "j - i" in exI)
- apply (simp (no_asm_simp) add: add_diff_inverse less_not_sym)
- done
+lemma less_imp_add_positive:
+ assumes "i < j"
+ shows "\<exists>k::nat. 0 < k & i + k = j"
+proof
+ from assms show "0 < j - i & i + (j - i) = j"
+ by (simp add: add_diff_inverse less_not_sym)
+qed
lemma zero_induct_lemma: "P k ==> (!!n. P (Suc n) ==> P n) ==> P (k - i)"
apply (induct k i rule: diff_induct)
@@ -989,33 +991,39 @@
subsection {* Monotonicity of Multiplication *}
lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k"
- by (simp add: mult_right_mono)
+ by (simp add: mult_right_mono)
lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j"
- by (simp add: mult_left_mono)
+ by (simp add: mult_left_mono)
text {* @{text "\<le>"} monotonicity, BOTH arguments *}
lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l"
- by (simp add: mult_mono)
+ by (simp add: mult_mono)
lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k"
- by (simp add: mult_strict_right_mono)
+ by (simp add: mult_strict_right_mono)
text{*Differs from the standard @{text zero_less_mult_iff} in that
there are no negative numbers.*}
lemma nat_0_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)"
apply (induct m)
- apply (case_tac [2] n, simp_all)
+ apply simp
+ apply (case_tac n)
+ apply simp_all
done
lemma one_le_mult_iff [simp]: "(Suc 0 \<le> m * n) = (1 \<le> m & 1 \<le> n)"
apply (induct m)
- apply (case_tac [2] n, simp_all)
+ apply simp
+ apply (case_tac n)
+ apply simp_all
done
lemma mult_eq_1_iff [simp]: "(m * n = Suc 0) = (m = 1 & n = 1)"
- apply (induct m, simp)
- apply (induct n, simp, fastsimp)
+ apply (induct m)
+ apply simp
+ apply (induct n)
+ apply auto
done
lemma one_eq_mult_iff [simp]: "(Suc 0 = m * n) = (m = 1 & n = 1)"
@@ -1034,10 +1042,10 @@
by (simp add: mult_commute [of k])
lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k --> m \<le> n)"
-by (simp add: linorder_not_less [symmetric], auto)
+ by (simp add: linorder_not_less [symmetric], auto)
lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k --> m \<le> n)"
-by (simp add: linorder_not_less [symmetric], auto)
+ by (simp add: linorder_not_less [symmetric], auto)
lemma mult_cancel2 [simp]: "(m * k = n * k) = (m = n | (k = (0::nat)))"
apply (cut_tac less_linear, safe, auto)
@@ -1068,24 +1076,23 @@
subsection {* Code generator setup *}
-lemma one_is_Suc_zero [code inline]:
- "1 = Suc 0"
+lemma one_is_Suc_zero [code inline]: "1 = Suc 0"
by simp
instance nat :: eq ..
lemma [code func]:
- "(0\<Colon>nat) = 0 \<longleftrightarrow> True"
- "Suc n = Suc m \<longleftrightarrow> n = m"
- "Suc n = 0 \<longleftrightarrow> False"
- "0 = Suc m \<longleftrightarrow> False"
+ "(0\<Colon>nat) = 0 \<longleftrightarrow> True"
+ "Suc n = Suc m \<longleftrightarrow> n = m"
+ "Suc n = 0 \<longleftrightarrow> False"
+ "0 = Suc m \<longleftrightarrow> False"
by auto
lemma [code func]:
- "(0\<Colon>nat) \<le> m \<longleftrightarrow> True"
- "Suc (n\<Colon>nat) \<le> m \<longleftrightarrow> n < m"
- "(n\<Colon>nat) < 0 \<longleftrightarrow> False"
- "(n\<Colon>nat) < Suc m \<longleftrightarrow> n \<le> m"
+ "(0\<Colon>nat) \<le> m \<longleftrightarrow> True"
+ "Suc (n\<Colon>nat) \<le> m \<longleftrightarrow> n < m"
+ "(n\<Colon>nat) < 0 \<longleftrightarrow> False"
+ "(n\<Colon>nat) < Suc m \<longleftrightarrow> n \<le> m"
using Suc_le_eq less_Suc_eq_le by simp_all
@@ -1100,14 +1107,12 @@
by (auto simp: le_eq_less_or_eq dest: less_imp_Suc_add)
lemma pred_nat_trancl_eq_le: "((m, n) : pred_nat^*) = (m \<le> n)"
-by (simp add: less_eq reflcl_trancl [symmetric]
- del: reflcl_trancl, arith)
+ by (simp add: less_eq reflcl_trancl [symmetric] del: reflcl_trancl, arith)
lemma nat_diff_split:
- "P(a - b::nat) = ((a<b --> P 0) & (ALL d. a = b + d --> P d))"
+ "P(a - b::nat) = ((a<b --> P 0) & (ALL d. a = b + d --> P d))"
-- {* elimination of @{text -} on @{text nat} *}
- by (cases "a<b" rule: case_split)
- (auto simp add: diff_is_0_eq [THEN iffD2])
+ by (cases "a<b" rule: case_split) (auto simp add: diff_is_0_eq [THEN iffD2])
lemma nat_diff_split_asm:
"P(a - b::nat) = (~ (a < b & ~ P 0 | (EX d. a = b + d & ~ P d)))"
@@ -1117,7 +1122,6 @@
lemmas [arith_split] = nat_diff_split split_min split_max
-
lemma le_square: "m \<le> m * (m::nat)"
by (induct m) auto
@@ -1128,81 +1132,81 @@
text{*Subtraction laws, mostly by Clemens Ballarin*}
lemma diff_less_mono: "[| a < (b::nat); c \<le> a |] ==> a-c < b-c"
-by arith
+ by arith
lemma less_diff_conv: "(i < j-k) = (i+k < (j::nat))"
-by arith
+ by arith
lemma le_diff_conv: "(j-k \<le> (i::nat)) = (j \<le> i+k)"
-by arith
+ by arith
lemma le_diff_conv2: "k \<le> j ==> (i \<le> j-k) = (i+k \<le> (j::nat))"
-by arith
+ by arith
lemma diff_diff_cancel [simp]: "i \<le> (n::nat) ==> n - (n - i) = i"
-by arith
+ by arith
lemma le_add_diff: "k \<le> (n::nat) ==> m \<le> n + m - k"
-by arith
+ by arith
(*Replaces the previous diff_less and le_diff_less, which had the stronger
second premise n\<le>m*)
lemma diff_less[simp]: "!!m::nat. [| 0<n; 0<m |] ==> m - n < m"
-by arith
+ by arith
(** Simplification of relational expressions involving subtraction **)
lemma diff_diff_eq: "[| k \<le> m; k \<le> (n::nat) |] ==> ((m-k) - (n-k)) = (m-n)"
-by (simp split add: nat_diff_split)
+ by (simp split add: nat_diff_split)
lemma eq_diff_iff: "[| k \<le> m; k \<le> (n::nat) |] ==> (m-k = n-k) = (m=n)"
-by (auto split add: nat_diff_split)
+ by (auto split add: nat_diff_split)
lemma less_diff_iff: "[| k \<le> m; k \<le> (n::nat) |] ==> (m-k < n-k) = (m<n)"
-by (auto split add: nat_diff_split)
+ by (auto split add: nat_diff_split)
lemma le_diff_iff: "[| k \<le> m; k \<le> (n::nat) |] ==> (m-k \<le> n-k) = (m\<le>n)"
-by (auto split add: nat_diff_split)
+ by (auto split add: nat_diff_split)
text{*(Anti)Monotonicity of subtraction -- by Stephan Merz*}
(* Monotonicity of subtraction in first argument *)
lemma diff_le_mono: "m \<le> (n::nat) ==> (m-l) \<le> (n-l)"
-by (simp split add: nat_diff_split)
+ by (simp split add: nat_diff_split)
lemma diff_le_mono2: "m \<le> (n::nat) ==> (l-n) \<le> (l-m)"
-by (simp split add: nat_diff_split)
+ by (simp split add: nat_diff_split)
lemma diff_less_mono2: "[| m < (n::nat); m<l |] ==> (l-n) < (l-m)"
-by (simp split add: nat_diff_split)
+ by (simp split add: nat_diff_split)
lemma diffs0_imp_equal: "!!m::nat. [| m-n = 0; n-m = 0 |] ==> m=n"
-by (simp split add: nat_diff_split)
+ by (simp split add: nat_diff_split)
text{*Lemmas for ex/Factorization*}
lemma one_less_mult: "[| Suc 0 < n; Suc 0 < m |] ==> Suc 0 < m*n"
-by (case_tac "m", auto)
+ by (cases m) auto
lemma n_less_m_mult_n: "[| Suc 0 < n; Suc 0 < m |] ==> n<m*n"
-by (case_tac "m", auto)
+ by (cases m) auto
lemma n_less_n_mult_m: "[| Suc 0 < n; Suc 0 < m |] ==> n<n*m"
-by (case_tac "m", auto)
+ by (cases m) auto
text{*Rewriting to pull differences out*}
lemma diff_diff_right [simp]: "k\<le>j --> i - (j - k) = i + (k::nat) - j"
-by arith
+ by arith
lemma diff_Suc_diff_eq1 [simp]: "k \<le> j ==> m - Suc (j - k) = m + k - Suc j"
-by arith
+ by arith
lemma diff_Suc_diff_eq2 [simp]: "k \<le> j ==> Suc (j - k) - m = Suc j - (k + m)"
-by arith
+ by arith
(*The others are
i - j - k = i - (j + k),
@@ -1245,8 +1249,9 @@
val diff_Suc_diff_eq2 = thm "diff_Suc_diff_eq2";
*}
-subsection{*Embedding of the Naturals into any @{text
-semiring_1_cancel}: @{term of_nat}*}
+
+subsection{*Embedding of the Naturals into any
+ @{text semiring_1_cancel}: @{term of_nat}*}
consts of_nat :: "nat => 'a::semiring_1_cancel"
@@ -1255,74 +1260,72 @@
of_nat_Suc: "of_nat (Suc m) = of_nat m + 1"
lemma of_nat_1 [simp]: "of_nat 1 = 1"
-by simp
+ by simp
lemma of_nat_add [simp]: "of_nat (m+n) = of_nat m + of_nat n"
-apply (induct m)
-apply (simp_all add: add_ac)
-done
+ by (induct m) (simp_all add: add_ac)
lemma of_nat_mult [simp]: "of_nat (m*n) = of_nat m * of_nat n"
-apply (induct m)
-apply (simp_all add: add_ac left_distrib)
-done
+ by (induct m) (simp_all add: add_ac left_distrib)
lemma zero_le_imp_of_nat: "0 \<le> (of_nat m::'a::ordered_semidom)"
-apply (induct m, simp_all)
-apply (erule order_trans)
-apply (rule less_add_one [THEN order_less_imp_le])
-done
+ apply (induct m, simp_all)
+ apply (erule order_trans)
+ apply (rule less_add_one [THEN order_less_imp_le])
+ done
lemma less_imp_of_nat_less:
- "m < n ==> of_nat m < (of_nat n::'a::ordered_semidom)"
-apply (induct m n rule: diff_induct, simp_all)
-apply (insert add_le_less_mono [OF zero_le_imp_of_nat zero_less_one], force)
-done
+ "m < n ==> of_nat m < (of_nat n::'a::ordered_semidom)"
+ apply (induct m n rule: diff_induct, simp_all)
+ apply (insert add_le_less_mono [OF zero_le_imp_of_nat zero_less_one], force)
+ done
lemma of_nat_less_imp_less:
- "of_nat m < (of_nat n::'a::ordered_semidom) ==> m < n"
-apply (induct m n rule: diff_induct, simp_all)
-apply (insert zero_le_imp_of_nat)
-apply (force simp add: linorder_not_less [symmetric])
-done
+ "of_nat m < (of_nat n::'a::ordered_semidom) ==> m < n"
+ apply (induct m n rule: diff_induct, simp_all)
+ apply (insert zero_le_imp_of_nat)
+ apply (force simp add: linorder_not_less [symmetric])
+ done
lemma of_nat_less_iff [simp]:
- "(of_nat m < (of_nat n::'a::ordered_semidom)) = (m<n)"
-by (blast intro: of_nat_less_imp_less less_imp_of_nat_less)
+ "(of_nat m < (of_nat n::'a::ordered_semidom)) = (m<n)"
+ by (blast intro: of_nat_less_imp_less less_imp_of_nat_less)
text{*Special cases where either operand is zero*}
-lemmas of_nat_0_less_iff = of_nat_less_iff [of 0, simplified]
-lemmas of_nat_less_0_iff = of_nat_less_iff [of _ 0, simplified]
-declare of_nat_0_less_iff [simp]
-declare of_nat_less_0_iff [simp]
+
+lemma of_nat_0_less_iff [simp]: "((0::'a::ordered_semidom) < of_nat n) = (0 < n)"
+ by (rule of_nat_less_iff [of 0, simplified])
+
+lemma of_nat_less_0_iff [simp]: "\<not> of_nat m < (0::'a::ordered_semidom)"
+ by (rule of_nat_less_iff [of _ 0, simplified])
lemma of_nat_le_iff [simp]:
- "(of_nat m \<le> (of_nat n::'a::ordered_semidom)) = (m \<le> n)"
-by (simp add: linorder_not_less [symmetric])
+ "(of_nat m \<le> (of_nat n::'a::ordered_semidom)) = (m \<le> n)"
+ by (simp add: linorder_not_less [symmetric])
text{*Special cases where either operand is zero*}
-lemmas of_nat_0_le_iff = of_nat_le_iff [of 0, simplified]
-lemmas of_nat_le_0_iff = of_nat_le_iff [of _ 0, simplified]
-declare of_nat_0_le_iff [simp]
-declare of_nat_le_0_iff [simp]
+lemma of_nat_0_le_iff [simp]: "(0::'a::ordered_semidom) \<le> of_nat n"
+ by (rule of_nat_le_iff [of 0, simplified])
+lemma of_nat_le_0_iff [simp]: "(of_nat m \<le> (0::'a::ordered_semidom)) = (m = 0)"
+ by (rule of_nat_le_iff [of _ 0, simplified])
text{*The ordering on the @{text semiring_1_cancel} is necessary
to exclude the possibility of a finite field, which indeed wraps back to
zero.*}
lemma of_nat_eq_iff [simp]:
- "(of_nat m = (of_nat n::'a::ordered_semidom)) = (m = n)"
-by (simp add: order_eq_iff)
+ "(of_nat m = (of_nat n::'a::ordered_semidom)) = (m = n)"
+ by (simp add: order_eq_iff)
text{*Special cases where either operand is zero*}
-lemmas of_nat_0_eq_iff = of_nat_eq_iff [of 0, simplified]
-lemmas of_nat_eq_0_iff = of_nat_eq_iff [of _ 0, simplified]
-declare of_nat_0_eq_iff [simp]
-declare of_nat_eq_0_iff [simp]
+lemma of_nat_0_eq_iff [simp]: "((0::'a::ordered_semidom) = of_nat n) = (0 = n)"
+ by (rule of_nat_eq_iff [of 0, simplified])
+lemma of_nat_eq_0_iff [simp]: "(of_nat m = (0::'a::ordered_semidom)) = (m = 0)"
+ by (rule of_nat_eq_iff [of _ 0, simplified])
lemma of_nat_diff [simp]:
- "n \<le> m ==> of_nat (m - n) = of_nat m - (of_nat n :: 'a::ring_1)"
-by (simp del: of_nat_add
- add: compare_rls of_nat_add [symmetric] split add: nat_diff_split)
+ "n \<le> m ==> of_nat (m - n) = of_nat m - (of_nat n :: 'a::ring_1)"
+ by (simp del: of_nat_add
+ add: compare_rls of_nat_add [symmetric] split add: nat_diff_split)
instance nat :: distrib_lattice
"inf \<equiv> min"
@@ -1332,7 +1335,7 @@
subsection {* Size function *}
-lemma nat_size[simp]: "size (n::nat) = n"
+lemma nat_size [simp]: "size (n::nat) = n"
by (induct n) simp_all
end